## Envision Math Common Core 7th Grade Answers Key Topic 4 Generate Equivalent Expressions

Topic 4 Essential Question
How can properties of operations help to generate equivalent expressions that can be used in solving problems?

3-ACT MATH

### Topic 4 enVision STEM Project

Did You Know?
In 2013, just over 30% of American consumers knew about activity trackers. By 2015, about 82% recognized them.

Continued research and development leads to technological advances and breakthroughs, such as the use of biosensing apparel to track activity.

Your Task: Analyze Activity Tracker Data
The ways that data are communicated and presented to the user are just as important as the types of data collected. You and your classmates will continue your exploration of activity trackers and use data to develop models based on individual fitness goals.

Topic 4 Get Ready

Review What You Know

Vocabulary

Choose the best term from the box to complete each definition.
evaluate
expression
factor
order of operations
substitute
term

Question 1.
When you __________ an expression, you replace each variable with a given value.

Answer:
When you evaluate an expression, you replace each variable with a given value.

Explanation:
In the above-given question,
given that,
when we evaluate an expression, we replace each variable with a given value.
for example:
Evaluate 3a-2b.
for a = 6 and b = 4.
3(6) – 2(4).
18 – 8.
10.

Question 2.
To evaluate a + 3 when a = 7, you can _________ 7 for a in the expression.

Answer:
To evaluate a + 3 when a = 7, you can substitute 7 for a in the expression.

Explanation:
In the above-given question,
given that,
if we evaluate a + 3 when a = 7.
we will substitute 7 for a in the expression.
a + 3.
7 + 3.
10.

Question 3
The set of rules used to determine the order in which operations are performed is called the _________

Answer:
The set of rules used to determine the order in which operations are performed is called the order of operations.

Explanation:
In the above-given question,
given that,
The set of rules used to determine the order in which operations are performed is called the order of operations.
for example:
3 + [6(11 + 1 – 4)]/8 x 2.
3+[6(8)]/8 x 2.
3 + 48 / 8 x 2.
3 + 6 x 2.
3 + 12.
15.

Question 4.
Each part of an expression that is separated by a plus or minus sign is a(n) __________.

Answer:
Each part of an expression that is separated by a plus or minus sign is the term.

Explanation:
In the above-given question,
given that,
Each part of an expression that is separated by a plus or minus sign is the term.
for example:
2x + 4y – 9.
where x and y are variables.
9 is the constant.
2 and 4 are coefficients.
terms are 2x, 4y, and 9.

Question 5.
A(n) __________ is a mathematical phrase that can contain numbers, variables, and operation symbols.

Answer:
An expression is a mathematical phrase that can contain numbers, variables, and operation symbols.

Explanation:
In the above-given question,
given that,
An expression is a mathematical phrase that can contain numbers, variables, and operation symbols.
for example:
n + 7 = 10.
x – 5 = 3.
3p = 15.
y/2 = 5.

Question 6.
When two numbers are multiplied to get a product, each number is called a(n) _________.

Answer:
When two numbers are multiplied to get a product, each number is called a factor.

Explanation:
In the above-given question,
given that,
When two numbers are multiplied to get a product, each number is called a factor.
for example:
3 x 5 = 15.
3 and 5 are the factors.
15 is the product.

Order of Operations

Evaluate each expression using the order of operations.
Question 7.
3(18 – 7) + 2

Answer:
3(18 – 7) + 2 = 35.

Explanation:
In the above-given question,
given that,
3(18 – 7) + 2.
3(11) + 2.
33 + 2.
35.
3(18 – 7) + 2 = 35.

Question 8.
(13 + 2) ÷ (9 – 4)

Answer:
(13 + 2) ÷ (9 – 4) = 3.

Explanation:
In the above-given question,
given that,
(13 + 2) ÷ (9 – 4).
(13 + 2) ÷ (9 – 4).
15 / 5.
3.
(13 + 2) ÷ (9 – 4) = 3.

Question 9.
24 ÷ 4 • 2 – 2

Answer:
24 ÷ 4 • 2 – 2 = 10.

Explanation:
In the above-given question,
given that,
24 ÷ 4 • 2 – 2.
6 . 2 – 2.
12 – 2.
10.
24 ÷ 4 • 2 – 2 = 10.

Equivalent Expressions

Evaluate each expression when a = -4 and b = 3.
Question 10.
ab

Answer:
ab = -12.

Explanation:
In the above-given question,
given that,
a = -4 and b = 3.
– 4 x 3.
-12.
ab = -12.

Question 11.
2a + 3b

Answer:
2a + 3b = 1.

Explanation:
In the above-given question,
given that,
a = -4 and b = 3.
2(-4) + 3(3).
-8 + 9.
1.
2a + 3b = 1.

Question 12.
2(a – b)

Answer:
2(a – b) = -14.

Explanation:
In the above-given question,
given that,
a = -4 and b = 3.
2(-4 – 3).
2(-7).
-14.
2(a – b) = -14.

Question 13.
Explain the difference between evaluating 3 • 7 – 4 ÷ 2 and evaluating 3(7 – 4) ÷ 2.

Answer:
The two expressions are different.

Explanation:
In the above-given question,
given that,
3 . 7 – 4 ÷ 2.
3 . 7 – 2.
21 – 2.
19.
3(7 – 4) ÷ 2.
3(3) / 2.
9 / 2.

Language Development

Complete each math statement using the word bank.

To evaluate an algebraic expression, substitute a __________ for the variable in the expression.

Answer:
To evaluate an algebraic expression, substitute properties of operations for the variable in the expression.

Explanation:
In the above-given question,
given that,
To evaluate an algebraic expression, substitute properties of operations for the variable in the expression.
for example:
n + 7 = 10.
x – 5 = 3.
3p = 15.
y/2 = 5.

In the algebraic expression 3(x – 2), 3 and x – 2 are ___________

Answer:
In the algebraic expression 3(x – 2), 3, and x – 2 are coefficients.

Explanation:
In the above-given question,
given that,
In the algebraic expression 3(x – 2), 3 and x – 2 are coefficients.
for example:
3(x – 2).
3 and x-2 are coefficients.

To generate equivalent expressions, you can use the __________

Answer:
To generate equivalent expressions, you can use the order of operations.

Explanation:
In the above-given question,
given that,
To generate equivalent expressions, you can use the order of operations.
for example:
5(x – 1) + 7.
5(x) + 5(-7) + 7.
5x – 5 + 7.
5x + 2.

In the expression 4x + 2x – 6y, you first need to __________

Answer:
In the expression 4x + 2x – 6y, you first need to add.

Explanation:
In the above-given question,
given that,
In the expression 4x + 2x – 6y, you first need to add.
4x + 2y – 6y.
4x – 4y.

You can use the Distributive Property to __________ the algebraic expression 5(x – 7).

Answer:
You can use the Distributive property to find the algebraic expression.

Explanation:
In the above-given question,
given that,
we can use the distributive property to find the algebraic expression.
for example:
5(x – 7).
5x – 35.

In the algebraic expression, 6x + 10, x is the ________ , 6 is the ________ and 10 is the ___________

Answer:
In the algebraic expression 6x + 10, x is the coefficient, 6 is the variable, and 10 is the constant.

Explanation:
In the above-given question,
given that,
In the algebraic expression 6x + 10, x is the coefficient, 6 is the variable, and 10 is the constant.
for example:
6x + 10.
where 6 is the variable.
x is coefficient.
10 is constant.

Four words that describe operations that can be used with expressions are _________, and ________, _________ and __________.

Answer:
The words that describe the operations are constants, terms, variables, and coefficients.

Explanation:
In the above-given question,
given that,
The words that describe the operations are constants, terms, variables, and coefficients.
for example:
2x + 4y – 9.
where x and y are variables.
9 is the constant.
2 and 4 are coefficients.
terms are 2x, 4y, and 9.

In the algebraic expression 5x + 4 + 6x – 3, you use the Commutative Property to _________ like terms next to each other and the Associative Property to _________ like terms together.

Answer:

Pick A Project

PROJECT 4A
Which emojis would you use to tell the story of your day so far?
PROJECT: WRITE AND ILLUSTRATE A CHILDREN’S BOOK

PROJECT 4B
How many different ways can you represent a dollar?
PROJECT: GENERATE EQUIVALENCE

PROJECT 4C
If you wrote a song, what would it sound like?
PROJECT: COMPOSE A SONG

PROJECT 4D
What was your favorite structure at a playground when you were younger?
PROJECT: BUILD A MODEL PLAYGROUND

### Lesson 4.1 Write and Evaluate Algebraic Expressions

Solve & Discuss It!
Mr. Ramirez’s class was playing a game in which students need to match sticky notes that have equivalent expressions. How can you sort the expressions into groups?
I can… write and evaluate algebraic expressions.

Focus on math practices
Reasoning is there more than one way to group the expressions? Give an example.

Answer:
Yes, there are more than one way to group the expressions.

Explanation:
In the above-given question,
given that,
A numerical expression in mathematics can be a combination of numbers, integers combined using.
for example:
16 is an numerical expression.

Essential Question
How can algebraic expressions be used to represent and solve problems?

Answer:
We can use algebra to solve mathematical problems.

Explanation:
In the above-given question,
given that,
we can also interpret the solution in the context of the original problem.
for example:
2x + 5 = 43.
where 43 is the constant.
always has an equal symbol.
2x + 5 = algebraic expression.

Try It!

Misumi started with $217 in her bank account. She deposits$25.50 each week and never withdraws any money. What expression can Misumi use to determine her account balance after w weeks?

Answer:
The expression can Misumi use to determine her account balance after w weeks = 8.5 weeks.

Explanation:
In the above-given question,
given that,
Misumi started with $217 in her bank account. She deposits$25.50 each week and never withdraws any money.
$217 / 25.50 = 8.5. so the expression can Misumi use to determine her account balance after w weeks = 8.5. Convince Me! How did you determine which value to use for the constant and which value to use for the coefficient? Answer: x is coefficient and 3 is constant. Explanation: In the above-given question, given that, 2x + 3 is the expression. x is the coefficient. 3 is the constant. Try It! The cost to rent a scooter is$15.50 per hour and the cost to rent a watercraft is $22.80 per hour. Use the expression 15.5s + 22.8w to determine how much it would cost to rent a scooter for 3$$\frac{1}{2}$$ hours and a watercraft for 1$$\frac{3}{4}$$ hours. Answer: The cost would cost to rent a scooter for 3(1/2) hours and watercraft for 1(3/4) hours =$54.25 and $40. Explanation: In the above-given question, given that, The cost to rent a scooter is$15.50 per hour and the cost to rent a watercraft is $22.80 per hour. Use the expression 15.5s + 22.8w to determine how much. 15.5s + 22.8w. 3. 1/2 = 7/2. 1. 3/4 = 7/4. 15.5(7/2) + 22.8(7/4). 108.5/2 + 159.6/4. 54.25 + 39.9. Try It! Emelia earns$8.74 per hour plus a gas allowance of $3.50 per day at her job. How much does Emelia’s job pay in a day when she works 5$$\frac{1}{2}$$ hours? Write an expression and evaluate for 5$$\frac{1}{2}$$ hours. Answer: Emelia’s job pay in a day when she works 5(1/2) hours =$67.32.

Explanation:
In the above-given question,
given that,
Emelia earns $8.74 per hour plus a gas allowance of$3.50 per day at her job.
$8.74 +$3.50.
$12.24. 5(1/2) = 5.5. 5.5 x$12.24 = $67.32. so Emelia’s job pay in a day when she works 5(1/2) hours =$67.32.

KEY CONCEPT
Algebraic expressions can be used to represent problems with unknown or variable values. Values can be substituted for variables to evaluate the expression.

Do You Understand?
Question 1.
Essential Question How are algebraic expressions used to represent and solve problems?

Answer:
Algebraic expressions are used to represent problems with unknowns or variable values.

Explanation:
In the above-given question,
given that,
Algebraic expressions are used to represent problems with unknowns or variable values.
Values can be substituted for variables to evaluate the expression.
for example:
2x + 3y = a.
where x = 2 and y = 3.
2 x 2 + 3 x 3 = a.
4 + 9 = a.
13 = a.

Question 2.
Use Structure How is a constant term different than a variable term for an expression that represents a real-world situation?

Answer:
a = 13.

Explanation:
In the above-given question,
given that,
2x + 3y = a.
where x = 2 and y = 3.
2 x 2 + 3 x 3 = a.
4 + 9 = a.
13 = a.

Question 3.
Look for Relationships Explain why you can have different values when evaluating an algebraic expression.

Answer:
To evaluate an algebraic expression we have to substitute a number for each variable and perform the arithmetic operations.

Explanation:
In the above-given question,
given that,
To evaluate an algebraic expression we have to substitute a number for each variable and perform the arithmetic operations.
for example:
x + 6.
where x = 6.
6 + 6 = 12.
if we know the value of our variables, we can replace the variables with their values and then evaluate the expression.

Do You Know How?
Question 4.
A tank containing 35 gallons of water is leaking at a rate of $$\frac{1}{4}$$ gallon per minute. Write an expression to determine the number of gallons left in the tank after m minutes.

Answer:
The number of gallons left in the tank after m minutes = 8.75 gallons.

Explanation:
In the above-given question,
given that,
A tank containing 35 gallons of water is leaking at a rate of $$\frac{1}{4}$$ gallon per minute.
35 x 1/4 = 35/4.
8.75.
so the number of gallons left in the tank after m minutes = 8.75 gallons.

Question 5.
Write an algebraic expression that Marshall can use to determine the total cost of buying a watermelon that weighs w pounds and some tomatoes that weigh t pounds. How much will it cost to buy a watermelon that weighs 18$$\frac{1}{2}$$ pounds and 5 pounds of tomatoes?

Answer:
The much will it cost to buy a watermelon that weighs 18(1/2) pounds and 5 pounds of tomatoes = $29.25 and$3.4.

Explanation:
In the above-given question,
given that,
the cost of tomatoes is $3.25 per lb. the cost of watermelons is$0.68 per lb.
18/2 = 9.
$3.25 x 9 =$29.25.
0.68 x 5 = $3.4. so the much will it cost to buy a watermelon that weighs 18(1/2) pounds and 5 pounds of tomatoes =$29.25 and $3.4. Question 6. What is the value of $$\frac{3}{8}$$x – 4.5 when x = 0.4? Answer: (3/8)x – 4.5 = 4.35. Explanation: In the above-given question, given that, (3/8)x – 4.5. where x = 0.4. (3/8)0.4 – 4.5. 0.375 x 0.4 – 4.5. 0.15 – 4.5. 4.35. (3/8)x – 4.5 = 4.35. Question 7. What is the value of 8.4n – 3.2p when n = 2 and p = 4? Answer: 8.4n – 3.2p = 4. Explanation: In the above-given question, given that, 8.4n – 3.2p. 8.4 (2) – 3.2 (4). 16.8 – 12.8. 4. 8.4n – 3.2p = 4. Practice & Problem Solving Leveled Practice For 8-10, fill in the boxes to complete the problems. Question 8. Evaluate 10.2x + 9.4y when x = 2 and y = 3. 10.2 (_______) + 9.4 (_______) = _______ + 28.2 = _______ Answer: 10.2 x + 9.4 y = 48.6. Explanation: In the above-given question, given that, 10.2 x + 9.4 y. where x = 2 and y = 3. 10.2 x 2 + 9.4 x 3. 20.4 + 28.2. 48.6. 10.2 x + 9.4 y = 48.6. Question 9. Evaluate $$\frac{1}{2}$$t + $$\frac{3}{8}$$ when t = $$\frac{1}{4}$$ $$\frac{1}{2}$$(________) + $$\frac{3}{8}$$ = ______ + $$\frac{3}{8}$$ = ______ Answer: 1/2 x 1/4 + 3/8 = 0.5. Explanation: In the above-given question, given that, 1/2 = 0.5. 1/4 = 0.25. 0.5 x 0.25 + 3/8. 0.125 + 0.375. 0.5. 1/2 x 1/4 + 3/8 = 0.5. Question 10. Write an expression that represents the height of a tree that began at 6 feet and increases by 2 feet per year. Let y represent the number of years. _____ + ______ y Answer: 6x + 2y. Explanation: In the above-given question, given that, the height of a tree that began at 6 feet and increases by 2 feet per year. 6x + 2y. where y represents the number of years. so the expression is 6x + 2y. For 11-14, evaluate each expression for the given value of the variable(s). Question 11. 3d – 4 d = 1.2 Answer: 3d – 4 = 0.4. Explanation: In the above-given question, given that, 3d – 4. where d = 1.2. 3(1.2) – 4. 3.6 – 4. 0.4. 3d – 4 = 0.4. Question 12. 0.5f – 2.39 f = 12, 9 = 2 Answer: 0.5f – 2.39 = 3.68. Explanation: In the above-given question, given that, 0.5f – 2.39. where f = 12 and 9 = 2. 0.5 x 12 – 2.32. 6 – 2.32. 3.68. 0.5f – 2.39 = 3.68. Question 13. p + 3 p = $$\frac{3}{5}$$ Answer: p + 3 = 3.6. Explanation: In the above-given question, given that, p + 3. where p = 3/5. 3/5 = 0.6. 0.6 + 3. 3.6. p + 3 = 3.6. Question 14. 34 + $$\frac{4}{9}$$w w = –$$\frac{1}{2}$$ Answer: 34 + 4/9x w = 33.9. Explanation: In the above-given question, given that, 34 + 4/9x w. where w = -1/2. 34 + 4/9(-1/2). 34 + 0.4(-0.5). 34 – 0.1. 33.9. Question 15. Model with Math What expression can be used to determine the total cost of buying g pounds of granola for$3.25 per pound and f pounds of flour for $0.74 per pound? Answer:$3.25g and $0.74f. Explanation: In the above-given question, given that, the total cost of buying g pounds of granola for$3.25 per pound.
f pounds of flour for $0.74 per pound.$3.25g + $0.74f Question 16. Model with Math Which expression can be used to determine the total weight of a box that by itself weighs 0.2 kilogram and contains p plaques that weigh 1.3 kilograms each? A. 1.3p +0.2 B. 0.2p + 1.3 C. 0.2 – 1.3p D. 1.2p Answer: Option A is correct. Explanation: In the above-given question, given that, total weight of a box that by itself weighs 0.2 kilogram. and contains p plaques that weigh 1.3 kilograms each. 0.2 + 1.3p. so option A is correct. Question 17. The expression -120 + 13n represents a submarine that began at a depth of 120 feet below sea level and ascended at a rate of 13 feet per minute. What was the depth of the submarine after 6 minutes? Answer: The depth of the submarine after 6 minutes = – 42 feet. Explanation: In the above-given question, given that, The expression -120 + 13n represents a submarine that began at a depth of 120 feet below sea level. ascended at a rate of 13 feet per minute. -120 + 13(6). -120 + 78. -42. so the depth of the submarine after 6 minutes = -42 feet. Question 18. Be Precise A full grain silo empties at a constant rate. Write an expression to determine the amount of grain left after s seconds. Answer: The amount of grain left after 5 seconds = 2982.5 cubic feet. Explanation: In the above-given question, given that, A full grain silo empties at a constant rate. the capacity of food grain is 3000 cubic feet. 3000 – 3.5/s. 3000 – 3.5(5). 3000 – 17.5. 2982.5. so the amount of grain left after 5 seconds = 2982.5 cubic feet. Question 19. Higher Order Thinking For the expression 5 – 5x to have a negative value, what must be true about the value of x? Answer: The value of x = 4. Explanation: In the above-given question, given that, the expression is 5 – 5x. where x = 4. 5 – 5(4). 5 – 20. -15. Assessment Practice Question 20. Joe bought g gallons of gasoline for$2.85 per gallon and c cans of oil for $3.15 per can. PART A What expression can be used to determine the total amount Joe spent on gasoline and oil? Answer: The total amount joe spend on gasoline and oil =$2.85g + $3.15c. Explanation: In the above-given question, given that, Joe bought g gallons of gasoline for$2.85 per gallon.
c cans of oil for $3.15 per can.$2.85g + $3.15c. so the total amount joe spends on gasoline and oil =$2.85g + $3.15c. PART B Joe spent$15. He bought 2 cans of oil. About how many gallons of gasoline did he buy?
A. 2.5
B. 3
C. 3.5
D. 4

Answer:
The gallons of gasoline did he buy = 3 gallons.

Explanation:
In the above-given question,
given that,
Joe spent $15. He bought 2 cans of oil. 1.5 x 2. 3. the gallons of gasoline did he buy = 3. so option B is correct. Question 21. The outside temperature was 73°F at 1 P.M. and decreases at a rate of 1.5°F each hour. What expression can be used to determine the temperature h hours after 1 P.M.? Answer: The expression can be used to determine the temperature h hours after 1 P.M = 71.5°F. Explanation: In the above-given question, given that, The outside temperature was 73°F at 1 P.M. and decreases at a rate of 1.5°F each hour. 73 – 1.5. 71.5°F. so the expression can be used to determine the temperature h hours after 1 P.M = 71.5°F. ## Lesson 4.2 Generate Equivalent Expressions Explore It! A shipment of eggs contains some cartons with a dozen eggs and some cartons with a half-dozen eggs. I can… write equivalent expressions for given expressions. A. How can you represent the total number of eggs in the shipment using diagrams or images? Explain your diagram. Answer: 1 dozen + 1/2 dozen eggs. Explanation: In the above-given question, given that, A shipment of eggs contains some cartons with a dozen eggs and some cartons with a half-dozen eggs. 1 dozen = 12. 1/2 doxen = 12/2. 12/2 = 6. 12 + 1/2 eggs. B. How can you represent the total number of eggs in the shipment using expressions? What variables do you use? What do they represent? Answer: 1 dozen + 1/2 dozen eggs. Explanation: In the above-given question, given that, A shipment of eggs contains some cartons with a dozen eggs and some cartons with a half-dozen eggs. 1 dozen = 12. 1/2 doxen = 12/2. 12/2 = 6. 12 + 1/2 eggs. Focus on math practices Construct Arguments How do the two representations compare? How are they different? Essential Question What are equivalent expressions? Try It! Nancy wrote the expression 3x – 12 to represent the relationship in a table of values. Use properties of operations to write two equivalent expressions. 3(x – _____) _____ + 3x Answer: The two equivalent expressions are 36 + 3x. Explanation: In the above-given question, given that, 3x – 12. 3(x – 12). 3x – 36. 36 + 3x. Convince Me! What property can you use to write an equivalent expression for -5(x – 2)? Explain. Answer: -5(x – 2) = -5x – 10. Explanation: In the above-given question, given that, -5(x – 2). -5x – 10. we can use the distributive property. -5x -10. Try It! Use properties of operations to write two expressions that are equivalent to $$\frac{3}{4}$$n + (8 + $$\frac{1}{3}$$z). Answer: 3/4n + (8 + {1/3}) = 0.75n + 8.3z. Explanation: In the above-given question, given that, 3/4n + (8 + {1/3})z. 3/4n + 8 + 0.3z. 0.75n + 8.3z. 3/4n + (8 + {1/3}) = 0.75n + 8.3z. Try It! Write two expressions that are equivalent to –$$\frac{5}{4}$$x – $$\frac{3}{4}$$ Answer: -5/4 – 3/4 = 2. Explanation: In the above-given question, given that, -5/4 – 3/4. -5/4 = 1.25. 3/4 = 0.75. -1.25 – 0.75. 2. KEY CONCEPT You can use properties of operations to write equivalent expressions. Do You Understand? What are equivalent Question 1. Essential Question expressions? Answer: -1/2(x + 8), -1/2x + (-4) and -4 +(-1/2x) are equivalent. Explanation: In the above-given question, given that, -1/2(x + 8). -1/2x + (-1/2) . 8. -1/2x + (-4). -4 + (-1/2x). the three expressions are true. Question 2. Make Sense and Persevere For which operations is the Commutative Property true? Answer: (-4) + -1/2x. Explanation: In the above-given question, given that, (-4) + -1/2x. we can use the commutative property for the expression. -1/2x + (-4). Question 3. How can the Associative Property be applied when writing equivalent expressions with variables? Answer: Do You Know How? Question 4. Write an expression equivalent to -3 + $$\frac{2}{3}$$y – 4 – $$\frac{1}{3}$$y. Answer: -3 + (2/3)y – 4 – (1/3)y. Explanation: In the above-given question, given that, -3 + $$\frac{2}{3}$$y – 4 – $$\frac{1}{3}$$y. -3 + (2/3)y – 4 – (1/3)y. -3 – 4 + (2/3)y – (1/3)y. -7 + 1/3y. -3 + (2/3)y – 4 – (1/3)y = -7 + 1/3y. Question 5. Complete the tables to determine if the expressions are equivalent. If the expressions are equivalent, name the property or properties that make them equivalent. Answer: 3(x – 5) = 3x – 15. Explanation: In the above-given question, given that, 3(x – 5). 3x – 15. x = 1. 3 – 15 = -12. x = 2. 6 – 15 = -9. x = 3. 9 – 15 = -6. Question 6. Use the properties of operations to write an expression equivalent to 4x + $$\frac{1}{2}$$ + 2x – 3. Answer: 4x +[{1/2}] + 2x – 3 = 3x -3. Explanation: In the above-given question, given that, 4x + (1/2) + 2x – 3. 4x + 2x + (1/2) – 3. 6x + (1/2) -3. 3x – 3. 4x +[{1/2}] + 2x – 3 = 3x -3. Practice & Problem Solving For 7-9, write an equivalent expression. Question 7. -3(7 + 5g) Answer: -36g. Explanation: In the above-given question, given that, -3(7 + 5g). -3 x 7 = -21. -3 x 5 = 15. -21 + (-15g). -36g. Question 8. (x + 7) + 3y Answer: 24xy. Explanation: In the above-given question, given that, (x + 7) + 3y. 3y x X + 3y x 7. 3xy + 21y. 24xy. Question 9. $$\frac{2}{9}$$ – $$\frac{1}{5}$$ • x Answer: 2/9 – 1/5 . X = Explanation: In the above-given question, given that, 2/9 x X – (1/5)x. 2/9 x – 1/5 x. Question 10. Which expression is equivalent to t + 4 + 3 – 2t? A. t + 7 B. -t + 7 C. 6t D. 10t Answer: t + 4 + 3 – 2t = -t + 7. Explanation: In the above-given question, given that, t + 4 + 3 – 2t. t + 7 – 2t. t – 2t + 7. -t + 7. t + 4 + 3 – 2t = -t + 7. Question 11. The distance in feet that Karina swims in a race is represented by 4d – 4, where d is the distance for each lap. What is an expression equivalent to 4d – 4? Answer: The expression equal to 4d – 4 = 4(d – 4). Explanation: In the above-given question, given that, 4d – 4. 4(d – 4). 4d – 16. 4d – 4 = 4d – 16. Question 12. Use the Associative Property to write an expression equivalent to (w + 9) + 3. Answer: The expression is (3 + w) + 9. Explanation: In the above-given question, given that, the expression is (w + 9) + 3. (3 + w) + 3. 6 + w. Question 13. Nigel is planning his training schedule for a marathon over a 4-day period. He is uncertain how many miles he will run on two d Answer: The number of miles he will run on two days = 14.5 miles. Explanation: In the above-given question, given that, Nigel is planning his training schedule for a marathon over a 4-day period. on day 1 he will run 12 miles. on day 2 he will run 14.5 miles. on day 3 he will run 17 miles. 12 + 17 = 29. 29/2 = 14.5. on day 2 he will run 14.5 miles. Question 14. Maria said the expression -4n+ 3 + 9n – 4 is equivalent to 4n. What error did Maria likely make? Answer: -4n + 3 + 9n – 4 = 5n – 1. Explanation: In the above-given question, given that, -4n + 3 + 9n – 4. -4n + 9n = 5n. 5n + 3 – 4. 5n – 1. -4n + 3 + 9n – 4 = 5n – 1. Question 15. Write an expression equivalent to x – 3y + 4. Answer: x – 3y + 4 = 4 + x – 3y. Explanation: In the above-given question, given that, x – 3y + 4. 4 + x – 3y. x + 4 – 3y. Question 16. Andre wrote the expression -2 + 4x = 3 to represent the relationship shown in the table. Write two other expressions that also represent the relationship shown in the table. Answer: -2 + 4x = 3. Explanation: In the above-given question, given that, -2 + 4x. x = 0. -2 + 0. -2. x = 6. -2 + 4(6). -2 + 24. 22. x = 12. -2 + 4(12). -2 + 48. 46. Question 17. Higher Order Thinking to rent a car for a trip, four friends are combining their money. The group chat shows the amount of money that each puts in. One expression for their total amount of money is 189 plus p plus 224 plus q. a. Use the Commutative Property to write two equivalent expressions. Answer: The expressions are 189 + p + 224q. Explanation: In the above-given question, given that, 189 + p + 224q. p + 189 + 224q. 224q + p + 189. b. If they need$500 to rent a car, find at least two different pairs of numbers that p and q could be.

Answer:
$500 + p + 224q. Explanation: In the above-given question, given that, 500 – 224 = 276. 276 + p + 224q. p + 276 + 224q. 224q + 276 + p. Assessment Practice Question 18. Select all expressions equivalent to $$\frac{3}{5}$$x + 3. Answer: The expressions equivalent to (3/5)x + 3 = 1 + 2/5x + 3 and 4/5x – 1/5x + 3. Explanation: In the above-given question, given that, (3/5)x + 3. 1 + 2/5x + 3. 4/5x – 1/5x + 3. 3/5x + 3. so the expressions equivalent to (3/5)x + 3 = 1 + 2/5x + 3 and 4/5x – 1/5x + 3. ### Lesson 4.3 Simplify Expressions Solve & Discuss It! How can the tiles below be sorted? I can… use properties of operations to simplify expressions. Focus on math practices Reasoning Would sorting the tiles with positive coefficients together and tiles with negative coefficients together help to simplify an expression that involves all the tiles? Explain. Answer: The positive coefficients are 4.25, 3/5y, 3/8, 1/5, 2.1x, and 1/2x. The negative coefficients are -0.5, -0.5x, -2.1y, and -2. Explanation: In the above-given question, given that, the coefficients are 4.25, 3/5y, 3/8, 1/5, 2.1x, -0.5, -0.5x, -2.1y, -2, and 1/2x. the positive coefficients are 4.25, 3/5y, 3/8, 1/5, 2.1x, and 1/2x. the negative coefficients are -0.5, -0.5x, -2.1y, and -2. Essential Question How are properties of operations used to simplify expressions? Try It! Simplify the expression – 6 – 6f + 7 – 3f – 9. ______ – 3f – _____ + 7 – ______ _____ – ______ Answer: -6 – 6f + 7 – 3f – 9 = -9f – 23. Explanation: In the above-given question, given that, -6 – 6f + 7 – 3f – 9. -6f – 3f – 6 + 7 – 9. -9f – 13 – 9. -9f – 23. -6 – 6f + 7 – 3f – 9 = -9f – 23. Convince Me! How do you decide in what way to reorder the terms of an expression when simplifying it? Try It! Simplify each expression. a. 59.95m – 30 + 7.95m + 45 + 9.49m Answer: 59.95m – 30 + 7.95m + 45 + 9.49m = 52m – 5.51. Explanation: In the above-given question, given that, 59.95m – 30 + 7.95m + 45 + 9.49. 59.95m + 7.95m – 30 + 45 + 9.49. 52m – 15 + 9.49. 52m – 5.51. b. -0.5p + $$\frac{1}{2}$$p – 2.75 + $$\frac{2}{3}$$p Answer: -0.5p + (1/2)p – 2.75 + (2/3)p = 0.6p – 2.75. Explanation: In the above-given question, given that, -0.5p + (1/2)p – 2.75 + (2/3)p. -0.5p + 0.5p – 2.75 + 0.6p. 0.6p – 2.75. -0.5p + (1/2)p – 2.75 + (2/3)p = 0.6p – 2.75. Try It! Simplify the expression -3.7 +59 + 4k + 11.1 – 10g. (______ – 10g) + 4k + (______ + 11.1) = _______ + 4k + ______ The simplified expression is ________. Answer: -3.7 + 59 + 4k + 11.1 – 10g = 55.3 – 10g. Explanation: In the above-given question, given that, -3.7 + 59 + 4k + 11.1 – 10g. -3.7 + 11.1 – 11.1 + 59 – 10g. -3.7 + 59 – 10g. 55.3 – 10g. KEY CONCEPT When simplifying algebraic expressions, use properties of operations to combine like terms. To simplify the expression below, group like terms. $$\frac{3}{10}$$y – 3.5x – $$\frac{3}{8}$$ +0.53x + 5.25 – 2.75y – 12 (-3.5x + 0.53x) + ($$\frac{3}{10}$$y – 2.75y) + (-$$\frac{3}{8}$$ + 5.25 – 12) Then combine like terms. -2.97x – 2.45y – 7.125 Do You Understand? Question 1. Essential Question How are properties of operations used to simplify expressions? Answer: The properties of operations are used to combine like terms. Explanation: In the above-given question, given that, the properties of operations used to combine like terms. for example: $$\frac{3}{10}$$y – 3.5x – $$\frac{3}{8}$$ +0.53x + 5.25 – 2.75y – 12 (-3.5x + 0.53x) + ($$\frac{3}{10}$$y – 2.75y) + (-$$\frac{3}{8}$$ + 5.25 – 12) Then combine like terms. -2.97x – 2.45y – 7.125 Question 2. Make Sense and Persevere Explain why constant terms expressed as different rational number types can be combined. Answer: The constant terms remain the same. Explanation: In the above-given question, given that, constant terms expressed as different rational number types can be combined. for example: $$\frac{3}{10}$$y – 3.5x – $$\frac{3}{8}$$ +0.53x + 5.25 – 2.75y – 12 (-3.5x + 0.53x) + ($$\frac{3}{10}$$y – 2.75y) + (-$$\frac{3}{8}$$ + 5.25 – 12) Then combine like terms. -2.97x – 2.45y – 7.125 Question 3. Reasoning How do you know when an expression is in its simplest form? Answer: The expression is in its simplest form when it has only limited expressions. Explanation: In the above-given question, given that, for example: -3.7 + 59 + 4k + 11.1 – 10g. -3.7 + 11.1 – 11.1 + 59 – 10g. -3.7 + 59 – 10g. 55.3 – 10g. Do You Know How? Question 4. Simplify -4b + (-9k) – 6 – 3b + 12. Answer: -7b – 9k + 6. Explanation: In the above-given question, given that, -4b + (-9k) – 6 – 3b + 12. -4b – 3b -9k – 6 + 12. -7b – 9k + 6. Question 5. Simplify -2 + 6.45z – 6+ (-3.25z). Answer: -8 + 3.2z. Explanation: In the above-given question, given that, -2 + 6.45z – 6+ (-3.25z). -2 + 6.45z – 6 – 3.25z. -8 + 3.2z. Question 6. Simplify –9 + (-$$\frac{1}{3}$$y) +6 – $$\frac{4}{3}$$y. Answer: -3 – 5/3y. Explanation: In the above-given question, given that, –9 + (-$$\frac{1}{3}$$y) +6 – $$\frac{4}{3}$$y. -9 – 1/3y + 6 – 4/3y. -9 + 6 – 5/3y. -3 – 5/3y. Practice & Problem Solving In 7-10, simplify each expression. Question 7. –2.8f +0.96 – 12 – 4 Answer: -2.8f + 0.96 – 12 – 4 = -2.8f – 15.04. Explanation: In the above-given question, given that, -2.8f + 0.96 – 12 – 4. -2.8f + 0.96 – 16. -2.8f – 15.04. -2.8f + 0.96 – 12 – 4 = -2.8f – 15.04. Question 8. 3.2 – 5.1n – 3n + 5 Answer: 3.2 – 5.1n – 3n + 5 = 8.2 – 8.1n. Explanation: In the above-given question, given that, 3.2 – 5.1n – 3n + 5. 3.2 – 8.1n + 5. 8.2 – 8.1n. 3.2 – 5.1n – 3n + 5 = 8.2 – 8.1n. Question 9. 2n + 5.5 – 0.9n – 8 + 4.5p Answer: 2n + 5.5 – 0.9n – 8 + 4.5p = 4.5p – 2.5 + 1.1n. Explanation: In the above-given question, given that, 2n + 5.5 – 0.9n – 8 + 4.5p. 2n – 0.9n + 5.5 – 8 + 4.5p. 1.1n – 2.5 + 4.5p. 4.5p – 2.5 + 1.1n. 2n + 5.5 – 0.9n – 8 + 4.5p = 4.5p – 2.5 + 1.1n. Question 10. 12 + (-4) – $$\frac{2}{5}$$j – $$\frac{4}{5}$$j + 5 Answer: 12 + (-4) – 2/5j – 4/5j + 5 = 13 – 6/5j. Explanation: In the above-given question, given that, 12 + (-4) – 2/5j – 4/5j + 5. 12 -4 – 6/5j + 5. 8 – 6/5j + 5. 13 – 6/5j. 12 + (-4) – 2/5j – 4/5j + 5 = 13 – 6/5j. Question 11. Which expression is equivalent to -5v + (-2) + 1 + (-2v)? A. -9v B. -4v C. -7v – 1 D. -7V + 3 Answer: Option C is correct. Explanation: In the above-given question, given that, -5v + (-2) + 1 + (-2v). -5v -2 + 1 -2v. -7v -1. so option C is correct. Question 12. Which expression is equivalent to $$\frac{2}{3}$$x + (-3) + (-2) – $$\frac{1}{3}$$x? A. x + 5 B. –$$\frac{1}{3}$$x + 5 C. $$\frac{1}{3}$$x – 1 D. $$\frac{1}{3}$$x – 5 Answer: Option D is correct. Explanation: In the above-given question, given that, $$\frac{2}{3}$$x + (-3) + (-2) – $$\frac{1}{3}$$x. 2/3x -3 -2 -1/3x. 1/3x -5 so option D is correct. Question 13. The dimensions of a garden are shown. Write an expression to find the perimeter. Answer: The perimeter of the garden = x – 7. Explanation: In the above-given question, given that, the length of the garden = x. the width of the garden = 1/2x – 7. area of the garden = l x b. x + 1/2x – 7. 2/2x – 7. x – 7. so the perimeter of the garden = x – 7. Question 14. Simplify the expression 8h + (-7.3d) – 14 + 5d – 3.2h. Answer: 8h + (-7.3d) – 14 + 5d – 3.2h = 5.2h – 2.3d – 14. Explanation: In the above-given question, given that, 8h + (-7.3d) – 14 + 5d – 3.2h. 8h – 7.3d – 14 + 5d – 3.2h. 5.2h – 2.3d – 14. 8h + (-7.3d) – 14 + 5d – 3.2h = 5.2h – 2.3d – 14. Question 15. Simply 4 – 2y + (-8y) + 6.2. Answer: 4 – 2y + (-8y) + 6.2 = 10.2 – 10y. Explanation: In the above-given question, given that, 4 – 2y + (-8y) + 6.2. 4 – 2y – 8y + 6.2. 4 – 10y + 6.2. 10.2 – 10y. 4 – 2y + (-8y) + 6.2 = 10.2 – 10y. Question 16. Simplify $$\frac{4}{9}$$z – $$\frac{3}{9}$$z + 5 – $$\frac{5}{9}$$z – 8. Answer: 4/9z – 3/9z + 5 – 5/9z – 8 = -4/9z – 3. Explanation: In the above-given question, given that, 4/9z – 3/9z + 5 – 5/9z – 8. 1/9z + 5 – 5/9z – 8. -4/9z + 5 – 8. -4/9z – 3. 4/9z – 3/9z + 5 – 5/9z – 8 = -4/9z – 3. Question 17. Construct Arguments Explain whether 11t – 4t is equivalent to 4t – 11t. Support your answer by evaluating the expression for t = 2. Answer: The values are the same but 11t – 4t is positive and 4t – 11t is negative. Explanation: In the above-given question, given that, 11t – 4t is equivalent to 4t – 11t. t = 2. 11(2) – 4(2). 22 – 8. 14. 4t – 11t. 4(2) – 11(2). 8 – 22. -14. Question 18. The signs show the costs of different games at a math festival. How much would it cost n people to play Decimal Decisions and Ratio Rage? Answer: The cost would take to n people to play Decimal Decisions and Ratio Rage = 6.6n/4 – 3. Explanation: In the above-given question, given that, the cost of 1 Game is 5.5n – 3. the cost of 1 Game is n/4. 5.5n – 3 + n/4. 6.6n/4 – 3. the cost would take to n people to play Decimal Decisions and Ratio Rage = 6.6n/4 – 3. Question 19. Higher Order Thinking in the expression ax + bx, a is a decimal and b is a fraction. How do you decide whether to write a as a fraction or b as a decimal? Answer: Yes, we can write an as a fraction and b as a decimal. Explanation: In the above-given question, given that, in the expression ax + bx, a is a decimal and b is a fraction. for example: a = 1.1. b = 1/2. ax + bx. 1.1x + 1/2x. so 1.1 is a decimal and 1/2 is a fraction. Assessment Practice Question 20. Select all expressions equivalent to -6z + (-5.5) + 3.5z + 5y – 2.5. ☐ -8 + 5y + 2.52 ☐ -2.5z + 5y – 8 ☐ -8 + 5y +(-2.5z) ☐ 2.5y + (-2.5z) – 5.5 ☐ 5y – 8 – 2.5z Answer: Option B and C are correct. Explanation: In the above-given question, given that, -6z + (-5.5) + 3.5z + 5y – 2.5. -6z + 3.5z – 5.5 – 2.5 + 5y. -2.5z -8 + 5y. so options B and C are correct. ### Lesson 4.4 Expand Expressions Solve & Discuss It! The school is planning to add a weight room to the gym. If the total area of the gym and weight room should stay under 5,500 square feet, what is one possible length for the new weight room? Show your work. Are there other lengths that would work? Why or why not? -90 ft I can… expand expressions using the Distributive Property. Look for Relationships What is the relationship between the areas of the gym and weight room? Answer: The relationship between the areas of the gym and weight room = 550 ft. Explanation: In the above-given question, given that, The school is planning to add a weight room to the gym. If the total area of the gym and weight room should stay under 5,500 square feet. the area of the school is l x b. where l = 90 ft and b = 55 ft. area = l x b. area = 90 x 55. area = 4950. 5500 – 4950 = 550. so the relationship between the areas of the gym and weight room = 550 ft. Focus on math practices Model with Math What is an expression using x that represents the total area of the gym and the weight room? Answer: The relationship between the areas of the gym and weight room = 550 ft. Explanation: In the above-given question, given that, The school is planning to add a weight room to the gym. If the total area of the gym and weight room should stay under 5,500 square feet. the area of the school is l x b. where l = 90 ft and b = 55 ft. area = l x b. area = 90 x 55. area = 4950. 5500 – 4950 = 550. so the relationship between the areas of the gym and weight room = 550 ft. Essential Question How does the value of an expression change when it is expanded? Try It! What is the expanded form of the expression 3.6(t + 5)? 3.6(t + 5) = ________t + _______ • 5 = _______ + _______ The expanded expression is _______. Answer: 3.6t + 18. Explanation: In the above-given question, given that, 3.6(t + 5). 3.6 x t = 3.6t. 3.6t + 3.6 x 5. 3.6t + 18. Convince Me! If you know the value of t, would the evaluated expression be different if you added the known value of t and 5 and then multiplied by 3.6? Explain. Try It! Expand the expression t(-1.2w + 3). Answer: The expression is -1.2tw + 3t. Explanation: In the above-given question, given that, the expression is t(-1.2w + 3). -1.2tw + 3t. so the expanded expression is -1.2tw + 3t. Try It! Simplify the expression –$$\frac{2}{5}$$(10 + 15m – 20n). Answer: The expression is -4 -6m – 8n. Explanation: In the above-given question, given that, –$$\frac{2}{5}$$(10 + 15m – 20n). -2/5 (10 + 15m – 20n). -2/5(10 + 15m – 20n). -20/5 – 30/5m – 40/5n. -4 – 6m – 8n. KEY CONCEPT You can expand an expression using the Distributive Property. Multiply, or distribute, the factor outside the parentheses with each term inside the parentheses. -7(3y – 1) = (-7)(3y) + (-7)(-1) = -21y + 7 The sign of each term is included in all calculations. Do You Understand? Question 1. Essential Question How does the value of an expression change when it is expanded? Answer: The value of an expression change when it is expanded. Explanation: In the above-given question, given that, the value of an expression change when it is expanded. for example: -8(2y – 2). -8(-2y) + (-8) (-2). -16y + -16. Question 2. Use Structure How does the subtraction part of the expression change when a(b – c) is expanded? Answer: The subtraction part of the expression change. Explanation: In the above-given question, given that, a(b – c). ax b – a x c. ab – ac. the product of in terms is multiplied with outterms. Question 3. Make Sense and Persevere When does expanding and simplifying a(b + c) result in a positive value for ac? Answer: ab + ac. Explanation: In the above-given question, given that, the expression is a(b + c). a x b + a x c. ab + ac. the sign is positive. so the value for ac is also positive. Do You Know How? Question 4. Shoes and hats are on sale. The expression $$\frac{1}{4}$$(s + 24.80) can be used to determine the discount when you buy shoes with a retail price of s dollars and a hat with a retail price of$24.80. Write another expression that can be used to determine the discount.

Answer:
Another expression is $1.55. Explanation: In the above-given question, given that, Shoes and hats are on sale. The expression $$\frac{1}{4}$$(s + 24.80). when you buy shoes with a retail price of s dollars and a hat with a retail price of$24.80.
1/4 (s + 24.80).
s/4 + 24.80/4.
s/4 + 6.2.
s/4 = – 6.2.
s = -6.2/4.
s = 1.55.
so the retail price of the shoes = $1.55. Question 5. Expand x(4 – 3.4y). Answer: The expression is 4x – 3.4xy. Explanation: In the above-given question, given that, x(4 – 3.4y). 4x X – 3.4 x X x Y. 4x – 3.4xy. so the expanded expression is 4x – 3.4xy. Question 6. Expand –$$\frac{2}{10}$$(1 – 2x + 2). Answer: The expanded expression is -3/5 – 2/5x. Explanation: In the above-given question, given that, –$$\frac{2}{10}$$(1 – 2x + 2). -2/10 (1 – 2x + 2). -1/5 (1 – 2x + 2). -1/5 – 2/5x – 2/5. -3/5 – 2/5x. Practice & Problem Solving Leveled Practice For 7-8, fill in the boxes to expand each expression. Question 7. 3(n + 7) = (3) (_____) + (3) (_____) = ____ + _____ Answer: 3n + 21. Explanation: In the above-given question, given that, 3(n + 7). 3 x n + 3 x 7. 3n + 21. Question 8. 4(x – 3) = ______ x – ______ (3) = ______ – ______ Answer: 4x – 12. Explanation: In the above-given question, given that, 4(x – 3). 4 x X – 4 x 3. 4x – 12. For 9-14, write the expanded form of the expression. Question 9. y(0.5 + 8) Answer: y(0.5 + 8) = 8.5y. Explanation: In the above-given question, given that, y(0.5 + 8). 0.5y + 8y. 8.5y. y(0.5 + 8) = 8.5y. Question 10. 4(3 + 4x – 2) Answer: 4(3 + 4x – 2) = 4 + 16x. Explanation: In the above-given question, given that, 4(3 + 4x – 2). 4 x 3 + 4x x 4 – 2 x 4. 12 + 16x – 8. 4 + 16x. 4(3 + 4x – 2) = 4 + 16x. Question 11. 6(y + x) Answer: 6(y + x) = 6y + 6x. Explanation: In the above-given question, given that, 6(y + x). 6 x y + 6 x x. 6y + 6x. 6(y + x) = 6y + 6x. Question 12. -2.5(-3 + 4n + 8) Answer: -2.5 (-3 + 4n + 8) = -14.5 – 10n. Explanation: In the above-given question, given that, -2.5 (-3 + 4n + 8). -2.5 x -3 – 2.5 x 4n – 2.5 x 8. -5.5 – 10n – 20. -14.5 – 10n. -2.5 (-3 + 4n + 8) = -14.5 – 10n. Question 13. –$$\frac{1}{3}$$(y – x) Answer: –$$\frac{1}{3}$$(y – x) = -1/3y + x/3. Explanation: In the above-given question, given that, –$$\frac{1}{3}$$(y – x). -1/3(y – x). -1/3y + x/3. –$$\frac{1}{3}$$(y – x) = -1/3y + x/3. Question 14. 8(6x – 4) Answer: 8(6x – 4) = 48x – 32. Explanation: In the above-given question, given that, 8(6x – 4). 8 x 6x – 4 x 8. 48x – 32. 8(6x – 4) = 48x – 32. Question 15. Higher Order Thinking A grocery store has a 13%-off sale on all bread. You decide to purchase 6 loaves of bread. Let b be the original price of a loaf of bread. Expand the expression 6(b – 0.13b). Once the expression is expanded, what do the terms represent? Answer: 6(b – 0.13b) = -5.22b. Explanation: In the above-given question, given that, A grocery store has a 13%-off sale on all bread. You decide to purchase 6 loaves of bread. Let b be the original price of a loaf of bread. 6(b – 0.13b). 6 x b – 0.13b x 6. 6b – 0.78b. -5.22b. 6(b – 0.13b) = -5.22b. Question 16. A gardener plans to extend the length of a rectangular garden. Let x represent the garden’s original length. The expression 4(x + 7) represents the area of the extended garden. When asked for the area of the extended portion, the gardener incorrectly said it was 11 square feet. Describe the error the gardener made. Answer: The error the gardener made = 4x + 28. Explanation: In the above-given question, given that, Let x represent the garden’s original length. The expression 4(x + 7) represents the area of the extended garden. 4(x + 7). 4 x x + 4 x 7. 4x + 28. Question 17. Find a difference equivalent to the product 11(x – y). Answer: 11(x – y) = 11x – 11y. Explanation: In the above-given question, given that, 11(x – y). 11 x x – 11 x y. 11x – 11y. 11(x – y) = 11x – 11y. Question 18. Use the Distributive Property to write an expression equivalent to 0.4(-5 – 7y – 13.8). Answer: 0.4(-5 – 7y – 13.8) = -10.12 – 2.8y. Explanation: In the above-given question, given that, the expression is 0.4(-5 – 7y – 13.8). 0.4 x (-5) – 0.4 (7y) – 0.4 (-13.8). -4.6 – 2.8y – 5.52. -10.12 – 2.8y. Question 19. Make Sense and Persevere Use the Distributive Property to expand 7(7x – 3y) – 6. Answer: 7(7x – 3y) – 6 = 49x – 21y – 6. Explanation: In the above-given question, given that, 7(7x – 3y) – 6. 7 x 7x – 7 (3y) – 6. 49x – 21y – 6. 7(7x – 3y) – 6 = 49x – 21y – 6. Question 20. Use the Distributive Property to write an expression equivalent to y(-3 – 8x). Answer: y(-3 – 8x) = -3y – 8xy. Explanation: In the above-given question, given that, y(-3 – 8x). -3 x y – 8x (y). -3y -8xy. Question 21. An architect plans to build an extension to Meiling’s rectangular deck. Let x represent the increase, in meters, of her deck’s length. The expression 5(X + 8) represents the area of the deck, where 5 is the width, in meters, and (x + 8) represents the extended length, in meters. Use the Distributive Property to write an expression that represents the total area of Meiling’s new deck. Answer: The total area of Meiling’s new deck = 5x + 40. Explanation: In the above-given question, given that, An architect plans to build an extension to Meiling’s rectangular deck. Let x represent the increase, in meters, of her deck’s length. The expression 5(X + 8) represents the area of the deck, where 5 is the width, in meters, and (x + 8) represents the extended length in meters. 5(x + 8). 5 x x + 5 x 8. 5x + 40. Assessment Practice Question 22. Select all expressions equivalent to –$$\frac{1}{2}$$(4 – 2 + 8x). ☐ -4x – 1 ☐ 4x – 1 ☐ 3x ☐ -2 + 1 – 4x ☐ 2 + 1 – 4x ☐ 4x + 1 Answer: -4x -1 and -2 + 1 – 4x. Explanation: In the above-given question, given that, –$$\frac{1}{2}$$(4 – 2 + 8x). -1/2(4 – 2 + 8x). -4/2 – 2/2 + 8/2x. -2 + 1 – 4x. -1 – 4x. Question 23. An expression is shown. $$\frac{1}{5}$$(5 – 7y + 10) Create an equivalent expression without parentheses. Answer: 1/5(5 – 7y + 10) = 1 – 7/5y + 2. Explanation: In the above-given question, given that, 1/5(5 – 7y + 10). 5/5 – 7/5y + 10/5. 1 – 7/5y + 2. ### Lesson 4.5 Factor Expressions Explain It! Tasha is packing gift bags that include the same items. She has 72 glow sticks, 36 markers, and 24 bottles of bubbles. Tasha believes that she can pack no more than 6 bags using all of her supplies. Answer: Yes, she can pack 44 bags. Explanation: In the above-given question, given that, Tasha is packing gift bags that include the same items. She has 72 glow sticks, 36 markers, and 24 bottles of bubbles. 72 + 36 + 24 = 132. 132/3 = 44. she can pack 44 bags. I can… use common factors and the Distributive Property to factor expressions. Make Sense and Persevere How can you use what you know about common factors to solve the problem? A. Critique Reasoning Do you agree with Tasha? Explain. Answer: No, Tasha is wrong. Explanation: In the above-given question, given that, Tasha is packing gift bags that include the same items. She has 72 glow sticks, 36 markers, and 24 bottles of bubbles. 72 + 36 + 24 = 132. 132/3 = 44. she can pack 44 bags. B. If Tasha creates the greatest number of gift bags, how many of each item is in each bag? Explain how you know. Answer: Tasha packs 44 bags. Explanation: In the above-given question, given that, Tasha is packing gift bags that include the same items. She has 72 glow sticks, 36 markers, and 24 bottles of bubbles. 72 + 36 + 24 = 132. 132/3 = 44. she can pack 44 bags. Focus on math practices Reasoning Tasha added more markers and now has a total of 48 markers. Does this change the possible number of gift bags? Explain. Essential Question How does the Distributive Property relate to factoring expressions? Try It! Use factoring to write an expression for the length of the pool with the given width. 4x + 20 = _____ (x + _____) So, the length of the pool is ______ meters. Answer: The length of the pool is 4(x + 5) meters. Explanation: In the above-given question, given that, 4x + 20. 4(x + 5). the length of the pool is 4(x + 5) meters. Convince Me! How can you use the Distributive Property to check the factored expression? Use the factored expression for Example 1 in your explanation. Try It! Show two different ways to factor -(4x – 28). Answer: -(4x – 28) = -4(x – 7). Explanation: In the above-given question, given that, -(4x – 28). -4x + 28. -4(x – 7). Try It! Write an equivalent expression for the expression above using a negative factor. Answer: The equivalent expression for the expression is -4(x – 7). Explanation: In the above-given question, given that, the equivalent expression for the expression is -4(x – 7). -(4x – 28). -4x + 28. -4(x – 7). KEY CONCEPT The greatest common factor (GCF) can be used to factor expressions. The Distributive Property can be applied to factor an expression. Factoring an expression creates an equivalent expression. Do You Understand? Question 1. Essential Question How does the Distributive Property relate to factoring expressions? Answer: 2x + 8 = 2(x + 4). Explanation: In the above-given question, given that, the distributive property can be applied to factor an expression. factoring an expression creates an equivalent expression. 2x + 8. 2(x + 4). Question 2. Susan incorrectly factored the expression below. 12a – 156 + 6 3(4a + 5b + 3) a. Explain any errors Susan may have made when factoring. Answer: 12a + 15b + 9. Explanation: In the above-given question, given that, 3(4a + 5b + 3). (3 x 4a) + (3 x 5b) + (3 x 3). 12a + 15b + 9. b. Factor the expression correctly. Answer: 12a + 15b + 9. Explanation: In the above-given question, given that, 3(4a + 5b + 3). (3 x 4a) + (3 x 5b) + (3 x 3). 12a + 15b + 9. Do You Know How? Question 3. Sahil is putting together supply kits and has 36 packs of x pencils, 12 packs of y crayons, and 24 erasers. a. Write an expression to show the total number of items. Answer: The expression to show the total number of items = 36x + 12y + 24. Explanation: In the above-given question, given that, Sahil is putting together supply kits and has 36 packs of x pencils, 12 packs of y crayons, and 24 erasers. 36 x X + 12 x Y + 24, 36x + 12y + 24. so the expression to show the total number of items = 36x + 12y + 24. b. Use factoring to show many kits Sahil can make while putting every type of item in each kit. Answer: The expression to show the total number of items = 36x + 12y + 24. Explanation: In the above-given question, given that, Sahil is putting together supply kits and has 36 packs of x pencils, 12 packs of y crayons, and 24 erasers. 36 x X + 12 x Y + 24, 36x + 12y + 24. so the expression to show the total number of items = 36x + 12y + 24. c. Use the factored expression to find the number of each item in each kit. Answer: The expression to show the total number of items = 36x + 12y + 24. Explanation: In the above-given question, given that, Sahil is putting together supply kits and has 36 packs of x pencils, 12 packs of y crayons, and 24 erasers. 36 x X + 12 x Y + 24, 36x + 12y + 24. so the expression to show the total number of items = 36x + 12y + 24. Question 4. Show two different ways to factor – 12x + 24 – 18y. Answer: -12x + 24 – 18y and -2(6x – 12 + 9y). Explanation: In the above-given question, given that, the expression is – 12x + 24 – 18y. -2(6x – 12 + 9y). Question 5. How can you use the Distributive Property to factor the expression 6x + 15? Answer: 3(2x + 5). Explanation: In the above-given question, given that, the given expression is 6x + 15. 3x + 3x +15. 3(2x + 5). Practice & Problem Solving Leveled Practice In 6-9, factor the expression. Question 6. 16a + 10. The GCF of 16a and 10 is 2. 2 × ______ = 16a 2 × _______ = 10 The factored expression is ________ Answer: 2 x 8a = 16a. 2 x 5 = 10. The factored expression is 16a + 10. Explanation: In the above-given question, given that, the expression is 16a + 10. 2(8a + 5). 2 x 8a = 16a. 2 x 5 = 10. Question 7. -9y – 3. The positive GCF of -9y and -3 is 3. 3 × ______ = -9y 3 × ______ = -3 The factored expression is ________ Answer: 3 x -3y = -9y. 3 x 1 = -3. Explanation: In the above-given question, given that, the expression is -9y – 3. -3(3y + 1). -3 x 3y = -9y. -3 x 1 = -3. Question 8. 14x + 49 Answer: 7 x 2x = 14x. 7 x 7 = 49. Explanation: In the above-given question, given that, the expression is 14x + 49. 7(2x + 7). 7 x 2x = 14x. 7 x 7 = 49. Question 9. 12y – 16 Answer: 2 x 6y = 12y. 2 x 8 = 16. Explanation: In the above-given question, given that, the expression is 12y – 16. 2(6y – 8). 2 x 6y = 12y. 2 x 8 = 16. Question 10. This model shows the area of a garden. Write two expressions that represent the area. Answer: The area of the garden = 5x + 10. Explanation: In the above-given question, given that, the expression is 5x + 10. 5(x + 2). 5 x x = 5x. 5 x 2 = 10. so the area of the garden = 5x + 10. Question 11. Use the GCF to write the factored form of the expression 18x + 24y. Answer: The factored form of the expression is 2(9x + 12y). Explanation: In the above-given question, given that, the expression is 18x + 24y. 2(9x + 12y). 6(3x + 4y). so the factored form of the expression is 2(9x + 12y). Question 12. Find the dimensions of the sports field at the right if the width is at least 60 yards. Answer: The dimensions of the sports field = -160 and 60. Explanation: In the above-given question, given that, the expression is 240 – 400x. -160x. -160(60). -9600. so the dimensions of the sports field = -160 and 60. Question 13. Your friend incorrectly factors the expression 15x – 20xy as 5x( 3 – 4xy). a. Factor the expression correctly. Answer: The expression correctly = 5(3x – 4xy). Explanation: In the above-given question, given that, the expression is 15x – 20xy. 5(3x – 4xy). 5 x 3x = 15x. 5 x 4xy = 20xy. so the expression correctly is 5(3x – 4xy). b. What error did your friend likely make? Answer: The expression correctly = 5(3x – 4xy). Explanation: In the above-given question, given that, the expression is 15x – 20xy. 5(3x – 4xy). 5 x 3x = 15x. 5 x 4xy = 20xy. so the expression correctly is 5(3x – 4xy). Question 14. You are given the expression 12x + 18y + 26. a. Make Sense and Persevere What is the first step in factoring the expression? Answer: The first step in factoring is 2(6x + 9y + 18). Explanation: In the above-given question, given that, the expression is 12x + 18y + 26. 2(6x + 9y + 13). 2 x 6x = 12x. 2 x 9y = 18y. 2 x 13 = 26. b. Factor the expression. Answer: The expression is 2(6x + 9y + 18). Explanation: In the above-given question, given that, the expression is 12x + 18y + 26. 2(6x + 9y + 13). 2 x 6x = 12x. 2 x 9y = 18y. 2 x 13 = 26. Question 15. A hotel manager is adding a tile border around the hotel’s rectangular pool. Let x represent the width of the pool, in feet. The length is 3 more than 2 times the width, as shown. Write two expressions that give the perimeter of the pool. Answer: The perimeter of the pool = 2XxX + 3x. Explanation: In the above-given question, given that, Let x represent the width of the pool, in feet. The length is 3 more than 2 times the width, as shown. the perimeter of the rectangle = length x width. perimeter = 2x + 3 x X. perimeter = 2×2 x 3x. Question 16. Higher Order Thinking Use the expressions below. 14m + mn 2y + 2x + 4 –$$\frac{3}{4}$$m + 8m + m 4 – 3p 5.75t + 7.75t – t 8xy – 6xy a. Circle the expressions that have like terms. Answer: The expressions that have the like terms = 14m + mn, 5.75t + 7.75t -t, and 8xy – 6xy. Explanation: In the above-given question, given that, the expressions are 14m + mn 2y + 2x + 4. –$$\frac{3}{4}$$m + 8m + m. 4 – 3p. 5.75t + 7.75t – t. 8xy – 6xy. the like terms are 8xy – 6xy = 2xy. 5.75t + 7.75t – t = 13.5t – t. 12.5 t. b. Explain why the other expressions do not have like terms. Answer: The other expressions that do not have like terms are 4 – 3p, 14m + mn. Explanation: In the above-given question, given that, the expressions are 14m + mn 2y + 2x + 4. –$$\frac{3}{4}$$m + 8m + m. 4 – 3p. 5.75t + 7.75t – t. 8xy – 6xy. the unlike terms are 14m + mn. 2y + 2x + 4. Question 17. Construct Arguments Ryan says the expression 3 + 5y cannot be factored using a GCF. Is he correct? Explain why or why not. Answer: Yes, Ryan was correct. Explanation: In the above-given question, given that, the expression is 3 + 5y. we cannot be factored in using a GCF. so Ryan was correct. Assessment Practice Question 18. Select all the expressions equivalent to 12 + 30y. ☐ 3(4 + 10y) ☐ 4(3 + 10y) ☐ 6(2 + 5y) ☐ 2(6 + 30y) ☐ 6(3 + 10y) Answer: The expressions equivaent to 12 + 30y is 3(4 + 10y), 6(2 + 5y). Explanation: In the above-given question, given that, the expression is 12 + 30y. 3(4 + 10y). 6(2 + 5y). 3 x 4 = 12 + 3 x 10y = 30y. 6 x 2 = 12 + 6 x 5y = 30y. Question 19. Write an expression that is the product of two factors and is equivalent to -2x – 10. Answer: The expression is -2(x + 5). Explanation: In the above-given question, given that, the expression is -2x – 10. -2(x + 5). -2x + 10. ### Topic 4 Mid-Topic Checkpoint Question 1. Vocabulary If you write an expression to represent the following situation, how can you determine which is the constant and which is the coefficient of the variable? Lesson 4-1 The zoo charges the Garcia family an admission fee of$5.25 per person and a one-time fee of $3.50 to rent a wagon for their young children. Answer: The admission fee is the coefficient and rents a wagon is constant. Explanation: In the above-given question, given that, The zoo charges the Garcia family an admission fee of$5.25 per person.
a one-time fee of $3.50 to rent a wagon for their young children.$5.25x + $3.50.$5.25 is the coefficient of the variable.
$3.50 is the constant. Question 2. An online photo service charges$20 to make a photo book with 16 pages. Each extra page costs $1.75. The cost to ship the completed photo book is$5. Write an expression to determine the total cost in dollars to make and ship a photo book with x extra pages. Lesson 4-1

Answer:
The total cost in dollars to make and ship a photo book with X extra pages is $20x +$5 = $1.75. Explanation: In the above-given question, given that, An online photo service charges$20 to make a photo book with 16 pages.
Each extra page costs $1.75. The cost to ship the completed photo book is$5.
$20x +$5 = $1.75.$25x -$1.75 =$5.

Question 3.
Write an expression equivalent to 2a + ($$\frac{3}{4}$$a + $$\frac{1}{5}$$b) by combining like terms. Lesson 4-3

Answer:
The expression equivalent to 2a + 3/4a + 1/5b is 5/4a + 1/5b.

Explanation:
In the above-given question,
given that,
the given expression is 2a + ($$\frac{3}{4}$$a + $$\frac{1}{5}$$b).
2a + 3/4a + 1/5b.
5/4a + 1/5b.

Question 4.
Which expression is equivalent to 3.2y – $$\frac{1}{3}$$ + (-7y) + $$\frac{2}{3}$$? Lesson 4-2
A. -10.2y + $$\frac{1}{3}$$
B. -3.8y + $$\frac{1}{3}$$
C. -3$$\frac{7}{15}$$y
D. -3y

Answer:
Option B is the correct.

Explanation:
In the above-given question,
given that,
the expression is 3.2y – $$\frac{1}{3}$$ + (-7y) + $$\frac{2}{3}$$.
3.2y – 1/3 -7y + 2/3.
-3.8y + 1/3.
so option B is the correct.

Question 5.
Ray wants to buy a hat that costs $10 and some shirts that cost$12 each. The sales tax rate is 6.5%. Write an expression to determine the amount of sales tax that Ray will pay on his entire purchase. Expand to simplify the expression. Lesson 4-4

Answer:
The expression to determine the amount of sales tax that Ray will pay on his entire purchase = $22 + 6.5%. Explanation: In the above-given question, given that, Ray wants to buy a hat that costs$10 and some shirts that cost $12 each. The sales tax rate is 6.5%.$10 + $12 + 6.5%.$22 + 6.5%.

Question 6.
Factor the expression 28r + 425 – 35. Lesson 4-5

Answer:
The expression is 7(4r + 61 – 5).

Explanation:
In the above-given question,
given that,
the expression is 28r + 425 – 35.
7(4r + 61 – 5).
7 x 4r = 28r.
61 x 7 = 425.
7 x 5 = 35.

Question 7.
Describe two ways the Distributive Property can be used to write equivalent expressions. Lessons 4-4 and 4-5

Answer:
The two ways the distributive property can be used to write equivalent expressions

Explanation:
In the above-given question,
given that,
-1/2(x + 8), -1/2x + (-4) and -4 +(-1/2x) are equivalent.
-1/2(x + 8).
-1/2x + (-1/2) . 8.
-1/2x + (-4).
-4 + (-1/2x).
the three expressions are true.

### Topic 4 Mid-Topic Performance Task

Alison is a buyer for a chain of 6 flower shops. This means that she buys flowers in bulk from a supplier and then distributes them to the 6 flower shops in the chain.
PART A
This week Alison bought 108 bunches of carnations and 96 bunches of roses from the supplier. Let c represent the number of carnations in each bunch, and let r represent the number of roses in each bunch. Write an expression to show the total number of carnations and roses that Alison bought.

Answer:
The total number of carnations and roses that Alison bought = 12(9c + 8r).

Explanation:
In the above-given question,
given that,
This week Alison bought 108 bunches of carnations and 96 bunches of roses from the supplier.
Let c represent the number of carnations in each bunch, and let r represent the number of roses in each bunch.
108c + 96r.
12(9c + 8r).
12 x 9c = 108c.
12 x 8r = 96r.

PART B
Alison wants to distribute the carnations and roses equally among the 6 flower shops. Factor the expression from Part A using 6 as the common factor. How does the factored expression help Alison determine how many carnations and how many roses each flower shop should get?

Answer:
The common factor is 3(c + r).

Explanation:
In the above-given question,
given that,
Alison wants to distribute the carnations and roses equally among the 6 flower shops.
3c + 3r.
3(c + r).
1(3c + 3r).
so the common factor is 3(c + r).

PART C
There are 24 carnations in each bunch and 12 roses in each bunch. Use your answer to Part B to determine the total number of carnations and the total number of roses Alison will distribute to each flower shop this week.

Answer:
The total number of carnations and roses = 6(4c + 2r).

Explanation:
In the above-given question,
given that,
There are 24 carnations in each bunch and 12 roses in each bunch.
24 c + 12 r.
2(12c + 6r).
6(4c + 2r).
6 x 4c = 24c.
6 x 2r = 12r.

PART D
Jake manages one of the flower shops. He wants to use the carnations and roses to make bouquets. He wants each bouquet to have the same combination of carnations and roses, with no flowers left over. Determine a way that Jake can divide the flowers to make the bouquets. How many bouquets will there be?

Answer:
The number of bouquets will there be = 6(c + r).

Explanation:
In the above-given question,
given that,
Jake manages one of the flower shops.
He wants to use the carnations and roses to make bouquets.
He wants each bouquet to have the same combination of carnations and roses, with no flowers left over.
6 and 6.
6 x 6 = 36.
6c + 6r.
6(c + r).
so the number of bouquets will there be = 6(c + 1).

3-Act Mathematical Modeling:
I’ve Got You Covered

ACT 1
Question 1.
After watching the video, what is the first question that comes to mind?
Answer:

Question 2.
Write the Main Question you will answer.
Answer:

Question 3.
Construct Arguments Predict an answer to this Main Question. Explain your prediction.

Answer:

Question 4.
On the number line below, write a number that is too small to be the answer. Write a number that is too large.

Answer:
The two numbers are 1 and 10.

Explanation:
In the above-given question,
given that,
the number line is 10 cm long.
the short is 1 cm.
the large is 10 cm.
so the two numbers that are too small and too large is 1 and 10.

Question 5.
Plot your prediction on the same number line.

Answer:
The two numbers are 1 and 10.

Explanation:
In the above-given question,
given that,
the number line is 10 cm long.
the short is 1 cm.
the large is 10 cm.
so the two numbers that are too small and too large is 1 and 10.

ACT 2
Question 6.
What information in this situation would be helpful to know? How would you use that information?

Answer:
The figure contain 7 objects.

Explanation:
In the above-given question,
given that,
the figure contains 7 objects.
3 objects on the left side.
4 objects on the right side.
so the figure contain 7 objects.

Question 7.
Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.

Answer:
The figure contain 7 objects.

Explanation:
In the above-given question,
given that,
the figure contains 7 objects.
3 objects on the left side.
4 objects on the right side.
so the figure contain 7 objects.

Question 8.
Model with Math Represent the situation using mathematics. Use your representation to answer the Main Question.
Answer:

Question 9.
What is your answer to the Main Question? Is it higher or lower than your prediction? Explain why.

Answer:

ACT 3
Question 10.
Write the answer you saw in the video.

Answer:

Question 11.
Reasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
Answer:

Question 12.
Make Sense and Persevere Would you change your model now that you know the answer? Explain.

Answer:

ACT 3

Extension
Reflect
Question 13.
Model with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
Answer:

Question 14.
Generalize What pattern did you notice in your calculations? How did that pattern help you solve the problem?
Answer:

SEQUEL
Question 15.
Reasoning A classmate says that another object needs 512 tiles. What do you know about the dimensions of the object?

Answer:
The dimensions of the object = 22 and 26.

Explanation:
In the above-given question,
given that,
A classmate says that another object needs 512 tiles.
22 x 26 = 512.
the length = 22.
the width = 26.
area = l x b.
22 x 26.
512.

### Lesson 4.6 Add Expressions

Solve & Discuss It!
The Smith family took a 2-day road trip. On the second day, they drove the distance they traveled on the first day. What is a possible distance they could have traveled over the 2 days? Is there more than one possible distance? Justify your response.
I can… add expressions that represent real-world problems.

Make Sense and Persevere
How are the quantities in the problem related?

Answer:
The quantities in the problem are the smith family took a 2-day road trip.

Explanation:
In the above-given question,
given that,
The Smith family took a 2-day road trip.
On the second day, they drove the distance they traveled on the first day.
2 + 2 = 4.

Focus on math practices
Use Structure How can two different expressions be used to represent the total distance?

Essential Question
How can properties of operations be used to add expressions?

Try It!

Sophia and Ollie each deposit $120 to open a joint account. They each make monthly deposits as shown. What expression represents the amount in the account after m months? The amount of money in the joint account after m months is ______ + _______. Answer: The amount of money in the joint account after m months is 120 + 120 =$525..

Explanation:
In the above-given question,
given that,
Sophia and Ollie each deposit $120 to open a joint account. 120 + 150 = 270. 120 + 135 = 255. 270 + 255 = 525. so the amount of money in the joint account after m months is$525.

Convince Me! Explain why the initial deposits and monthly deposits are not combined into one term?

Try It!

Find each sum.
a. (9.740 – 250.50) + (-5.48p + 185.70)

Answer:
Te sum is 60.54.

Explanation:
In the above-given question,
given that,
(9.740 – 250.50) + (-5.48p + 185.70).
(-240.76) + (180.22).
60.54.

b. ($$\frac{2}{11}$$x – 3 – 5y) + (-$$\frac{3}{11}$$ + 5y + 5.5)

Answer:
-1/11 x + 2.5 + 5y.

Explanation:
In the above-given question,
given that,
($$\frac{2}{11}$$x – 3 – 5y) + (-$$\frac{3}{11}$$ + 5y + 5.5)
(2/11)x – 3 – 5y + (-3/11) + 5y + 5.5.
-1/11x + 2.5 + 5y.
-1/11 x + 2.5 + 5y.

C. (-14.2b – 97.35) + (6.76d – 118.7 – 3.4d)

Answer:
-14.2b – 216.05 + 3.36d.

Explanation:
In the above-given question,
given that,
(-14.2b – 97.35) + (6.76d – 118.7 – 3.4d).
-14.2b – 97.35 + 3.36d – 118.7.
-14.2b – 216.05 + 3.36d.

d. ($$\frac{3}{8}$$ – $$\frac{1}{6}$$m + 5t) + ($$\frac{7}{10}$$m + 9t + $$\frac{1}{4}$$)

Answer:
1/3 + 3/2m + 14t.

Explanation:
In the above-given question,
given that,
$$\frac{3}{8}$$ – $$\frac{1}{6}$$m + 5t) + ($$\frac{7}{10}$$m + 9t + $$\frac{1}{4}$$)
3/8 – 1/6 m + 5t + 7/10 m + 9t + 1/4.
3/8 + 1/4 – 1/6 m + 7/10 m + 5t + 9t.
3/8 + 1/4 + 3/2 m + 14 t.
1/3 + 3/2m + 14t.

KEY CONCEPT

Adding expressions may require combining like terms.
Terms with the same variables are added together and constants are added together.
When adding terms with the same variables, the rules for adding rational numbers apply to their coefficients.

(3.6 + 22.4t) + (2 + 18.9t) = 5.6 + 41.3t

Do You Understand?
Question 1.
Essential Question How can properties of operations be used to add expressions?

Answer:
The properties of operations be used to add expressions with the like terms.

Explanation:
In the above-given question,
given that,
Adding expressions may require combining like terms.
Terms with the same variables are added together and constants are added together.
When adding terms with the same variables, the rules for adding rational numbers apply to their coefficients.
(3.6 + 22.4t) + (2 + 18.9t).
5.6 + 41.3t.

Question 2.
Reasoning Explain whether the coefficients of two terms with different variables can be added to make one new term.

Answer:
Yes the coefficients of two terms with different variables can be added to make the new term.

Explanation:
In the above-given question,
given that,
Adding expressions may require combining like terms.
Terms with the same variables are added together and constants are added together.
When adding terms with the same variables, the rules for adding rational numbers apply to their coefficients.
(3.6 + 22.4t) + (2 + 18.9t).
5.6 + 41.3t.

Question 3.
Be Precise which properties of operations could be used to show that (-5p + 9) + (-2 + p) is equivalent to (-5p) + p + 9 – 2?

Answer:
The equivalent expression is -4p + 7.

Explanation:
In the above-given question,
given that,
the expression is (-5p + 9) + (-2 + p).
-5p + 9 -2 + p.
-4p + 7.

Do You Know How?
Question 4.
Dillon says that 4b and -2b are not like terms because 4b is positive and -2b is negative. Is he correct? Explain.

Answer:
No, they are like terms.

Explanation:
In the above-given question,
given that,
Dillon says that 4b and -2b are not like terms because 4b is positive and -2b is negative.
4b and 2b are same.
4b – 2b = 2b.
so they are like terms.

Question 5.
Joel spent $28 for an Internet data service and pays$14.50 per month. He spent $24.50 to join an online movie streaming site and pays$13.25 per month. Write an expression to represent Joel’s total cost for both memberships after m months.

Answer:
The expression to represent Joel’s total cost for both memberships after m months = $27.75m +$52.50.

Explanation:
In the above-given question,
given that,
Joel spent $28 for an Internet data service and pays$14.50 per month.
He spent $24.50 to join an online movie streaming site and pays$13.25 per month.
$14.50m +$13.25m + $28 +$24.50.
$27.75m +$52.50.

Question 6.
Add $$\frac{1}{3}$$n + $$\frac{2}{3}$$ and –$$\frac{1}{6}$$n + $$\frac{1}{6}$$m.

Answer:
-1/3n + 2/3 + 1/6m.

Explanation:
In the above-given question,
given that,
$$\frac{1}{3}$$n + $$\frac{2}{3}$$ and –$$\frac{1}{6}$$n + $$\frac{1}{6}$$m.
1/3 n + 2/3 – 1/6 n + 1/6m.
-1/3n + 2/3 + 1/6m.

Question 7.
Find the sum.
(-3.5t – 4s +4.5) + (-7.1 – 0.3s + 4.1t)

Answer:
2.6 + 0.6t – 4.3s.

Explanation:
In the above-given question,
given that,
(-3.5t – 4s +4.5) + (-7.1 – 0.3s + 4.1t).
-3.5t – 4s + 4.5 -7.1 – 0.3s + 4.1t.
0.6 t – 4.3s – 2.6.
2.6 + 0.6t – 4.3s.

Practice & Problem Solving

Leveled Practice For 8-9, fill in the boxes to add the expressions.
Question 8.
(2a + 8) + (4a + 5)
= (2a + ______) + (8 + _____)
= _______ + 13

Answer:
6a + 13.

Explanation:
In the above-given question,
given that,
the expression is (2a + 8) + (4a + 5).
(2a + 4a) + ( 8 + 5).
6a + 13.

Question 9.
($$\frac{2}{7}$$x – 7) + ($$\frac{1}{7}$$x + 8)
= (_______ + _______) + (-7 + ______)
= ______x + ______

Answer:
3/7 + 1.

Explanation:
In the above-given question,
given that,
($$\frac{2}{7}$$x – 7) + ($$\frac{1}{7}$$x + 8).
(2/7 + 1/7)x – 7 + 8.
3/7x + 1.

Question 10.
Find the sum.
(8b + 7) + (6x – 4) + (5c + 8)

Answer:
8b + 6x + 11.

Explanation:
In the above-given question,
given that,
(8b + 7) + (6x – 4) + (5c + 8).
8b + 7 + 6x – 4 + 5c + 8.
8b + 3 + 6x + 8.
8b + 6x + 11.

Question 11.
Combine like terms.
(-3y – 5) + (5m + 7y) + (6 + 9m)

Answer:
14m + 4y + 1.

Explanation:
In the above-given question,
given that,
(-3y – 5) + (5m + 7y) + (6 + 9m)
-3y – 5 + 5m + 7y + 6 + 9m.
4y + 1 + 14m.
14m + 4y + 1.

Question 12.
Felipe is going to plant b sunflower seeds in one garden and 5b + 10 sunflower seeds in another. How many seeds is Felipe going to plant altogether?

Answer:
The number of seeds Felipe going to plant altogether = 5bsquare + 10b.

Explanation:
In the above-given question,
given that,
Felipe is going to plant b sunflower seeds in one garden and 5b + 10 sunflower seeds in another.
(5b + 10) b.
5b x b + 10b.
so the number of seeds Felipe goind to plant altogether = 5bsquare + 10b.

Question 13.
An art class is making a mural for the school that has a triangle drawn in the middle. The length of the bottom of the triangle is x. Another side is 1 more than three times the length of the bottom of the triangle. The last side is 2 more than the bottom of the triangle. Write and simplify an expression for the perimeter of the triangle.

Answer:
The perimeter of the triangle =

Explanation:
In the above-given question,
given that,
An art class is making a mural for the school that has a triangle drawn in the middle.
The length of the bottom of the triangle is x.
Another side is 1 more than three times the length of the bottom of the triangle.
The last side is 2 more than the bottom of the triangle.
x + 3x + 2x.
p = x + 2x + 3x.
p = 6x.
so the perimeter of the triangle = 6x.

Question 14.
On a math test, Sarah has to identify all the coefficients and constants of the expression 4 + n + 7m. Sarah identifies the only coefficient as 7 and the only constant as 4.
a. Identify all the coefficients of the expression.

Answer:
The coefficients of the expression is 7.

Explanation:
In the above-given question,
given that,
On a math test, Sarah has to identify all the coefficients and constants of the expression 4 + n + 7m.
Sarah identifies the only coefficient as 7.
so the coefficient is 7.

b. Identify all the constants of the expression.

Answer:
The constants of the expression is n.

Explanation:
In the above-given question,
given that,
On a math test, Sarah has to identify all the coefficients and constants of the expression 4 + n + 7m.
Sarah identifies the only constant as n.
so the constant is n.

c. What error did Sarah likely make?

Answer:
Sarah make the mistake was 4 is the constant.

Explanation:
In the above-given question,
given that,
On a math test, Sarah has to identify all the coefficients and constants of the expression 4 + n + 7m.
Sarah identifies the only constant as 4.
so the constant is 4.

Question 15.
The width of a rectangle is 5x – 2.5 feet and the length is 2.5x + 8 feet. Find the perimeter of the rectangle.

Answer:
The perimeter of the rectangle = 6.25x – 20.

Explanation:
In the above-given question,
given that,
The width of a rectangle is 5x – 2.5 feet and the length is 2.5x + 8 feet.
(5x – 2.5) x (2.5x + 8).
6.25x – 20.
so the perimeter of the rectangle = 6.25x – 20.

Question 16.
Nina has x coins. Clayton has 5 fewer coins than six times the number of coins Nina has. Write an expression for the total number of coins Nina and Clayton have altogether. Then simplify the expression.

Answer:
The expression for the total number of coins Nina and Clayton have altogether = 6x  – 30.

Explanation:
In the above-given question,
given that,
Nina has x coins.
Clayton has 5 fewer coins than six times the number of coins Nina has.
x – 5 x 6.
6x – 30.
so the expression for the total number of coins Nina and Clayton have altogether = 6x – 30.

Question 17.
Higher Order Thinking Use the expression (8x + 2) + (-9x + 7).
a. Find the sum.

Answer:
9 – x.

Explanation:
In the above-given question,
given that
(8x + 2) + (-9x + 7).
8x + 2 -9x + 7.
-x + 9.
9 – x.

b. Reasoning Explain how you know when to combine terms with variables.

Answer:
We have to combine the variables when they have like terms.

Explanation:
In the above-given question,
given that
(8x + 2) + (-9x + 7).
8x + 2 -9x + 7.
-x + 9.
9 – x.

Question 18.
Gabe went to the Florida Mall. He bought k model planes and spent $24 on books. Then he spent another$25 at another store.

a. Write an expression that represents the amount Gabe spent at the mall.

Answer:
The expression that represents the amount Gabe = $49k +$14.99.

Explanation:
In the above-given question,
given that,
Gabe went to the Florida Mall.
He bought k model planes and spent $24k on books. Then he spent another$25k at another store.
each model panel cost $14.99.$24k + $25k +$14.99.
$49k +$14.99.

b. How much did Gabe spend in all if he bought 3 model planes?

Answer:
The amount did Gabe spend in all if he bought 3 models planes = $93.97. Explanation: In the above-given question, given that, Gabe went to the Florida Mall. He bought k model planes and spent$24k on books.
Then he spent another $25k at another store. each model panel cost$14.99.
3 x $14.99 = 93.97. so the amount did Gabe spend in all if he bought 3 models planes =$93.97.

Assessment Practice

Question 19.
A middle school with x students conducted a survey to determine students’ Tuesday afternoon activities.
PART A
Write an expression for each activity.
25 more than one-tenth of the students dance.

20 fewer than three-tenths of the students play soccer.

21 more than one-tenth of the students play baseball

Answer:
25 + 1/10, 20 – 3/10, and 21 + 1/10.

Explanation:
In the above-given question,
given that,
25 more than one-tenth of the students dance.
20 fewer than three-tenths of the students play soccer.
21 more than one-tenth of the students play baseball.
25 + 1/10, 20 – 3/10, and 21/10.

PART B
Write a simplified expression to represent the number of students who either dance or play baseball on Tuesday afternoons.

Answer:
The number of students who either dance or play baseball on  Tuesday afternoons = 25 + 1/10 and 21/10.

Explanation:
In the above-given question,
given that,
25 more than one-tenth of the students dance.
21 more than one-tenth of the students play baseball.
25 + 1/10 and 21/10.

### Lesson 4.7 Subtract Expressions

Explore It!
The East Side Bulldogs and the West Side Bears are playing a football game. A fan is keeping score using T for a touchdown plus extra point, worth 7 points total, and F for a field goal, worth 3 points.

I can… subtract expressions using properties of operations.

A. How can you represent the score of each team using expressions?

Answer:
The score of the each team using expressions = 10 points.

Explanation:
In the above-given question,
given that,
East Side Bulldogs and the West Side Bears are playing a football game.
fan is keeping score using T for a touchdown.
plus extra point, worth 7 points total, and F for a field goal, worth 3 points.
7 + 5 = 12.

B. How can you represent the difference of the teams’ scores using an expression?

Answer:
The difference of the teams scores using an expression is 4.

Explanation:
In the above-given question,
given that,
East Side Bulldogs and the West Side Bears are playing a football game.
fan is keeping score using T for a touchdown.
plus extra point, worth 7 points total, and F for a field goal, worth 3 points.
12 – 1 = 11.

C. How can you determine how many more points the winning team had than the losing team?

Answer:
The number of points the winning team had than the losing team = 1.

Explanation:
In the above-given question,
given that,
East Side Bulldogs and the West Side Bears are playing a football game.
fan is keeping score using T for a touchdown.
plus extra point, worth 7 points total, and F for a field goal, worth 3 points.
12 – 1 = 11.
so the number of points the winning team had than the losing team = 1.

Focus on math practices
Look for Relationships How can looking at the coefficients help you determine which team scored the greater number of points?

Essential Question
How can properties of operations be used to subtract expressions?

Try It!

A frame holds a picture that is 15 inches long and x inches wide. The frame border is 3 inches wide around the picture. What expression represents the area of the frame border?
Area of frame border = Area of entire frame – Area of photo = ________ – ________
The area of the frame is _________ in2

Answer:
The expression represents the area of the frame border =

Explanation:
In the above-given question,
given that,
A frame holds a picture that is 15 inches long and x inches wide.
The frame border is 3 inches wide around the picture.
3 = 15 – x.
x = 15/3.
x = 5.
so the expression represents the area of the frame border is 5 square inches.

Convince Me! Why can you choose to add or subtract when subtracting an expression?

Try It!

Subtract (0.95x – 0.04) – (0.99x – 0.13).

Answer:
0.04x + 0.09.

Explanation:
In the above-given question,
given that,
(0.95x – 0.04) – (0.99x – 0.13).
0.95x – 0.04 – 0.99x + 0.13.
0.04x + 0.09.
0.04x + 0.09.

Try It!

Subtract (17 + 4.5m + 8k) – (7.5m – 9 + 4k).

Answer:
4k – 3m + 26.

Explanation:
In the above-given question,
given that,
(17 + 4.5m + 8k) – (7.5m – 9 + 4k).
17 + 4.5m + 8k – 7.5m + 9 – 4k.
26 – 3m + 4k.
4k – 3m + 26.

KEY CONCEPT
To subtract expressions, you can use properties of operations.
Write the subtraction as addition and use the Distributive Property to multiply – 1 to the terms in the expression being subtracted.
5 – (2x – 7)
= 5 – (-2x – 7)
= 5 +(-1)(-2x – 7)
= 5+ (-1)(-2)x + (-1)(-7)
= 5 + 2x + 7
You can use the Distributive Property to distribute the minus sign to the second expression, which changes the signs of the terms.
5 – (-2x – 7)
= 5 + 2x + 7

Do You Understand?
Question 1.
Essential Question How can properties of operations be used to subtract expressions?

Answer:
The properties be used to subtract expressions are Distributive property.

Explanation:
In the above-given question,
given that,
5 – (2x – 7)
= 5 – (-2x – 7)
= 5 +(-1)(-2x – 7)
= 5+ (-1)(-2)x + (-1)(-7)
= 5 + 2x + 7.

Question 2.
Use Structure How is subtracting – 4x from 9x similar to subtracting -4 from 9?

Answer:
The difference is 5x.

Explanation:
In the above-given question,
given that,
-4x and 9x.
9x – 4x.
5x.
so the difference is 5x.

Question 3.
Is adding the quantity – 12 + 8r to an expression the same as subtracting -8r+ 12 from the same expression? Explain your reasoning.

Answer:
No, the expressions are not equal.

Explanation:
In the above-given question,
given that,
-12 + 8r.
8r – 12.
-8r + 12.
12 – 8r.
so both the expressions are not equal.

Do You Know How?
Question 4.
Subtract.
a. (21x) – (-16 + 7x)

Answer:
28x + 16.

Explanation:
In the above-given question,
given that,
subtract the expressions.
21x – (-16 + 7x).
21x + (-1) (-16 + 7x).
21x + (-1)(-16) + (-1)(-7x).
21x + 16 + 7x.
28x + 16.

b. (-13n) – (17 – 5n)

Answer:
18n – 17.

Explanation:
In the above-given question,
given that,
subtract the expressions.
(-13n) – (17 – 5n).
13n – 17 + 5n.
18n – 17.

c. (4y – 7) – (y – 7)

Answer:
3y.

Explanation:
In the above-given question,
given that,
subtract the expressions.
(4y – 7) – (y – 7).
4y – 7 -y + 7.
3y.

d. (-w + 0.4) – (-w – 0.4)

Answer:
0.8.

Explanation:
In the above-given question,
given that,
subtract the expressions.
(-w + 0.4) – (-w – 0.4).
-w + 0.4 + w + 0.4.
0.8.

Question 5.
Jude has 5 pairs of sunglasses that cost the same in his online shopping cart but then decides to get only 2. Each pair of sunglasses is the same price. Let p represent the cost of each pair. Write an expression for the original cost, the updated cost, and the difference in cost.

Answer:
The expression for the original cost, the updated cost, and the difference in cost = $2.02. Explanation: In the above-given question, given that, Jude has 5 pairs of sunglasses that cost the same in his online shopping cart but then decides to get only 2. Each pair of sunglasses is the same price. Let p represent the cost of each pair. 5 – 2 = 3p. 3 x$1.49 = 4.47.
$6.49 –$4.47 = $2.02. so the difference in cost =$2.02.

Question 6.
Subtract and simplify.
$$\frac{1}{6}$$m – (-$$\frac{5}{8}$$m + $$\frac{1}{3}$$)

Answer:
53m + 1/3.

Explanation:
In the above-given question,
given that,
$$\frac{1}{6}$$m – (-$$\frac{5}{8}$$m + $$\frac{1}{3}$$).
1/6m + 5/8m + 1/3.
6 x 8 = 48.
48 + 5 = 53.
53m + 1/3.

Practice & Problem Solving

Multimedia Leveled Practice In 7-9, fill in the missing signs or numbers.
Question 7.
Rewrite the expression 14m – (5 + 8m)
14m 5 8m

Answer:
The missing signs are minus.

Explanation:
In the above-given question,
given that,
14m – (5 + 8m).
14m – 5 – 8m.
6m – 5.

Question 8.
Rewrite the expression 13d – (-9d – 4) without parentheses. without parentheses.
13d 9d 4

Answer:
The expression is 22d + 4.

Explanation:
In the above-given question,
given that,
the expression is 13d – (-9d – 4).
13d + 9d + 4.
22d + 4.
so the expression without parentheses is 22d + 4.

Question 9.
Write an equivalent expression to 8k – (5 + 2k) without parentheses. Then simplify.
8k – (5 + 2k) = 8k 5 2k
= 8k 2k 5
= k 5

Answer:
The equivalent expression is 6k – 5.

Explanation:
In the above-given question,
given that,
8k – (5 + 2k).
8k – 5 – 2k.
6k – 5.

Question 10.
A company has two manufacturing plants with daily production levels of 5x + 11 items and 2x – 3 items, respectively, where x represents a minimum quantity. The first plant produces how many more items daily than the second plant?

Answer:
The first plant produces 3 more items daily than the second plant.

Explanation:
In the above-given question,
given that,
A company has two manufacturing plants with daily production levels of 5x + 11 items and 2x – 3 items.
5x + 11 – 2x – 3.
3x + 8.
so the first plant produces 3 more items daily than the second plant.

Question 11.
Two communications companies offer calling plans. With Company X, it costs 35¢ to connect and then 5¢ for each minute. With Company Y, it costs 15¢ to connect and then 4¢ for each minute.
Write and simplify an expression that represents how much more Company X charges than Company Y, in cents, for n minutes.

Answer:
The expression that represents company X charges than company Y, in cents, for n minutes =

Explanation:
In the above-given question,
given that,
Two communications companies offer calling plans. With Company X, it costs 35¢ to connect and then 5¢ for each minute.
With Company Y, it costs 15¢ to connect and then 4¢ for each minute.
35 + 5 = 40¢.
15 + 4 = 19¢.
40 – 19 = 21¢.
so the expression that represents company x charges than company y, in cents, for n minutes = 21¢.

Question 12.
Make Sense and Persevere The base and height of a triangle are each extended 2 cm. What is the area of the shaded region? How do you know?

Answer:
The area of the shaded region = x cm.

Explanation:
In the above-given question,
given that,
The base and height of a triangle are each extended 2 cm.
area = 1/2 x b x h.
area = 1/2 x 2 x x.
area = 2/2x.
area = x cm.
so the area of the shaded region is x cm.

Question 13.
Two friends shop for fresh fruit. Jackson buys a watermelon for $7.65 and 5 pounds of cherries. Tim buys a pineapple for$2.45 and 4 pounds of cherries. Use the variable p to represent the price, in dollars, per pound of cherries. Write and simplify an expression to represent how much more Jackson spent.

Answer:
The expression to represent how much more jackson spent = $6.2p. Explanation: In the above-given question, given that, Two friends shop for fresh fruit. Jackson buys a watermelon for$7.65 and 5 pounds of cherries.
Tim buys a pineapple for $2.45 and 4 pounds of cherries.$2.45 + 4 = $6.45.$7.65 + 5 = $12.65.$12.65 – $6.45 =$6.2.
so the expression to represent how much more jacson spent = $6.2p. Question 14. Yu’s family wants to rent a car to go on vacation. Envocar charges$50.50 and 8¢ per mile. Freedomride charges $70.50 and 12¢ per mile. How much more does Freedomride charge for driving d miles than EnvoCar? Answer: The much does Freedomride charge for driving d miles than Envo Car =$20 and 4¢ .

Explanation:
In the above-given question,
given that,
Yu’s family wants to rent a car to go on vacation.
Envocar charges $50.50 and 8¢ per mile. Freedomride charges$70.50 and 12¢ per mile.
$50.50 and 8¢ .$70.50 and 12¢ .
70.50 – 50.50 and 12 – 8.
$20 and 4¢ . Question 15. A rectangular garden has a walkway around it. Find the area of the walkway. Answer: The area of the walkway = 168x + 136.5 sq ft. Explanation: In the above-given question, given that, the area of the walkway = l x b. area = (8x + 6.5 ft ) x 21 ft. area = 168x + 136.5 square ft. Question 16. Critique Reasoning Tim incorrectly rewrote the expression $$\frac{1}{2}$$p – ($$\frac{1}{4}$$p + 4) as $$\frac{1}{2}$$p + $$\frac{1}{4}$$p – 4. Rewrite the expression without parentheses. What was Tim’s error? Answer: The Tim’s error = 1/2p – 4. Explanation: In the above-given question, given that, Tim incorrectly rewrote the expression $$\frac{1}{2}$$p – ($$\frac{1}{4}$$p + 4). $$\frac{1}{2}$$p + $$\frac{1}{4}$$p – 4. 1/2p – 1/4p – 4. 1/2p – 4. Question 17. Higher Order Thinking Find the difference. (7x – 6$$\frac{2}{3}$$) – (-3x +4$$\frac{3}{4}$$) Answer: The difference is 10x – 7. Explanation: In the above-given question, given that, (7x – 6$$\frac{2}{3}$$) – (-3x +4$$\frac{3}{4}$$) 7x – 6 x 2/3 – (-3x + 4 (3/4). 7x – 6 (2/3) + 3x – 4 (-3/4). 10x -4 – 3. 10x – 7. Question 18. Each month, a shopkeeper spends 5x + 11 dollars on rent and electricity. If he spends 2x – 3 dollars on rent, how much does he spend on electricity? Answer: The much he spend on electricity = 3x + 8. Explanation: In the above-given question, given that, Each month, a shopkeeper spends 5x + 11 dollars on rent and electricity. If he spends 2x – 3 dollars on rent. 5x + 11 – 2x – 3. 3x + 8. Question 19. Use the expression $$\frac{1}{4}$$p – (1 – $$\frac{1}{3}$$p). a. Rewrite the expression without parentheses. Simplify. Show your work. Answer: The expression is 1/7p – 1. Explanation: In the above-given question, given that, $$\frac{1}{4}$$p – (1 – $$\frac{1}{3}$$p). 1/4 p – (1 – 1/3p). 1/4p – 1 + 1/3p. 1/7p – 1. b. Use a different method to write the expression without parentheses. Do not simplify. Answer: The expression is 1/7p – 1. Explanation: In the above-given question, given that, $$\frac{1}{4}$$p – (1 – $$\frac{1}{3}$$p). 1/4 p – (1 – 1/3p). 1/4p – 1 + 1/3p. 1/7p – 1. Assessment Practice Question 20. An expression is shown. (0.25n – 0.3) – (0.8n – 0.25) Create an equivalent expression without parentheses. Answer: The equivalent expression without parantheses = 0.05 – 0.55n. Explanation: In the above-given question, given that, (0.25n – 0.3) – (0.8n – 0.25). 0.25n – 0.3 – 0.8n + 0.25. -0.55n + 0.05. ### Lesson 4.8 Analyze Equivalent Expressions Solve & Discuss It! How many toothpicks make a triangle? Two triangles? Write an expression that represents the number of toothpicks needed to make x triangles that appear side-by-side in a single row, as shown. Explain your reasoning. I can… use an equivalent expression to find new information. Look for Relationships What do you notice about the number of toothpicks needed for more than 1 triangle? Focus on math practices Reasoning Can there be more than one expression that represents the total number of toothpicks needed to make x triangles in the arrangement shown? Explain. Essential Question How can writing equivalent expressions show how quantities are related? Try It! Joe is buying gift cards that are on sale for 15% off. He uses c – 0.15c to determine the sale price of gift cards. What is an equivalent expression that Joe could also use to determine the sale price of a gift card? Answer: The equivalent expression that Joe could use to determine the sale price = 0.15 – 0.15c. Explanation: In the above-given question, given that, Joe is buying gift cards that are on sale for 15% off. He uses c – 0.15c to determine the sale price of gift cards. 15/100 – 0.15c. 0.15 – 0.15c. Convince Me! How do you know if an expression is describing a percent increase or a percent decrease? Try It! The total area, in square feet, of a rectangular stage that has been widened by x feet is represented by 1,900 + 76x. Use the Distributive Property to factor the expression. What does each factor in the equivalent expression tell you about the stage? Answer: The equivalent expression about the stage = 1900x + 76 x square. Explanation: In the above-given question, given that, The total area, in square feet, of a rectangular stage that has been widened by x feet is represented by 1,900 + 76x. 1900 + 76x x x. 1900x + 76xsquare. KEY CONCEPT Rewriting expressions can clarify relationships among quantities or variables. When you rewrite an expression, you are writing an equivalent expression. 4x + 12 is equivalent to 4(x + 3) is equivalent to x + x + x + x + 3 + 3 + 3 + 3 Do You Understand? Question 1. Essential Question How can writing equivalent expressions show how quantities are related? Answer: The equivalent expressions are 4x + 12. Explanation: In the above-given question, given that, 4x + 12. 4(x + 3). x + x + x + x + 3 + 3 + 3 + 3. Question 2. Use Structure The total area, in square feet, of a rectangular mural that has been extended by x feet is represented by 5.5(7.5 + x). Expand the expression using the Distributive Property. What do each of the terms in the equivalent expression tell you about the mural? Answer: The expressions tell you about the mural = 41.25 + 5.5x. Explanation: In the above-given question, given that, the expression is 5.5(7.5 + x). 5.5 x 7.5 + 5.5 x x. 41.25 + 5.5x. so the equivalent expression is 41.25 + 5.5x. Question 3. The expression (2x + 6) + x represents the perimeter of an isosceles triangle. If x represents the length of one side of the triangle, explain how you can use the Distributive Property to find the length of each of the two equivalent sides? Answer: The Distributive property to find the length of each of the two equivalent sides = 2x square + 6x. Explanation: In the above-given question, given that, The expression (2x + 6) + x represents the perimeter of an isosceles triangle. (2x + 6) + x. 2x X x + 6x. so the distributive property to find the length of each of the two equivalent sides = 2x x x + 6x. Do You Know How? Question 4. Rewrite the expression 12x + 8 to find an equivalent expression. Show three possible expressions. What do the rewritten expressions tell you about the relationships among the quantities? Answer: The expression are 2(6x + 4) and 4(3x + 2). Explanation: In the above-given question, given that, Rewrite the expression 12x + 8 to find an equivalent expression. 2 (6x + 4). 4 (3x + 2). Question 5. A rope is used to make a fence in the shape of an equilateral triangle around a newly planted tree. The length of the rope is represented with the expression 9x + 15 a. Rewrite the expression to represent the three side lengths of the rope fence. Answer: The length of the rope is represented with the expression = Explanation: In the above-given question, given that, 9x + 15. 3(3x + 5). 3 x 3x = 9x. 3 x 5 = 15. so the length of the rope is 3(3x + 5). b. What is the length of one side? Answer: The length of one side = 3x. Explanation: In the above-given question, given that, 9x + 15. 3(3x + 5). so the length of one side = 3x. Question 6. The expression (x – 0.35x) represents 35% off the cost of an item x. How is this equivalent to multiplying x by 0.65? Answer: The expression equivalent to multiplying = 0.4225. Explanation: In the above-given question, given that, The expression (x – 0.35x) represents 35% off the cost of an item x. x = 0.65. 0.65 – 0.35(0.65). 0.65 – 0.2275. 0.4225. Practice & Problem Solving Question 7. Reasoning Eric is planning an event at a hotel. Let g stand for the number of Eric’s guests. The two expressions represent the difference between the cost of the rooms. Expression 1: (326 + 37g) – (287 + 23g). Expression 2: 39 + 14g. What can you tell about Expression 2 and Expression 1? Answer: The two expressions are same. Explanation: In the above-given question, given that, Eric is planning an event at a hotel. Let g stand for the number of Eric’s guests. The two expressions represent the difference between the cost of the rooms. Expression 1: (326 + 37g) – (287 + 23g). Expression 2: 39 + 14g. 326 + 37g – 287 – 23g. 39 + 14g. so the expressions 1 and 2 are same. Question 8. A student received a coupon for 17% off the total purchase price at a clothing store. Let b be the original price of the purchase. Use the expression b-0.17b for the new price of the purchase. Write an equivalent expression by combining like terms. Answer: The equivalent expression is 0.16b. Explanation: In the above-given question, given that, A student received a coupon for 17% off the total purchase price at a clothing store. Let b be the original price of the purchase. Use the expression b-0.17b for the new price of the purchase. 0.17b – b. 0.16b. so the equivaent expression is 0.16b. Question 9. Kirana buys boxes of crackers that each have the same cost, c. She represents the cost of 3 boxes of cheese crackers, 2 boxes of poppy seed crackers, and 2 boxes of plain crackers using the expression 3c + 2c + 2c. What equivalent expression can represent the cost? Answer: The equivalent epression that represent the cost = 7c. Explanation: In the above-given question, given that, Kirana buys boxes of crackers that each have the same cost c. She represents the cost of 3 boxes of cheese crackers. 2 boxes of poppy seed crackers, and 2 boxes of plain crackers using the expression 3c + 2c + 2c. 3c + 2c + 2c. 5c + 2c. 7c. so the equivalent expression that represent the cost = 7c. Question 10. A student received a coupon for 14% off the total purchase price at a clothing store. Let c be the original price of the purchase. The expression c – 0.14c represents the new price of the purchase. Write an equivalent expression to show another way to represent the new price. Answer: The equivalent expression to show another way to represent the new price = 0.13c. Explanation: In the above-given question, given that, A student received a coupon for 14% off the total purchase price at a clothing store. Let c be the original price of the purchase. The expression c – 0.14c represents the new price of the purchase. c – 0.14c. 0.13c. so the equivalent expression to show another way to represent the new price = 0.13c. Question 11. A farmer recently sold a large plot of land. The sale decreased his total acreage by 8%. Let v be the original acreage. a. Find two equivalent expressions that will give the new acreage. Answer: The two equivalent expressions that will give the new acreage = v – 0.08v and 0.07v. Explanation: In the above-given question, given that, A farmer recently sold a large plot of land. The sale decreased his total acreage by 8%. v – 0.08v. 0.07v. b. Use the expressions to describe two ways to find the new acreage. Answer: The two equivalent expressions that will give the new acreage = v – 0.08v and 0.07v. Explanation: In the above-given question, given that, A farmer recently sold a large plot of land. The sale decreased his total acreage by 8%. v – 0.08v. 0.07v. Question 12. An art teacher enlarged the area of a copy of a painting by 49%. Let d represent the area of the original painting. The expression d + 0.49d is one way to represent the area of the new painting. Write two additional expressions that will give the area of the new painting. Answer: The two additional expressions that will give the area of the new painting = d + 0.49d and 0.50d. Explanation: In the above-given question, given that, An art teacher enlarged the area of a copy of a painting by 49%. Let d represent the area of the original painting. The expression d + 0.49d is one way to represent the area of the new painting. so the expressions are d nd d + 0.49d. 0.50d. Question 13. Use Structure The area of a rectangular playground has been extended on one side. The total area of the playground, in square meters, can be written as 352 + 22x. Rewrite the expression to give a possible set of dimensions for the playground. Answer: The expression to give a possible set of dimensions for the playground = 22(16 + x). Explanation: In the above-given question, given that, The area of a rectangular playground has been extended on one side. The total area of the playground, in square meters, can be written as 352 + 22x. 352 + 22x. 22(16 + x). 22 x 16 = 352. 22 x x = 22x. Question 14. The manager of a store increases the price of the bathing suits by 7%. Let t be the original price of a bathing suit. The new price is t + 0.07t. a. Find an expression equivalent to t + 0.07t. Answer: The expression equivalent to t + 0.07t = 0.08t. Explanation: In the above-given question, given that, The manager of a store increases the price of the bathing suits by 7%. Let t be the original price of a bathing suit. The new price is t + 0.07t. 0.08t. b. If the original price of a bathing suit was$19.99, estimate the new price by first rounding the original price to the nearest dollar.

Answer:
The original price to the nearest dollar = $20 +$1.4t.

Explanation:
In the above-given question,
given that,
If the original price of a bathing suit was $19.99. 19.99 = 20. 20 + 0.07 x 20. 20 + 1.4t. Question 15. Higher Order Thinking A customer at a clothing store is buying a pair of pants and a shirt. The customer can choose between a sale that offers a discount on pants, or a coupon for a discount on the entire purchase. Let n represent the original price of the pants and s represent the price of the shirt a. Write two expressions that represent the “15% off sale on all pants” option. Answer: The expressions that represents is n – 0.015. Explanation: In the above-given question, given that, A customer at a clothing store is buying a pair of pants and a shirt. The customer can choose between a sale that offers a discount on pants, or a coupon for a discount on the entire purchase. n – 0.015. b. Write two expressions that represent the “10% off her entire purchase” option. Answer: The expressions that represents is n – 0.010. Explanation: In the above-given question, given that, A customer at a clothing store is buying a pair of pants and a shirt. The customer can choose between a sale that offers a discount on pants, or a coupon for a discount on the entire purchase. n – 0.010. c. If the original cost of the pants is$25 and the shirt is $10, which option should the customer choose? Explain. Answer: The customer choose the both pants and shirts =$35.

Explanation:
In the above-given question,
given that,
If the original cost of the pants is $25 and the shirt is$10.
$25 +$10.
$35. so the customer choose the both pants and shirts =$35.

Assessment Practice

Question 16.
At a college, the cost of tuition increased by 10%. Let b represent the former cost of tuition. Use the expression b + 0.10b for the new cost of tuition.
PART A
Write an equivalent expression for the new cost of tuition.

Answer:
The new cost of tuition is 0.11b.

Explanation:
In the above-given question,
given that,
At a college, the cost of tuition increased by 10%.
Let b represent the former cost of tuition.
b + 0.10b.
0.11b.
so the new cost of tution is 0.11b.

PART B
What does your equivalent expression tell you about how to find the new cost of tuition?
Answer:

250
4-8 Analyze Equivalent Expressions

### Topic 4 Review

Topic Essential Question
How can properties of operations help to generate equivalent expressions that can be used in solving problems?

Vocabulary Review
Complete each definition and then provide an example of each vocabulary word.

Vocabulary
coefficient
constant
variable
factor
expression

Answer:
A term that contains only a number is constant.
The number part of the term that contains a variable is coefficient.
A variable is a letter that represents an unknown value.

Explanation:
In the above-given question,
given that,
A term that contains only a number is constant.
The number part of the term that contains a variable is coefficient.
A variable is a letter that represents an unknown value.
for example:
2x + 4y – 9.
where 2 and 4 are coefficients.
x and y are variables.
9 is the constant.

Use Vocabulary in Writing
Membership in a digital library has a $5 startup fee and then costs$9.95 per month. Membership in a video streaming service costs $7.99 per month with no startup fee. Use vocabulary words to explain how this information could be used to write an expression for the total cost of both memberships after m months. Answer: The total cost of memberships after m months =$49.75 + $7.99m. Explanation: In the above-given question, given that, Membership in a digital library has a$5 startup fee and then costs $9.95 per month. Membership in a video streaming service costs$7.99 per month with no startup fee.
5 x $9.95 +$7.99m.
$49.75 +$7.99m.
so the total cost of memberships after m months = $49.75 +$7.99m.

Concepts and Skills Review

Lesson 4.1 Write and Evaluate Algebraic Expressions

Quick Review
You can use an algebraic expression to represent and solve a problem with unknown values. The expression can consist of coefficients, constants, and variables. You can substitute values for variables to evaluate expressions.

Example
A farm charges $1.75 for each pound of strawberries picked and$2 for a basket to hold the strawberries. What is the total cost to pick 5 pounds of strawberries?

Answer:
The total cost to pick 5 pounds of strawberries = $10.75. Explanation: In the above-given question, given that, A farm charges$1.75 for each pound of strawberries picked and $2 for a basket to hold the strawberries. ($1.75 x $5) + 2. ($8.75) + 2.
$10.75. Write an expression to represent the total cost in dollars to pick p pounds of strawberries. 1.75p + 2 Substitute 5 for p. 1.75(5) + 2 = 8.75 + 2 = 10.75 It costs$10.75 to pick 5 pounds of strawberries.

Practice
Question 1.
Haddie makes and sells knit scarves. Next week she will pay a $25 fee for the use of a booth at a craft fair. She will charge$12 for each scarf she sells at the fair. Write an expression to determine Haddie’s profit for selling s scarves after paying the fee for the use of the booth.

Answer:
The expression to determine Haddie’s profit for selling scarves = $37. Explanation: In the above-given question, given that, Haddie makes and sells knit scarves. Next week she will pay a$25 fee for the use of a booth at a craft fair.
She will charge $12 for each scarf she sells at the fair.$25 + $12.$37.
so the expression to determine Haddie’s profit for selling scarves = $37. Question 2. The cost to buy p pounds of potatoes at$0.32 per pound and n pounds of onions at $0.48 per pound can be determined by using the expression 0.32p + 0.48n. How much will it cost to buy 4.5 pounds of potatoes and 2.5 pounds of onions? Answer: The cost to buy 4.5 pounds of potatoes and 2.5 pounds of onions = 0.4608 + 1.2n. Explanation: In the above-given question, given that, The cost to buy p pounds of potatoes at$0.32 per pound and
n pounds of onions at $0.48 per pound can be determined by using the expression 0.32p + 0.48n. 0.32 x 4.5p = 0.4608p. 0.48 x 2.5n = 1.2. 0.4608 + 1.2n. Lessons 4-2 AND 4-3 Generate Equivalent Expressions and Simplify Expressions Quick Review You can use properties of operations and combine like terms to simplify expressions. Like terms are terms that have the same variable part. Example Simplify the expression below. -7 + $$\frac{1}{3}$$n – $$\frac{4}{3}$$ + 2n Use the Commutative Property to put like terms together, $$\frac{1}{3}$$n + 2n – 7 – $$\frac{4}{3}$$ Combine like terms. 2$$\frac{1}{3}$$n – 8$$\frac{1}{3}$$ Practice Simplify each expression below. Question 1. $$\frac{5}{8}$$m + 9 – $$\frac{3}{8}$$m – 15 Answer: 3/8 – 6. Explanation: In the above-given question, given that, $$\frac{5}{8}$$m + 9 – $$\frac{3}{8}$$m – 15. Use the Commutative Property to put like terms together, 5/8 – 3/8 + 9 – 15. combine like terms. 3/8 – 6. Question 2. -8w + (-4z) + 2 + 6w + 9z – 7 Answer: 5z – 2w – 5. Explanation: In the above-given question, given that, -8w + (-4z) + 2 + 6w + 9z – 7. Use the Commutative Property to put like terms together, 6w – 8w -4z + 9z -7 + 2. combine like terms. -2w + 5z -5. 5z – 2w – 5. Question 3. -6 + (-2d) + (-4d) + 3d Answer: -3(d + 2). Explanation: In the above-given question, given that, -6 + (-2d) + (-4d) + 3d. Use the Commutative Property to put like terms together, -6 – 2d – 4d + 3d. combine like terms. 3d – 6d -6. -3d – 6. -3(d + 2). Lesson 4.4 Expand Expressions Quick Review The Distributive Property allows you to multiply each term inside parentheses by a factor that is outside the parentheses. This means that you can use the Distributive Property to expand expressions. Example Expand the expression (6 + 7). ($$\frac{1}{4}$$ × h) + ($$\frac{1}{4}$$ × 7) = $$\frac{1}{4}$$h + 1.75 Practice Question 1. Expand the expression 3.5(-3n + 4). Answer: The expression is -10.5n + 14. Explanation: In the above-given question, given that, the expression is 3.5(-3n + 4). 3.5 x -3n + 3.5 x 4. -10.5n + 14. so the expression is -10.5n + 14. Question 2. Simplify the expression –$$\frac{3}{5}$$(-8 + $$\frac{5}{9}$$x – 3). Answer: 5/9 + 24/5 – 3. Explanation: In the above-given question, given that, the expression –$$\frac{3}{5}$$(-8 + $$\frac{5}{9}$$x – 3). -3/5 -8 + 5/9 x – 3. 24/5 + 5/9x – 3. 5/9x + 24/5 – 3. Lesson 4.5 Factor Expressions Quick Review When you factor an expression, you write it as a product of two expressions. The new expression is equivalent to the original expression. The greatest common factor (GCF) and the Distributive Property are tools that you use when you need to factor an expression. Example Factor the expression 12x – 9y + 15. The GCF of 12x, 15, and -9y is 3. Rewrite each term using 3 as a factor. 12x = 3 • 4x -9y = 3 • (-3y) 15= 3 • 5 Use the Distributive Property to factor the expression. 3(4x – 3y + 5) Practice Factor each expression. Question 1. 63a – 42b Answer: 3(21a – 14b). Explanation: In the above-given question, given that, the expression is 63a – 42b. 3(21a – 14b). 3 x 21a = 63a. 3 x 14b = 42b. Question 2. 81y + 54 Answer: 9(9y + 6). Explanation: In the above-given question, given that, the expression is 81y + 54. 9(9y + 6). 9 x 9y = 81y. 9 x 6 = 54. Question 3. Which show a way to factor the expression 32t – 48? Select all that apply. ☐ 2(16t – 24) ☐ 4(12t – 48) ☐ 6(26 – 42) ☐ 8(4t – 6) ☐ 16(2t – 3) Answer: 2(16t – 24) and 8(4t – 6). Explanation: In the above-given question, given that, the expression is 32t – 48. 2(16t – 24). 2 x 16t = 32t. 2 x 24 = 48. 8(4t – 6). 8 x 4t = 32t. 8 x 6 = 48. Lessons 4.6 AND 4.7 Add and Subtract Expressions Quick Review Adding and subtracting expressions may require combining like terms. This means that you must use the Commutative and Associative Properties to reorder and group terms as needed. Example Kerry has n markers. Rachel has 1 marker fewer than twice the number of markers Kerry has. Write and simplify an expression for the total number of markers they have. Number of markers Kerry has: n Number of markers Rachel has: 2n – 1 Total number of markers: n + (2n – 1) (n + 2n) – 1 3n – 1 Practice Add the expressions. Question 1. 5.2C – 7.35) + (-3.9C + 2.65) Answer: 1.3c – 4.7. Explanation: In the above-given question, given that, 5.2C – 7.35) + (-3.9C + 2.65). 5.2c – 7.35 – 3.9c + 2.65. 5.2c – 3.9c + 2.65 – 7.35. 1.3c – 4.7. Question 2. (6x – 2y – 5) – (-5 + 9y – 8x) Answer: 14x – 11y. Explanation: In the above-given question, given that, (6x – 2y – 5) – (-5 + 9y – 8x). 6x – 2y – 5 + 5 – 9y + 8x. 14x – 11y. Question 3. Last week Jean ran 2 fewer than 4m miles. This week she ran 0.5 miles more than last week. Write and simplify an expression for the total number of miles Jean ran in the two weeks. Answer: The total number of miles Jean ran in the two weeks = 2.5m. Explanation: In the above-given question, given that, Last week Jean ran 2 fewer than 4m miles. This week she ran 0.5 miles more than last week. 2m + 0.5m. 2.5m. so the total number of miles Jean ran in the two weeks = 2.5m. Lesson 4.8 Analyze Equivalent Expressions Quick Review Equivalent expressions can help to show new information about a problem. Sometimes the equivalent expression will be an expanded expression. In other cases, it will be a factored expression. Example The perimeter of a square is represented with the expression 84 + 44s. What is the length of one side of the square? A square has 4 sides, so factor 4 out of each term in the expression for the perimeter. 84 + 445 = 4 • 21 + 4 • 11s = 4(21 + 115) The factor within the parentheses represents the length of one side of the square. The length of one side is 21 + 11s. Practice Question 1. Hal earns n dollars per hour. Next month he will receive a 2% raise in pay per hour. The expression n + 0.02n is one way to represent Hal’s pay per hour after the raise. Write an equivalent simplified expression that will represent his pay per hour after the raise. Answer: The equivalent expression that will represent his pay per hour after the raise = 0.03n. Explanation: In the above-given question, given that, Hal earns n dollars per hour. Next month he will receive a 2% raise in pay per hour. The expression n + 0.02n is one way to represent Hal’s pay per hour after the raise. n + 0.02n = 0.03n. Question 2. The area of a garden plot can be represented by the expression 84z – 54. The garden will be divided into six sections for planting six different vegetables. The sections will be equal in area. Write an expression that represents the area of each section. Answer: The expression that represents the area of each section = Explanati ### Topic 4 Fluency Practice Hidden Clue For each ordered pair, solve the percent problems to find the coordinates. Then locate and label the corresponding point on the graph. Draw line segments to connect the points in alphabetical order. Use the completed picture to help you answer the riddle below. I can… represent and solve percent problems. #### enVision Math Common Core Grade 7 Answer Key ## Envision Math Common Core Grade 7 Answer Key Topic 7 Probability ## Envision Math Common Core 7th Grade Answers Key Topic 7 Probability GET READY! Review What You Know! Vocabulary Choose the best term from the box to complete each definition. Question 1. A(n) _________ is a drawing that can be used to visually represent information. Answer: A(n) diagram is a drawing that can be used to visually represent information. Explanation: In the above-given question, given that, A(n) diagram is a drawing that can be used to visually represent information. for example: a diagram is a visual representation of information. diagrams can be both two-dimensional and three-dimensional. some of the most common types of diagrams are flowcharts. Question 2. The number of times a specific value occurs is referred to as _________. Answer: The number of times a specific value occurs is referred to as frequency. Explanation: In the above-given question, given that, the number of times a specific value occurs is referred to as frequency. for example: 3, 3, 5, 5, 5, 6, 7, 7, 7, 7, and 7. from the above data set – 3 occurs 2 times, then 2 is the frequency of 3. 5 occurs 3 times, then 3 is the frequency of 5. 6 occurs 1 time, then 1 is the frequency of 6. Question 3. A(n) _________ is a relationship between one quantity and another quantity. Answer: A(n) ratio is a relationship between one quantity and another quantity. Explanation: In the above-given question, given that, A(n) ratio is a relationship between one quantity and another quantity. for example: there are 3 boys and 6 girls in a class. therefore the ratio is 1: 2. Question 4. Quantities that have the same value are Answer: Quantities that have the same value are equivalent ratios. Explanation: In the above-given question, given that, Quantities that have the same value are equivalent ratios. for example: the cost of the order is 8. pizzas ordered is 1. 8/1 = 16/2 = 24/3 = 32/4 = 40/5 = 8. so$8 per pizza.

Operations with Fractions
Solve for x.

Question 5.
$$\frac{2}{5}$$ + x = 1

Answer:
2/5 + x = 1 is 0.6.

Explanation:
In the above-given question,
given that,
$$\frac{2}{5}$$ + x = 1.
2/5 + x = 1.
0.4 + x = 1.
x = 1 – 0.4.
x = 0.6.
so the value of x is 0.6.

Question 6.
225 • $$\frac{1}{3}$$= x

Answer:
225 (1/3) = x is 75.

Explanation:
In the above-given question,
given that,
225 • $$\frac{1}{3}$$= x
225(1/3) = x.
75 = x.
so the value of x is 75.

Question 7.
1 = $$\frac{1}{8}$$ + x + $$\frac{2}{8}$$

Answer:
1 = 1/8 + x + 2/8 is 0.625.

Explanation:
In the above-given question,
given that,
1 = $$\frac{1}{8}$$ + x + $$\frac{2}{8}$$.
1 = 0.125 + x + 0.25.
1 – 0.125 = x + 0.25.
0.875 – 0.25 = x.
x = 0.625.
so the value of x is 0.625.

Ratios
Write each ratio in fraction form. Then write the percent equivalent.

Question 8.
72 out of 96

Answer:
72/96 = 0.75.

Explanation:
In the above-given question,
given that,
72 out of 96.
72/96.
0.75.

Question 9.
88 out of 132

Answer:
88/132 = 0.66.

Explanation:
In the above-given question,
given that,
88 out of 132.
88/132.
0.66.

Question 10.
39 out of 104

Answer:
39/104 = 0.375.

Explanation:
In the above-given question,
given that,
39 out of 104.
39/104.
0.375.

Question 11.
23 out of 69

Answer:
23/69 = 0.33.

Explanation:
In the above-given question,
given that,
23 out of 69.
23/69.
0.33.

Question 12.
52 out of 208

Answer:
52/208 = 0.25.

Explanation:
In the above-given question,
given that,
52 out of 208.
52/208.
0.25.

Question 13.
25 out of 200

Answer:
25/200 = 0.125.

Explanation:
In the above-given question,
given that,
25 out of 200.
25/200.
0.125.

Order Fractions and Decimals
Plot the following fractions and decimals on the number line.

Answer:
The fractions and decimals are 0.7, 1/3, 7/8, 0.4, 0.125, and 5/6.

Explanation:
In the above-given question,
given that,
the fractions and decimals are 0.7, 1/3, 7/8, 0.4, 0.125, and 5/6.
1/3 = 0.3.
7/8 = 0.875.
5/6 = 0.83.
1/4 = 0.25.
1/2 = 0.5.
3/4 = 0.75.
0.7 lies in between 3/4 and 1.
0.3 lies in between 1/4 and 1/2.
7/8 lies in between 3/4 and 1.
0.4 lies in between 1/4 and 1/2.
5/6 lies in between 3/4 and 1.
0.125 lies in between 0 and 1/4.

Language Development
Sort the vocabulary words into categories. Explain your categories.

Category: Probability.

Explanation:
In the above-given question,
given that,
probability of a given event is an expression of the likelihood of occurrence of an event.
a probability is a number that ranges from 0 to 1.
there are so many types of probabilities are there.
they are theoretical probability and experimental probability.
for example:
0 for an event that cannot occur.
1 for an event that can occur.

Category: Event.

Explanation:
In the above-given question,
given that,
an event is a subject of the sample space.
that is any collection of outcomes from an event.
events will be denoted b capital letters.

Category: Frequency.

Explanation:
In the above-given question,
given that,
frequency is the rate at which something occurs over a particular period of time or in a given sample.
sample space and relative frequency come under this.
for example:
3, 3, 5, 5, 5, 6, 7, 7, 7, 7, and 7.
from the above data set – 3 occurs 2 times, then 2 is the frequency of 3.
5 occurs 3 times, then 3 is the frequency of 5.
6 occurs 1 time, then 1 is the frequency of 6.

Category: Simulation.

Pick A Project

PROJECT 7A
What makes a carnival game fun and successful?

PROJECT: DEVELOP A GAME OF CHANCE

PROJECT 7B
If you could invent a character for an adventure, what would that character be like?
Project: Design An Adventure

PROJECT 7C
What is the silliest sentence you can think of? Why is it silly?
Project: Generate A Funny Sentence

PROJECT 7D
How could you teach a math concept through a performance?
Project: Perform Your Knowledge

### Lesson 7.1 Understand Likelihood and Probability

Solve & Discuss It!
For a game show, Jared has to choose 1 of 8 boxes to win a prize. One of the boxes has a big prize, 3 boxes have a medium prize, 3 boxes have smaller prizes, and 1 box is empty. How confident should Jared be that whatever box he chooses, he will win a prize? Support your response with a mathematical argument.
I can… describe the likelihood that an event will occur.

Make Sense and Persevere
What are the chances that Jared will choose a box with a prize?

Answer:
The three chances are 3/8, 1/8, and 3/8.

Explanation:
In the above-given question,
given that,
Jared has to choose 1 of 8 boxes to win a prize.
One of the boxes has a big prize, 3 boxes have a medium prize, 3 boxes have smaller prizes, and 1 box is empty.
1 out of 8 boxes is 1/8.
3 out of 8 is 3/8 medium prize.
3/8 smaller prize.
1 box is empty is 1/8.
there are 3 chances.
they are 3/8, 1/8, and 3/8.

Focus on math practices
Construct Arguments Suppose the empty box is taken out of the game.
How confident should Jared be that he will win a prize? Explain.

Essential Question what is probability?

Try It!
How might the probability of the pointer landing on a given color change for the spinner shown at the right?

Convince Me! How would the probability of the pointer landing on a given color change if the spinner had six equal-sized sections with each section a different color?

Try It!
The game piece shown has 12 sides, labeled 1 to 12.

a. What is the probability of rolling an 11?

Answer:
The probability of rolling an 11 is 11/12.

Explanation:
In the above-given question,
given that,
The game piece shown has 12 sides, labeled 1 to 12.
the probability of rolling an 11 is 11/12.
so the event is on 11/12.

b. What is the probability of rolling a number greater than 5?

Answer:
The probability of rolling a number greater than 5 is 7/12.

Explanation:
In the above-given question,
given that,
The game piece shown has 12 sides, labeled 1 to 12.
the probability of rolling a number greater than 5 in 12 – 7.
12 – 7 = 5.
so the probability of rolling a number greater than 5 is 7/12.

c. What is the probability of rolling a number greater than 12?

Answer:
The probability of rolling a number greater than 12 is 1.

Explanation:
In the above-given question,
given that,
The game piece shown has 12 sides, labeled 1 to 12.
12/12 = 1.
so the probability of rolling a number greater than 12 is 1.

Try It!
Is the spinner shown a fair spinner? If yes, explain why. If not, describe a change that could make the spinner fair.

Answer:
Yes, the spinner is not a fair spinner.

Explanation:
In the above-given question,
given that,
the fair spinner is on the 12 hands.
12/6 = 2.
so the spinner is not a fair spinner.

KEY CONCEPT
The probability that something will occur is a value from 0 to 1, which describes its likelihood. You can write probability as a ratio, such as 1 out of 2, or 3, or as a percent, such as 50%.

Do You Understand?
Question 1.
Essential Question What is probability?

Answer:
The probability of a given event is an expression of the likelihood of occurrence of an event.

Explanation:
In the above-given question,
given that,
probability of a given event is an expression of the likelihood of occurrence of an event.
a probability is a number that ranges from 0 to 1.
there are so many types of probabilities are there.
they are theoretical probability and experimental probability.
for example:
0 for an event that cannot occur.
1 for an event that can occur.

Question 2.
Construct Arguments How can you use probability to draw conclusions about the likelihood that something will occur?

Answer:
0 for an event that cannot occur.
1 for an event that can occur.

Explanation:
In the above-given question,
given that,
probability of a given event is an expression of the likelihood of occurrence of an event.
a probability is a number that ranges from 0 to 1.
there are so many types of probabilities are there.
they are theoretical probability and experimental probability.
for example:
0 for an event that cannot occur.
1 for an event that can occur.

Question 3.
Reasoning Why is probability limited to numbers between 0 and 1?

Answer:
The probability limited to numbers between 0 and 1 is 0.

Explanation:
In the above-given question,
given that,
probability of a given event is an expression of the likelihood of occurrence of an event.
a probability is a number that ranges from 0 to 1.
there are so many types of probabilities are there.
they are theoretical probability and experimental probability.
for example:
0 for an event that cannot occur.
1 for an event that can occur.

Do You Know How?
Allie is going to select a card from the group of cards shown. Complete each statement.

Question 4.
The probability that Allie will select a card labeled 3 is _________ out of 10, or _________%.

Answer:
The probability that Allie will select a card labeled 3 is 2 out of 10, or 0.002.

Explanation:
In the above-given question,
given that,
Allie is going to select a card from the group of cards shown.
3 is 2 out of 10.
3 is 2/10.
3 is 1/5 = 0.2.

Question 5.
Because the probability that each number will be selected is not _________, the group of cards is not fair.

Answer:
Because the probability that each number will be selected is not fair, the group of cards is not fair.

Explanation:
In the above-given question,
given that,
Allie is going to select a card from the group of cards shown.
5 out of 10 is 5/10 = 1/2.
so the probability that each number will be selected is not fair.

Question 6.
It is _________ that Allie will select a card labeled with a number less than 6.

Answer:
5/10 = 1/2.

Explanation:
In the above-given question,
given that,
Allie is going to select a card from the group of cards shown.
It is 5 that Allie will select a card labeled with a number less than 6.
5/10 = 1/2.

Question 7.
It is _________ that Allie will select a card labeled 4.

Answer:
It is 4/10 that Allie will select a card labeled 4.

Explanation:
In the above-given question,
given that,
Allie is going to select a card from the group of cards shown
it is 4/10 that Allie will select a card labeled 4.
4/10 = 2/5.
2/5 = 0.4.

Practice & Problem Solving

Leveled Practice In 8-10, fill in the boxes to complete each statement.

Question 8.
A spinner has 8 equal-sized sections. Six of the sections are green.

Answer:
6 out of 8 sections are green.

Explanation:
In the above-given question,
given that,
A spinner has 8 equal-sized sections. Six of the sections are green.
6/8 = 3/4.
6 out of 8 sections are green.
3/4 = 0.75.

a. What is the probability that the spinner will land on green?

Answer:
The missing numbers are 6, 3, and 75%.

Explanation:
In the above-given question,
given that,
6 out of 8.
6/8 = 3/4.
3/4 = 0.75.
0.75/100 = 75.
so the missing numbers are 6, 3, and 75%.

b. Use words to describe the probability.
It is _________ that the spinner will land on green.

Answer:
It is 6 out of 8 that the spinner will land on the green.

Explanation:
In the above-given question,
given that,
6 out of 8.
6/8 = 3/4.
3/4 = 0.75.

Question 9.
Marcus is rolling a number cube with sides labeled 1 to 6.
a. The probability that the number cube will _________ show 10 is _________.

Answer:

b. It is _________ that the number cube will show 10.
Answer:

Question 10.
Of the marbles in a bag, 3 are yellow, 2 are red, and 2 are blue. Sandra will randomly choose one marble from the bag.
a. The probability that Sandra will choose a blue marble from the bag is _________ out of _________, or _________.

Answer:
The probability that Sandra will choose a blue marble from the bag is 1 out of 7 marbles or 1/7.

Explanation:
In the above-given question,
given that,
Of the marbles in a bag, 3 are yellow, 2 are red, and 2 are blue.
Sandra will randomly choose one marble from the bag.
3 + 2 + 2 = 7.
1 out of 7.
1/7 is 0.142.

b. It is _________ that sandra will choose a blue marble from the bag.

Answer:
It is 1/7 that Sandra will choose a blue marble from the bag.

Explanation:
In the above-given question,
given that,
Of the marbles in a bag, 3 are yellow, 2 are red, and 2 are blue.
Sandra will randomly choose one marble from the bag.
3 + 2 + 2 = 7.
1 out of 7.
1/7.

Question 11.
Suppose you have a bag with 20 letter tiles in it, and 3 of the tiles are labeled Y. Suppose a second bag has 500 letter tiles in it, and 170 of the tiles are labeled Y. From which bag are you more likely to pick a tile that is labeled Y? Explain.

Answer:
We will choose the bag with 20 letter tiles.

Explanation:
In the above-given question,
given that,
you have a bag with 20 letter tiles in it, and 3 of the tiles are labeled Y.
20/3 = 6.6.
Suppose a second bag has 500 letter tiles in it, and 170 of the tiles are labeled Y.
500/170 = 2.9.
so we will choose the bag with 20 letter tiles.

Question 12.
Make Sense and Persevere Suppose you have a bag of 40 marbles, and 20 of them are white. If you choose a marble without looking, the probability that you choose a white marble is $$\frac{20}{40}$$. Describe the probability.

Answer:
The probability is 20/40 is 0.5.

Explanation:
In the above-given question,
given that,
you have a bag of 40 marbles, and 20 of them are white.
If you choose a marble without looking, the probability that you choose a white marble is $$\frac{20}{40}$$.
20/40 = 1/2.
1/2 = 0.5.

Question 13.
Suppose Nigel has a bag of colored wristbands, and he chooses one without looking. The bag contains a total of 25 wristbands and 6 of the wristbands are blue.

Answer:
1 out of 25 = 0.04.

Explanation:
In the above-given question,
given that,
Nigel has a bag of colored wristbands, and he chooses one without looking.
The bag contains a total of 25 wristbands and 6 of the wristbands are blue.
1 out of 25.
1/25.

a. What is the probability that Nigel will choose a blue wristband?

Answer:
The probability is 6/25.

Explanation:
In the above-given question,
given that,
Nigel has a bag of colored wristbands, and he chooses one without looking.
The bag contains a total of 25 wristbands and 6 of the wristbands are blue.
6 out of 25.
6/25.
so the probability that Nigel will choose a blue wristband.

b. Is it likely, unlikely, or neither likely nor unlikely that Nigel will choose a blue wristband?

Answer:
It is neither likely nor unlikely that Nigel will choose a blue wristband.

Explanation:
In the above-given question,
given that,
Nigel has a bag of colored wristbands, and he chooses one without looking.
The bag contains a total of 25 wristbands and 6 of the wristbands are blue.
6 out of 25.
6/25.

Question 14.
A box contains four equal-sized cards labeled 1, 3, 5, and 7. Tim will select one card from the box.
a. What is the probability that Tim will select a card labeled 4?

Answer:
The probability that Tim will select a card labeled 4 is 1/4.

Explanation:
In the above-given question,
given that,
A box contains four equal-sized cards labeled 1, 3, 5, and 7.
1, 3, 5, and 7.
4 out of 16.
4/16 = 1/4.

b. What is the probability that Tim will select a card labeled with a number less than 6?

Answer:
The probability that Tim will select a card labeled with a number less than 6 is 5/16.

Explanation:
In the above-given question,
given that,
The probability that Tim will select a card labeled with a number less than 6.
5 out of 16.
5/16.
0.3125.

c. What is the probability that Tim will select a card labeled with an odd number?

Answer:
The probability that Tim will select a card labeled with an odd number is 5/16.

Explanation:
In the above-given question,
given that,
The probability that Tim will select a card labeled with an odd number.
5 out of 16.
5/16.
0.3125.

Question 15.
Model with Math Henry is going to color a spinner with 10 equal-sized sections. Three of the sections will be orange and 7 of the sections will be purple. Is this spinner fair? If so, explain why. If not, explain how to make it a fair spinner.

Answer:
Yes, It is fair.

Explanation:
In the above-given question,
given that,
Henry is going to color a spinner with 10 equal-sized sections.
Three of the sections will be orange and 7 of the sections will be purple.
3/10 and 7/10.
3 out of 10.
7 out of 10.
7 + 3 = 10.
so it is fair.

Question 16.
Higher Order Thinking Without being able to calculate probability, describe the likelihood that the following event will occur.
All 21 students in a class share the same birthday.

Answer:
The probability is 1.

Explanation:
In the above-given question,
given that,
Without being able to calculate probability,
describe the likelihood that the following event will occur.
All 21 students in a class share the same birthday.
21/ 21 = 1.
it is the like event.

Assessment Practice
Question 17.
After many studies, a researcher finds that the probability that a word recognition app correctly interprets a handwritten word is $$\frac{9}{10}$$. Which statement is true?
A It is impossible that the handwritten word will be correctly interpreted.
B It is unlikely that the handwritten word will be correctly interpreted.
C It is likely that the handwritten word will be correctly interpreted.
D It is certain that the handwritten word will be correctly interpreted.

Answer:
Option A is correct.

Explanation:
In the above-given question,
given that,
After many studies, a researcher finds that the probability that a word recognition app correctly interprets a handwritten word is 9/10.
9 out of 10 words.
9/10.
it is impossible that the handwritten word will be correctly interpreted.
so option A is correct.

Question 18.
A bag contains 8 letter tiles of the same size. The tiles are labeled either A, B, C, D, E, or F. Three of the tiles are labeled C. If Corey selects 1 tile from the bag without looking, is the selection of letters fair? Explain.

Answer:
Yes, the selection of letters is fair.

Explanation:
In the above-given question,
given that,
A bag contains 8 letter tiles of the same size.
The tiles are labeled either A, B, C, D, E, or F.
Three of the tiles are labeled C.
1 out of 8.
1/8.
so the selection of letters is fair.

Lesson 7.2 Understand Theoretical Probability

Solve & Discuss It!

Betty and Carl will conduct an experiment. They will flip a coin 100 times and record the result of each flip. What should they expect the results of their experiment to be? Justify your answer.

I can… determine the theoretical probability of an event.

Answer:
The theoretical probability of an event is 1/100, 2/100, ….. 100/100.

Explanation:
In the above-given question,
given that,
Betty and Carl will conduct an experiment.
They will flip a coin 100 times and record the result of each flip.
so the theoretical probability of an event is,
1/100, 2/100, .. 50/100, …. 100/100.

Focus on math practices
Look for Relationships How would their expected results change if Betty and Carl flipped a coin 500 times?

Essential Question
How can the probability of an event help make predictions?

Try It!
If Talia and Yoshi redesign their spinner to have 14 sections instead of 16 sections, would they likely have more or fewer winners? Explain why.

Answer:
They have fewer winners.

Explanation:
In the above-given question,
given that,
If Talia and Yoshi redesign their spinner to have 14 sections instead of 16 sections.
14/2 = 7.
16/2 = 8.
so they have fewer winners.

Convince Me! If there are always 2 red sections, how does the number of total sections in the spinner relate to the theoretical probability of winning this game?

Answer:
The probability of winning this game is 2 out of 14.

Explanation:
In the above-given question,
given that,
If there are always 2 red sections, how does the number of total sections in the spinner relate to the theoretical probability of winning this game?
2/14 = 1/7.
so the probability of the game is 2/14.

Try It!
Joaquin wants to reduce the number of winners so he does not have to prepare as many prizes. Choose another sum he could use as a winning sum, and predict the number of winners if 500 people play his game.

Answer:
The number of winners is 10.

Explanation:
In the above-given question,
given that,
Joaquin wants to reduce the number of winners so he does not have to prepare as many prizes.
500/50 = 10.
so the number of winners is 10.

KEY CONCEPT
You can determine the theoretical probability of an event, Plevent), if you know all the possible outcomes and they are equally likely.
+ number of favorable outcomes

You can use theoretical probability and proportional reasoning to make predictions, such as in a game situation.

Do You Understand?

Question 1.
Essential Question How can the probability of an event help make predictions?

Answer:
The number of favorable outcomes/ total number of possible outcomes is equal to the number of winning outcomes/ total number of possible outcomes.

Explanation:
In the above-given question,
given that,
the probability of an event help make predictions are:
the number of favorable outcomes/ total number of possible outcomes is equal to the number of winning outcomes/ total number of possible outcomes.

Question 2.
Construct Arguments A game board has a spinner with 10 equal-sized sections, of which 4 are green, 3 are blue, 2 are yellow, and 1 is red. What is the sum of the probabilities of the pointer landing in the green, blue, yellow, and red sections? Explain.

Answer:
p(event) = 1/5.

Explanation:
In the above-given question,
given that,
A game board has a spinner with 10 equal-sized sections, of which 4 are green, 3 are blue, 2 are yellow, and 1 is red.
the sum of the probabilities of the pointer landing in the green, blue, yellow, and red.
p(event) = number of favorable outcomes/total number of possible outcomes.
p(event) = 4 / 10 + 4 + 3 + 2 + 1.
p(event) = 4/20.
p(event) = 1/5.

Question 3.
Reasoning What does it mean that there is an equal theoretical probability of each outcome? Explain.

Answer:
The theoretical probability of each outcome = number of favorable outcomes/ total number of possible outcomes.

Explanation:
In the above-given question,
given that,
the probability of an event help make predictions are:
the number of favorable outcomes/ total number of possible outcomes is equal to the number of winning outcomes/ total number of possible outcomes.

Do You Know How?
In 4-6, Monique rolls a six-sided number cube labeled 1 to 6.

Question 4.
Find P(rolling a 4).

Answer:
p(rolling a 4) = 4/6.

Explanation:
In the above-given question,
given that,
Monique rolls a six-sided number cube labeled 1 to 6.
p(rolling a 4) = number of favorable outcomes/ total number of possible outcomes.
p(rolling a 4) = 4/6.

Question 5.
Find P(rolling an odd number).

Answer:
p(rolling an odd number) = 3/6.

Explanation:
In the above-given question,
given that,
Monique rolls a six-sided number cube labeled 1 to 6.
p(rolling an odd number) = number of favorable outcomes/ total number of possible outcomes.
p(rolling an odd number) = 3/6.

Question 6.
If Monique rolls the number cube 12 times, how many times would she expect a number greater than 4 to be rolled?

Answer:
The number of times would she expect a number greater than 4 to be rolled = 3.

Explanation:
In the above-given question,
given that,
If Monique rolls the number cube 12 times.
p(event) = 4/12.
p(event) = 1/3.
so the number of times would she expect a number greater than 4 to be rolled = 3.

Practice & Problem Solving
Leveled Practice In 7-9, complete each statement.

Question 7.
A spinner has 8 equal-sized sections. To win the game, the pointer must land on a yellow section.

Answer:
P(yellow) = 2/8 = 1/4.

Explanation:
In the above-given question,
given that,
A spinner has 8 equal-sized sections.
2/8 = 1/4.
so to win the game the pointer must land on a yellow section is 1/4.

Question 8.
Natalie is playing a game using a fair coin. Contestants win the game if the fair coin lands tails up.
The theoretical probability that the coin will land tails up is _________.
If 250 contestants play the game, about of them are expected to win.

Answer:
250/250 = 1.

Explanation:
In the above-given question,
given that,
Natalie is playing a game using a fair coin.
Contestants win the game if the fair coin lands tail up.
If 250 contestants play the game, about of them are expected to win.
so the theoretical probability = 1.

Question 9.
In a different game, the probability of correctly guessing which of 5 boxes contains a tennis ball is $$\frac{1}{5}$$. About how many winners would be expected if 60 contestants play the game?

X = _________ winners

Answer:
The number of winners would be expected if 60 contestants play the game = 12.

Explanation:
In the above-given question,
given that,
the probability of correctly guessing which of 5 boxes contains a tennis ball is 1/5.
1/5 = x/60.
x = 1/5 x 60.
x = 12.
so the number of winners would be expected if 60 contestants play the game = 12.

Question 10.
Make Sense and Persevere A 12-sided solid has equal-sized faces numbered 1 to 12.
a. Find P(number greater than 10).
b. Find P(number less than 5).
c. If the 12-sided solid is rolled 200 times, how many times would you expect either a 4, 6, or 9 to be rolled?

Answer:
a. p (number gteater than 10) = 10/12.
b. p(number less than 5) = 5/12.
c. 12-sided solid is rolled 200 times = 6.

Explanation:
In the above-given question,
given that,
A 12-sided solid has equal-sized faces numbered 1 to 12.
a. p (number gteater than 10) = 10/12.
b. p(number less than 5) = 5/12.
c. 12-sided solid is rolled 200 times = 6.

Question 11.
Tamara finds the sum of two number cubes rolled at the same time. The chart below shows all possible sums from the 36 possible combinations when rolling two number cubes. How many times should Tamara expect the sum of the two cubes to be equal to 5 if she rolls the two number cubes 180 times?

Answer:
The number of times should Tamara expect the sum of the two cubes to be equal to 5 is 36 times.

Explanation:
In the above-given question,
given that,
Tamara finds the sum of two number cubes rolled at the same time.
The chart below shows all possible sums from the 36 possible combinations when rolling two number cubes.
180/5 = 36.
so the number of times should Tamara expect the sum of the two cubes to be equal to 5 is 36 times.

Question 12.
Higher Order Thinking A store is giving every customer who enters the store a scratch-off card labeled with numbers from 1 to 10. It is equally likely that any of the numbers from 1 to 10 will be labeled on a given card. If the card is an even number, the customer gets a 15% discount on a purchase. If the card is an odd number greater than 6, the customer gets a 30% discount. Otherwise, the discount is 20%.

a. What is the probability for each discount?
15% discount: _________
20% discount: _________
30% discount: _________

Answer:
A card is an even number the customer gets a 15% discount.
if the card is an odd number greater than 6 the customer gets a 30% discount.
a card is an odd number the customer gets a 20% discount.

Explanation:
In the above-given question,
given that,
A store is giving every customer who enters the store a scratch-off card labeled with numbers from 1 to 10.
It is equally likely that any of the numbers from 1 to 10 will be labeled on a given card.
If the card is an even number, the customer gets a 15% discount on a purchase.
A card is an even number the customer gets a 15% discount.
if the card is an odd number greater than 6 the customer gets a 30% discount.
a card is an odd number the customer gets a 20% discount.

b. The store manager gives out 300 scratch-off cards. Which discount will the greatest number of customers likely receive? Explain.

Answer:
The greatest number of customers likely receive is 3%.

Explanation:
In the above-given question,
given that,
The store manager gives out 300 scratch-off cards.
300/100 = 3.
so the greatest number of customers likely receive is 3%.

Assessment Practice

Question 13.
A spinner is divided into 4 equal parts. 1 part is colored red, 2 parts are colored blue, and 1 part is colored yellow. The spinner is spun 1,000 times. Select all of the reasonable possible outcomes.
☐ The spinner lands on blue 445 times.
☐ The spinner lands on red 430 times.
☐ The spinner lands on blue 290 times.
☐ The spinner lands on yellow 200 times.
☐ The spinner lands on red 290 times.

Answer:
The spinner lands on blue 445 times.

Explanation:
In the above-given question,
given that,
A spinner is divided into 4 equal parts.
1 part is colored red, 2 parts are colored blue, and 1 part is colored yellow.
The spinner is spun 1,000 times.
1000/2 = 500.
so the spinner lands on blue 445 times.

Question 14.
One thousand five hundred runners have signed up for a marathon. The probability of a runner finishing the race is $$\frac{11}{12}$$. Approximately how many runners are expected to finish the race?

Answer:
The number of runners is expected to finish the race = 125.

Explanation:
In the above-given question,
given that,
One thousand five hundred runners have signed up for a marathon.
The probability of a runner finishing the race is $$\frac{11}{12}$$.
11/12 = 0.916.
1500 / 12 = 125.
so the number of runners are expected to finish the race = 125.

### Lesson 7.3 Understand Experimental Probability

Solve & Discuss It!
Kevin is awarded a penalty shot. He will either score a goal or not score a goal. Are both outcomes equally likely? Explain.

I can… determine the experimental probability of an event.

Look for Relationships
What might affect the outcome?

Answer:
Yes, both outcomes are equally likely.

Explanation:
In the above-given question,
given that,
Kevin is awarded a penalty shot.
He will either score a goal or not score a goal.
so both outcomes are equally likely.

Focus on math practices
Construct Arguments Lowe Senior High School’s soccer team won 12, lost 5, and tied in 3 of their first 20 games this season. Which outcome is most likely for the team’s next game? Explain your reasoning.

Answer:
The outcome is most likely for the team’s next game is 0.15.

Explanation:
In the above-given question,
given that,
Lowe Senior High School’s soccer team won 12, lost 5, and tied in 3 of their first 20 games this season.
12/5 = 122.6.
3/20 = 0.15.
so the outcome is most likely for the team’s next game is 0.15.

Essential Question
How is experimental probability similar to and different from theoretical probability?

Try It!

During the second day of the school fair, Talia and Yoshi recorded 43 winners out of a total of 324 players. How does the actual number of winners compare to the expected number of winners?
Theoretical Probability
P(red) = $$\frac{1}{8}$$ = 12.5%

Experimental Probability
$$\frac{1}{324}$$ ≈ ________ %
This experimental probability is ________ than the theoretical probability.
There were ________ winners than expected.

Convince Me! Will experimental probability always be close to theoretical probability? Explain.

Try It!

Amir and Marvin continue until they each flip a coin 200 times. How do you expect Amir’s results and Marvin’s results to compare? How will their results compare with expected results based on theoretical probability?

Answer:
The expected results are based on theoretical probability = 100.

Explanation:
In the above-given question,
given that,
Amir and Marvin continue until they each flip a coin 200 times.
200/2 = 100.
so the expected results are based on theoretical probability = 100.

KEY CONCEPT
Relative frequency, or experimental probability, is based on the actual results of an experiment, while theoretical probability is based on calculated results from the knowledge of the possible outcomes. Experimental probability and theoretical probability may be close but are rarely exactly the same.

The experimental probability tends to get closer to the theoretical probability of an experiment as more trials are conducted.

Do You Understand?
Question 1.
Essential Question How is experimental probability similar to and different from theoretical probability?

Answer:
The number of times an event occurs / total number of times the experiment is carried out.

Explanation:
In the above-given question,
given that,
Relative frequency, or experimental probability, is based on the actual results of an experiment,
while theoretical probability is based on calculated results from the knowledge of the possible outcomes.
Experimental probability = number of times an event occurs/ total number of times the experiment is carried out.
this value changes each time an experiment is carried out.

Question 2.
Construct Arguments How can experimental probability be used to make predictions?

Answer:
The number of times an event occurs / total number of times the experiment is carried out.

Explanation:
In the above-given question,
given that,
Relative frequency, or experimental probability, is based on the actual results of an experiment,
while theoretical probability is based on calculated results from the knowledge of the possible outcomes.
Experimental probability = number of times an event occurs/ total number of times the experiment is carried out.
this value changes each time an experiment is carried out.

Question 3.
Reasoning is experimental probability always close to theoretical probability? Explain.

Answer:
The number of times an event occurs / total number of times the experiment is carried out.

Explanation:
In the above-given question,
given that,
Relative frequency, or experimental probability, is based on the actual results of an experiment,
while theoretical probability is based on calculated results from the knowledge of the possible outcomes.
Experimental probability = number of times an event occurs/ total number of times the experiment is carried out.
this value changes each time an experiment is carried out.

Do You Know How?
In 4–6, complete each statement. Kelly flips a coin 20 times. The results are shown in the table, where “H” represents the coin landing heads up and “T” represents the coin landing tails up.

Question 4.
The theoretical probability that the coin will land heads up is _________.

Answer:
The theoretical probability that the coin will land heads up is 9/20 times.

Explanation:
In the above-given question,
given that,
Kelly flips a coin 20 times.
the results were shown in the table.
T represents the coin landing tails up.
so the theoretical probability that the coin will land heads up is 9/20 times.

Question 5.
Based on the data, the experimental probability that the coin will land heads up is __________.

Answer:
The experimental probability that the coin will land heads up is 11/20 times.

Explanation:
In the above-given question,
given that,
Kelly flips a coin 20 times.
the results were shown in the table.
H represents the coin landing the heads up.
so the experimental probability that the coin will land heads up is 11/20 times.

Question 6.
The experimental probability is _________ than the theoretical probability.

Answer:
The experimental probability is greater than the theoretical probability.

Explanation:
In the above-given question,
given that,
Relative frequency, or experimental probability, is based on the actual results of an experiment,
while theoretical probability is based on calculated results from the knowledge of the possible outcomes.
Experimental probability = number of times an event occurs/ total number of times the experiment is carried out.
this value changes each time an experiment is carried out.

Practice & Problem Solving

Leveled Practice In 7 and 8, complete each statement.
Question 7.
The table shows the results of spinning a wheel 80 times.
What is the relative frequency of the event “spin a 3”?

The relative frequency of the wheel landing on 3 is
$$\frac{\text { number of times an event occurs }}{\text { total number of trials }}$$ = $$\frac{}{}$$ = _________%

Answer:
The relative frequency of the wheel landing on 3 is 6 times.

Explanation:
In the above-given question,
given that,
The table shows the results of spinning a wheel 80 times.
The relative frequency of the wheel landing on 3 is
18/3 = 6.
so the relative frequency of the wheel landing on 3 is 6 times.

Question 8.
Liz flips a coin 50 times. The coin lands heads up 20 times and tails up 30 times. Complete each statement.
The theoretical probability of the coin landing heads up is __________
Based on Liz’s results, the experimental probability of the coin landing heads up is __________.
The theoretical probability is __________ than the experimental probability in this experiment.

Answer:
The theoretical probability of the coin landing heads up = 20/50.
the experimental probability of the coin landing heads up = 30/50.
the theoretical probability is less than the experimental probability in this experiment.

Explanation:
In the above-given question,
given that,
Liz flips a coin 50 times.
The coin lands head up 20 times and tails up 30 times.
20 out of 50 is 20/50.
30 out of 50 is 30/50.
The theoretical probability of the coin landing heads up = 20/50.
the experimental probability of the coin landing heads up = 30/50.
the theoretical probability is less than the experimental probability in this experiment.

Question 9.
Jess spins a pointer 25 times and finds an experimental probability of the pointer landing on 3 to be $$\frac{4}{25}$$, or 16%. The theoretical probability of the spinner landing on 3 is , or 25%. Why might there be a significant difference between the theoretical and experimental probabilities?

Answer:
The experimental probability is less than the theoretical probability.

Explanation:
In the above-given question,
given that,
Jess spins a pointer 25 times and finds an experimental probability of the pointer landing on 3 to be $$\frac{4}{25}$$, or 16%.
The theoretical probability of the spinner landing on 3 is, or 25%.
4/25 = 0.16.
0.16/100 = 16.
16 is less than 25.
so the experimental probability is less than the theoretical probability.

Question 10.
The table shows the results of a survey of 100 people randomly selected at an airport. Find the experimental probability that a person is going to City E.

Answer:
The experimental probability that a person is going to city E is 8/100.

Explanation:
In the above-given question,
given that,
The table shows the results of a survey of 100 people randomly selected at an airport.
city A is 28 responses.
City B is 34 responses.
city C is 16 responses.
city D is 14 responses.
city E is 8 responses.
so the experimental probability that a person is going to city E is 8/100.

Question 11.
The theoretical probability of selecting a consonant at random from a list of letters in the alphabet is $$\frac{21}{26}$$ Wayne opens a book, randomly selects a letter on the page, and records the letter. He repeats the experiment 200 times. He finds P(consonant) = 60%. How does the theoretical probability differ from the experimental probability? What are some possible sources for this discrepancy?

Answer:
Some possible sources for this discrepancy = 333.3.

Explanation:
In the above-given question,
given that,
The theoretical probability of selecting a consonant at random from a list of letters in the alphabet is $$\frac{21}{26}$$ Wayne opens a book, randomly selects a letter on the page and records the letter.
He repeats the experiment 200 times.
He finds P(consonant) = 60%.
21/26 = 0.8076.
60/100 = 6/10.
6/10 = 0.6.
200/ 0.6 = 333.3.

Question 12.
Higher Order Thinking Seven different names are written onto sticks and placed into a cup. A stick is chosen 100 times, out of which the name Grace is chosen 23 times. How do the theoretical probability and experimental probability compare? Explain why there is a discrepancy between them, if there is any.

Answer:
The discrepancy between them is 4.347.

Explanation:
In the above-given question,
given that,
Seven different names are written onto sticks and placed into a cup.
A stick is chosen 100 times, out of which the name Grace is chosen 23 times.
100/23 = 4.347.
so the discrepancy between them is 4.347.

Question 13.
Each of three friends flips a coin 36 times. Angel records “tails” 20 times. Michael records “tails” 17 times. Fernanda records “tails” 23 times.
a. Find the relative frequency with which each friend records “tails”.

Answer:
The relative frequency with which each friend records tails is 15.

Explanation:
In the above-given question,
given that,
Each of three friends flips a coin 36 times.
Angel records “tails” 20 times.
Michael records “tails” 17 times.
Fernanda records “tails” 23 times.
20 + 17 + 23 = 60.
60/4 = 15.
so the relative frequency with which each friend records tails is 15.

b. Which friend has a relative frequency that is closest to the theoretical probability of flipping “tails” 36 times? Explain.

Answer:
Fernanda has a relative frequency that is closest to the theoretical probability of flipping tails.

Explanation:
In the above-given question,
given that,
Each of three friends flips a coin 36 times.
Angel records “tails” 20 times.
Michael records “tails” 17 times.
Fernanda records “tails” 23 times.
20 + 17 + 23 = 60.
60/4 = 15.
so the relative frequency that is closest to the theoretical probability of flipping tails is Fernanda.

Assessment Practice
Question 14.
In a survey, 125 people were asked to choose one card out of five cards labeled 1 to 5. The results are shown in the table. Compare the theoretical probability and experimental probability of choosing a card with the number 1.

Answer:
The theoretical probability and experimental probability of choosing a card with the number 1 is 0.12 and 0.2.

Explanation:
In the above-given question,
given that,
In a survey, 125 people were asked to choose one card out of five cards labeled 1 to 5.
the theoretical property is 15/125.
15/125 = 0.12.
the experimental property is 25/125.
25/125 = 0.2.
so the theoretical probability and experimental probability of choosing a card with the number 1 are 0.12 and 0.2.

Question 15.
A basketball player makes 65% of all free throws in her first 5 seasons. In her 6th season she makes 105 out of 150 free throws. How does the observed frequency of her 6th season compare to the expected frequency? Provide a possible explanation for any similarities or differences in the frequencies.

Answer:
The difference is 0.57.

Explanation:
In the above-given question,
given that,
A basketball player makes 65% of all free throws in her first 5 seasons.
In her 6th season, she makes 105 out of 150 free throws.
105/150 = 0.7.
65/5 = 13%.
0.7 – 0.13 = 0.57.
so the difference is 0.57.

### Lesson 7.4 Use Probability Models

Explain It!

The Chess Club has 8 members. A new captain will be chosen by randomly selecting the name of one of the members. Leah and Luke both want to be captain. Leah says the chance that she will be chosen as captain is $$\frac{1}{2}$$ because she is either chosen for captain or she is not. Luke says the chance that he is chosen is $$\frac{1}{8}$$.
I can… use probability models to find probabilities of events.

A. Construct Arguments Do you agree with Leah’s statement? Use a mathematical argument to justify your answer.

Answer:
Yes, I agree with Leah’s statement.

Explanation:
In the above-given question,
given that,
The Chess Club has 8 members.
A new captain will be chosen by randomly selecting the name of one of the members.
Leah and Luke both want to be captains.
Leah says the chance that she will be chosen as captain is $$\frac{1}{2}$$ because she is either chosen for a captain or she is not.
so I agree with Leah’s statement.

B. Construct Arguments Do you agree with Luke’s statement? Use . a mathematical argument to justify your answer.

Answer:
Yes, Luke was also correct.

Explanation:
In the above-given question,
given that,
The Chess Club has 8 members.
A new captain will be chosen by randomly selecting the name of one of the members.
Leah and Luke both want to be captains.
Luke says the chance that he is chosen is $$\frac{1}{8}$$.
so I agree with Luke’s statement.

Focus on math practices
Look for Relationships How does the probability of Leah being chosen captain compare to the probability of Luke being chosen captain?

Essential Question
How can a model be used to find the probability of an event?

Try It!
Mr. Campbell decides that too many students are getting a pass on homework. He adds 10 yellow marbles to the jar. Tell whether each part of the probability model does or does not change.
The sample space _________ change. Each event within the sample space change. The probability of each event ________ change.
The new probability of drawing a red marble is P(R) = $$\frac{1}{}$$

Answer:
The sample space does not change.
Each event within the sample space change.
The probability of each event does not change.

Explanation:
In the above-given question,
given that,
Mr. Campbell decides that too many students are getting a pass on homework.
He adds 10 yellow marbles to the jar.
so the sample space does not change.
each event within the sample space change.
the probability of each event does not change.

Convince Me! How does a probability model help you predict how likely an event is to occur?

Try It!
To reduce the number of homework passes, which color of marble should Ms. Stillman use as the pass on homework? Explain.

Answer:
He should use red color marbles.

Explanation:
In the above-given question,
given that,
to reduce the number of homework passes,
he should use red color marbles.

KEY CONCEPT
A probability model can help you evaluate a chance process and its outcomes. You can develop a model using theoretical or experimental probability. A probability model consists of the sample space of an action, events within the sample space, and probabilities associated with each event.
For rolling a number cube labeled from 1 through 6: Sample space, S = {1, 2, 3, 4, 5, 6}

P(1) = $$\frac{1}{6}$$
P(2) = $$\frac{1}{6}$$
P(3) = $$\frac{1}{6}$$
P(4) = $$\frac{1}{6}$$
P(5) = $$\frac{1}{6}$$
P(6) = $$\frac{1}{6}$$

Do You Understand?
Question 1.
Essential Question How can a model be used to find the probability of an event?

Answer:
The probability of an event is to evaluate a chance process and its outcomes.

Explanation:
In the above-given question,
given that,
A probability model can help you evaluate a chance process and its outcomes. You can develop a model using theoretical or experimental probability. A probability model consists of the sample space of action, events within the sample space, and probabilities associated with each event.
For rolling a number cube labeled from 1 through 6: Sample space, S = {1, 2, 3, 4, 5, 6}.
so the probability of an event is to evaluate a chance process and its outcomes.

Question 2.
Construct Arguments How can you check the sample space of a probability model?

Answer:
The probability of an event is to evaluate a chance process and its outcomes.

Explanation:
In the above-given question,
given that,
A probability model can help you evaluate a chance process and its outcomes. You can develop a model using theoretical or experimental probability. A probability model consists of the sample space of action, events within the sample space, and probabilities associated with each event.
For rolling a number cube labeled from 1 through 6: Sample space, S = {1, 2, 3, 4, 5, 6}.
so the probability of an event is to evaluate a chance process and its outcomes.

Question 3.
Reasoning How does developing a probability model based on experimental probability help you evaluate a situation or make an estimate? Explain.

Answer:
The probability of an event is to evaluate a chance process and its outcomes.

Explanation:
In the above-given question,
given that,
A probability model can help you evaluate a chance process and its outcomes. You can develop a model using theoretical or experimental probability. A probability model consists of the sample space of action, events within the sample space, and probabilities associated with each event.
For rolling a number cube labeled from 1 through 6: Sample space, S = {1, 2, 3, 4, 5, 6}.
so the probability of an event is to evaluate a chance process and its outcomes.

Do You Know How?
Question 4.
Develop a probability model for the spinner shown.

Answer:
P(1) = $$\frac{1}{5}$$
P(2) = $$\frac{1}{5}$$
P(3) = $$\frac{1}{5}$$
P(4) = $$\frac{1}{5}$$
P(5) = $$\frac{1}{5}$$

Explanation:
In the above-given question,
given that,
the spinner is on the 2 and 5.
if the spinner is on the 1 through 1/5.
if the spinner is on the 2 through 2/5.
if the spinner is on the 3 through 3/5.
if the spinner is on the 4 through 4/5.
if the spinner is on the 5 through 5/5.

Question 5.
Mr. Henry has a basket full of fruit. He does not know how many pieces of fruit are in the basket or the types of fruit. Each of the 20 students in his class selects one piece of fruit from the basket without looking, notes its fruit type, and then puts it back in the basket. Based on the results shown in the table, what can the students conclude about the probability of selecting an apple?

Answer:
The probability of selecting an apple is 4.

Explanation:
In the above-given question,
given that,
Mr. Henry has a basket full of fruit.
He does not know how many pieces of fruit are in the basket or the types of fruit.
Each of the 20 students in his class selects one piece of fruit from the basket without looking, notes its fruit type, and then puts it back in the basket.
the number of pieces of fruit of apple is 5.
20/5 = 4.
4 x 5 = 20.
so the probability of selecting an apple is 4.

Question 6.
The probability model based on experimental probability for randomly selecting a marble from a bag is P(green) = $$\frac{18}{40}$$, P(blue) = $$\frac{14}{40}$$, and P(white) = $$\frac{8}{40}$$. About how many marbles of each color are in the bag if there are 60 total marbles?

Answer:
The number of marbles of each color is in the bag if there are 60 total marbles = 25, 20, and 15.

Explanation:
In the above-given question,
given that,
p(green) = 18/40.
p(blue) = 14/40.
p(white) = 8/40.
18 + 14 + 8 = 40.
25 + 20 + 15 = 60.
so p(green) = 25.
p(blue) = 20.
p(white) = 15.
so the number of marbles of each color is in the bag if there are 60 total marbles = 25, 20, and 15.

Practice & Problem Solving

Question 7.
Murray spins the pointer of the spinner shown at the right.

a. What is the sample space for the probability model?

Answer:
The sample space for the probability model is 3/8.

Explanation:
In the above-given question,
given that,
Murray spins the pointer of the spinner shown at the right.
the spinner is on the 3.
p(3/8) = 3/8.
so the sample space for the probability model is 3/8.

b. What is the probability of each event in the sample space?

Answer:
The probability of each event in the sample space = 1/8, 3/8, and 5/8.

Explanation:
In the above-given question,
given that,
Murray spins the pointer of the spinner shown at the right.
if the spinner is on the 1.
p(1/8) = 1/8.
if the spinner is on the 3.
p(3/8) = 3/8.
so the probability of each event in the sample space = 1/8, 3/8, and 5/8.

Question 8.
Rafael spins the pointers of the two spinners shown at the right. Find the probability of each possible sum.

P(sum 2) = __________
P(sum 3) = __________
P(sum 4) = __________
P(sum 5) = __________

Answer:
p(sum 2) =
p(sum 3) =
p(sum 4) =
p(sum 5) =

Explanation:
In the above-given question,
given that,

Question 9.
Be Precise An arts and crafts store has a crate that contains glass, wood, and brass beads. Friends take turns choosing a bead without looking, recording the bead type, and returning the bead to the crate. The table shows the results of 300 selections.

a. Write a probability model for choosing a bead.

Answer:
The model for choosing a bead = p(0.2), p(0.32), and p(0.48).

Explanation:
In the above-given question,
given that,
An arts and crafts store has a crate that contains glass, wood, and brass beads.
Friends take turns choosing a bead without looking, recording the bead type, and returning the bead to the crate.
glass = p(60/300) = p(0.2).
wood = p(96/300) = p(0.32).
brass = p(144/300) = p(0.48).
so the model for choosing a bead = p(0.2), p(0.32), and p(0.48).

b. Based on the frequencies in the table, estimate the number of each type of bead that will be chosen if the friends select a total of 450 beads from the crate.

Answer:
The friends select a total of 450 beads from the crate = p(0.2), p(0.32), and p(0.48).

Explanation:
In the above-given question,
given that,
An arts and crafts store has a crate that contains glass, wood, and brass beads.
Friends take turns choosing a bead without looking, recording the bead type, and returning the bead to the crate.
60 + 40 = 100.
96 + 54 = 150.
144 + 56 = 200.
100 + 150 + 200 = 450.
glass = p(100/450) = p(0.2).
wood = p(150/450) = p(0.32).
brass = p(200/450) = p(0.48).
so the friends select a total of 450 beads from the crate = p(0.2), p(0.32), and p(0.48).

Question 10.
A bag contains 14 green, 12 orange, and 19 purple tennis balls.
a. Create a probability model for choosing a tennis ball from the bag.

Answer:
The probability model for choosing a tennis ball from the bag = p(0.3), p(0.2), and p(0.4).

Explanation:
In the above-given question,
given that,
A bag contains 14 green, 12 orange, and 19 purple tennis balls.
14 + 12 + 19 = 45.
p(green) = p(14/45).
p(orange) = p(12/45).
p(purple) = p(19/45).
so the probability model for choosing a tennis ball from the bag = p(0.3), p(0.2), and p(0.4).

b. Suppose a tennis ball is randomly selected and then replaced 75 times. How many orange tennis balls do you expect? Explain.

Answer:
The number of orange tennis balls does you expert = p(20/45).

Explanation:
In the above-given question,
given that,
Suppose a tennis ball is randomly selected and then replaced 75 times.
14 + 11 = 25.
12 + 8 = 20.
19 + 11 = 30.
25 + 20 + 30 = 75.
so the number of orange tennis balls does you expert = p(20/45).

Question 11.
Given that Pred pepper) = $$\frac{3}{5}$$, write another probability statement to complete the probability model of a random pepper selection from the box below.

Answer:
The probability model of a random pepper is p(pepper) = 5/14.

Explanation:
In the above-given question,
given that,
P(red pepper) = $$\frac{3}{5}$$.
total there are 14 trays.
I chooses 5 trays.
p(pepper) = p(5/14).
so the probability model of a random pepper is p(pepper) = 5/14.

Question 12.
Higher Order Thinking A survey asked 600 people for their favorite genre of book. The table shows the number of people who preferred four possible genres.

a. How many people surveyed responded with a genre that is not listed in the table?

Answer:
The number of people surveyed responded with a genre that is not listed in the table = 126.

Explanation:
In the above-given question,
given that,
A survey asked 600 people for their favorite genre of book.
the people who selected Adventure is 90.
the people who selected comedy is 102.
the people who selected Mystery is 150.
the people who selected Romance is 132.
90 + 102 + 150 + 132 = 474.
600 – 474 = 126.
so the number of people surveyed responded with a genre that is not listed in the table = 126.

b. Find the probabilities and complete a probability model to describe each response, including “other genre”.

Answer:
The probabilities are p(0.18), p(0.21), p(0.31), and p(0.27).

Explanation:
In the above-given question,
given that,
A survey asked 600 people for their favorite genre of book.
the people who selected Adventure is 90.
the people who selected comedy is 102.
the people who selected Mystery is 150.
the people who selected Romance is 132.
p(90/474) = p(0.18).
p(102/474) = p(0.21).
p(150/474) = p(0.31).
p(132/474) = p(0.27).
so the probabilities are p(0.18), p(0.21), p(0.31), and p(0.27).

Assessment Practice
Question 13.
One hundred people buy gum balls from a gum ball machine. 45 of them get a red gum ball, 40 get a blue gum ball and 15 get a yellow gum ball.

PART A
Develop a probability model to predict the color of the next gum ball purchased. Compare the probability of getting a red gum ball to the probability of getting a yellow gum ball.

Answer:
The probabilities are p(0.45), p(0.4), and p(0.15).

Explanation:
In the above-given question,
given that,
One hundred people buy gumballs from a gumball machine.
45 of them get a red gumball, 40 get a blue gumball and 15 get a yellow gumball.
p(red) = p(45/100).
p(blue) = p(40/100).
p(yellow) = p(15/100).
so the probabilities are p(0.45), p(0.4), and p(0.15).

PART B
Of the next 10 people to buy gum balls, 7 get yellow, 1 gets red and 2 get blue. Explain a possible reason for this outcome.

Answer:
The probabilities are p(0.7), p(0.1), and p(0.2).

Explanation:
In the above-given question,
given that,
One hundred people buy gumballs from a gumball machine.
45 of them get a red gumball, 40 get a blue gumball and 15 get a yellow gumball.
p(red) = p(1/10).
p(blue) = p(2/10).
p(yellow) = p(7/10).
so the probabilities are p(0.7), p(0.2), and p(0.1).

### Topic 7 MID-TOPIC CHECKPOINT

Question 1.
Vocabulary How does the theoretical probability of the event “flip heads” change when a coin is flipped more times in an experiment? Lesson 7-2
A. increases; there are more chances for heads to be flipped
B. decreases; there are more chances for tails to be flipped
C. does not change
D. increases; all values increase

Answer:
Option A is correct.

Explanation:
In the above-given question,
given that,
the theoretical probability of the event “flip heads” changes when a coin is flipped more times in an experiment.
so it is increased.
there are more chances for heads to be flipped.
so option A is correct.

In 2-4, use the information given. Brianna has a bag of marbles that are all the same size. Of all the marbles in the bag, there are 6 red, 7 white, 3 black, and 4 green marbles.
Question 2.
Select all the likelihood statements that are true. Lesson 7-1
☐ It is impossible that Brianna will draw a blue marble.
☐ It is more likely that Brianna will draw a black marble than a green marble.
☐ It is certain that Brianna will draw either a red, white, black, or green marble.
☐ It is unlikely that Brianna will draw a black marble.
☐ It is neither likely nor unlikely that Brianna will draw a green marble.

Answer:
Options C, D, and E are correct.

Explanation:
In the above-given question,
given that,
Brianna has a bag of marbles that are all the same size.
Of all the marbles in the bag, there are 6 red, 7 white, 3 black, and 4 green marbles.
it is certain that Brianna will draw either a red, white, black, or green marble.
it is unlikely that Brianna will draw a black marble.
it is neither likely nor unlikely that Brianna will draw a green marble.
so options C, D, and E are correct.

Question 3.
Ryan asks 80 people to choose a marble, note the color, and replace the marble in Brianna’s bag. The results of the random marble selections in this experiment are: 34 red, 18 white, 9 black, and 19 green marbles. How does the theoretical probability compare with the experimental probability of drawing a white marble? Lessons 7-2 and 7-3

Answer:
The theoretical probability is equal to the experimental probability.

Explanation:
In the above-given question,
given that,
Ryan asks 80 people to choose marble, note the color, and replace the marble in Brianna’s bag.
The results of the random marble selections in this experiment are 34 red, 18 white, 9 black, and 19 green marbles.
34 + 18 + 9 + 19 = 80.
p(white) = p(18/80).
p(white) = p(0.225).
so the theoretical probability is equal to the experimental probability.

Question 4.
Write a probability model for this experiment, and use the probability model to predict how many times Brianna would pick a green marble if she chose a marble 50 times. Give the probabilities as simplified fractions. Lesson 7-4
Drawing a red marble: __________
Drawing a black marble: __________
Drawing a white marble: __________
Drawing a green marble: __________
Brianna would draw __________ green marbles in 50 tries.

Answer:
Drawing red marble is p(25/50) = p(0.5).

Explanation:
In the above-given question,
given that,
Brianna would pick a green marble if she chose a marble 50 times.
34 – 9 = 25.
18 – 8 = 10.
9 – 4 = 5.
19 – 9 = 10.
drawing red marble is p(red) = p(25/50).
25/50 = 1/2.
so drawing red marble is p(0.5).

Question 5.
Jewel spins the pointer of a spinner. The spinner has 7 equal-sized sections labeled 1 to 7. What is the probability that Jewel will spin a 7? Lessons 7-2 and 7-4

Answer:
The probability that Jewel will spin a 7 is p(1).

Explanation:
In the above-given question,
given that,
Jewel spins the pointer of a spinner.
The spinner has 7 equal-sized sections labeled 1 to 7.
p(7/7) = p(1).
so the probability that Jewel will spin a 7 is p(1).

### Topic 7 MID-TOPIC PERFORMANCE TASK

Viet, Quinn, and Lucy are going to play Bingo, using a standard game set. They make some predictions before the game begins. The table shows how the numbers match with the letters B, I, N, G, and O.

PART A
Viet makes a probability model to describe the probability of each number being called first. Quinn makes a probability model to describe the probability of any particular letter being called first. Compare the probability models.

Answer:
The probability models are (1/50), (16/30), (31/45), (46/60), and (61/75).

Explanation:
In the above-given question,
given that,
Viet, Quinn, and Lucy are going to play Bingo, using a standard game set.
B – 1 to 15.
I – 16 to 30.
N – 31 to 45.
G – 46 to 60.
O – 61 to 75.
so the probability models are 1/50), (16/30), (31/45), (46/60), and (61/75).

PART B
Lucy makes a probability model to determine whether the first number drawn will be even or odd. Compare the different probabilities.

Answer:
The first probability model is odd.

Explanation:
In the above-given question,
given that,
the first number drawn is odd.
1 – 15.
15/75.
so the first probability model is odd.

PART C
Suppose the game changed to have 90 numbers, instead of 75 numbers, matched with the letters B, I, N, G, and O. How would Viet’s, Quinn’s, and Lucy’s probability models change? Explain.

Answer:
Lucy’s probability model change is 21/90.

Explanation:
In the above-given question,
given that,
suppose the game changed to have 90 numbers., instead of 75 numbers.
so the B – 1 to 21.
I – 22 to 36.
N – 37 to 51.
G – 52 to 66.
O – 67 to 90.
so Lucy’s probability model change is 21/90.

### 3-Act Mathematical Modeling: Photo Finish

ACT 1
Question 1.
After watching the video, what is the first question that comes to mind?
Answer:

Question 2.
Write the Main Question you will answer.
Answer:

Question 3.
Construct Arguments Predict an answer to this Main Question. Explain your prediction.

Answer:

Question 4.
On the number line below, write a number that is too small to be the answer. Write a number that is too large. Plot your prediction on the same number line.

Answer:
The too small is 0.01 and the large is 1.

Explanation:
In the above-given question,
given that,
the number that is too small is 0.01.
it is the starting stage of the number line.
the number that is too large is 1.
it is the ending stage of the number line.

ACT 2
Question 5.
What information in this situation would be helpful to know? How would you use that information?

Answer:

Question 6.
Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
Answer:

Question 7.
Model with Math Represent the situation using mathematics. Use your representation to answer the Main Question.
Answer:

Question 8.
What is your answer to the Main Question? Is it higher or lower than your prediction? Explain why.

Answer:

ACT 3
Question 9.
Write the answer you saw in the video.
Answer:

Question 10.
Reasoning Does your answer match the answer in the video?
If not, what are some reasons that would explain the difference?

Answer:

Question 11.
Make Sense and Persevere Would you change your model now that you know the answer? Explain.

Answer:
The answer is 6.

Explanation:
In the above-given question,
given that,
the numbers on the figure are 1, 2, and 3.
1 + 2 + 3 = 6.
so the answer is 6.

ACT 3

Reflect
Question 12.
Model with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?

Answer:

Question 13.
Be Precise What vocabulary have you learned in this topic that helps you communicate the answer to the Main Question?
Answer:

SEQUEL
Question 14.
Generalize How would your answer change if a fifth person joined the race? A sixth person? If n people are running in the race?
Answer:

### Lesson 7.5 Determine Outcomes of Compound Events

Solve & Discuss It!

Cameron packed two pairs of shorts and three T-shirts for a weekend trip. What are some combinations of shirts and shorts that Cameron can wear while on his trip? How many days will he have a different outfit to wear?
I can… find all possible outcomes of a compound event.

Focus on math practices
Reasoning How would the number of different outfits change if Cameron packed a pair of khaki shorts? Explain.

Essential Question
How can all the possible outcomes, or sample space, of a compound event be represented?

Answer:
The number of times the possible outcomes are 3.

Explanation:
In the above-given question,
given that,
Cameron can wear 3 t-shirts in 3 days.
Cameron packed two pairs of shorts and three T-shirts for a weekend trip.
Cameron can wear 3 t-shirts in 3 days.
so the number of times the possible outcomes are 3.

Try It!
Jorge will flip two quarters at the same time. Complete the tree diagram, and then list the sample space of this compound event. Use H for heads and T for tails.

The sample space is:______________

Answer:
The missing numbers are H, H, and T.

Explanation:
In the above-given question,
given that,
Jorge will flip two quarters at the same time.
two heads give the answer head.
two tails give the answer head.
so the missing numbers are H, H, and T.

Convince Me! How does the sample space change when the number of quarters that Jorge flips is increased by 1?

Try It!
The bag contains tiles labeled with the letters A, B, and C. The box contains tiles labeled with the numbers 1, 2, and 3. June draws one letter tile and one number tile. Represent the sample space using either a table or an organized list.

Answer:
The letter tiles are C/3 and 2/3.

Explanation:
In the above-given question,
given that,
The bag contains tiles labeled with the letters A, B, and C.
The box contains tiles labeled with the numbers 1, 2, and 3.
C/3 and 2/3.
so the letter tiles are C/3 and 2/3.

KEY CONCEPT
A compound event is a combination of two or more events.

An organized list, table, or tree diagram can be used to represent the sample space of a compound event. The sample space for flipping two coins consists of 4 outcomes.

Do You Understand?
Question 1.
Essential Question How can all the possible outcomes, or sample space, of a compound event be represented?

Answer:
A compound event is a combination of two or more events.

Explanation:
In the above-given question,
given that,
two heads are equal to one tail.
one head and one tail are equal to head and tail.
two tails are equal to one head.
so compound event is a combination of two or more events.

Question 2.
Generalize Will a list, a table, and a tree diagram always give you the same number of outcomes for the same compound event? Explain.

Answer:
Yes, a table and a tree diagram always give the same number of outcomes.

Explanation:
In the above-given question,
given that,
two heads are equal to one tail.
one head and one tail are equal to head and tail.
two tails are equal to one head.
so the table and a tree diagram always give the same number of outcomes.

Question 3.
Use Structure Shari is drawing a tree diagram to represent the sample space of rolling a 12-sided game piece and spinning the pointer of a 4-section spinner. Does it matter if Shari starts the tree diagram with the game piece outcomes or the spinner outcomes? Explain.

Answer:
The game piece outcomes = 3.

Explanation:
In the above-given question,
given that,
Shari is drawing a tree diagram to represent the sample space of rolling a 12-sided game piece and spinning the pointer of a 4-section spinner.
12/4 = 3.
4 x 3 = 12.
so the game piece outcomes = 3.

Do You Know How?
Question 4.
Both Spinner A and Spinner B have equal-sized sections, as shown at the right. Make a table to represent the sample space when both spinners are spun.

Answer:
The spinner A = 1, 2, and 3.

Explanation:
In the above-given question,
given that,
Both Spinner A and Spinner B have equal-sized sections, as shown on the right.
spinner B has the numbers 1, 2, and 3.
so the spinner A has 1, 2, and 3.

Question 5.
Tiles labeled with the letters X, Y, and Z are in a bag. Tiles labeled with the numbers 1 and 2 are in a box.
Make a tree diagram to represent the sample space of the compound event of selecting one tile from each container.

Answer:
Two x’s gives Y.
Two Y’s gives Z.

Explanation:
In the above-given question,
given that,
Tiles labeled with the letters X, Y, and Z are in a bag.
Tiles labeled with the numbers 1 and 2 are in a box.
1 – x, y, and z.
2 – x, y, and z.
so two x’s give y.
two y’s gives z.

Practice & Problem Solving

Leveled Practice In 6 and 7, find the number of outcomes for each event.
Question 6.
Oliver is playing a game in which he has to choose one of two numbers (2 or 7) and then one of five vowels (a, e, i, o, or u). How many possible outcomes are there?

There are ________ possible outcomes.

Answer:
There are 7 possible outcomes.

Explanation:
In the above-given question,
given that,
Oliver is playing a game in which he has to choose one of two numbers (2 or 7).
and then one of five vowels (a, e, i, o, or u).
the events are 1/7, 2/7, 3/7, 4/7, 5/7, 6/7, and 1.
the vowels are a/7, e/7, i/7, o/7, and u/7.
so there are 7 possible outcomes.

Question 7.
There are four stores that sell school supplies (S1, S2, S3, and S4) and three stores that sell sporting goods (G1, G2, and G3) nearby. How many possible combinations of stores could you visit to buy a tennis racquet and then a backpack?

There are ________ possible combinations.

Answer:
There are 3 possible combinations.

Explanation:
In the above-given question,
given that,
There are four stores that sell school supplies (S1, S2, S3, and S4) and three stores.
that sell sporting goods (G1, G2, and G3) nearby.
G1 – s1, s2, s3, and s4.
G2 – s2, s3, s4, and s1.
G3 – s3, s4, s3, and s2.
so there are 3 possible combinations.

Question 8.
A bakery sells wheat, multigrain, rye, and oat bread. Each type of bread is available as a loaf or as dinner rolls.

a. Complete the table to show all the possible outcomes for the types and styles of bread sold by the bakery.

Answer:

b. Find the number of possible outcomes.

Answer:
The number of possible outcomes is 8.

Explanation:
In the above-given question,
given that,
A bakery sells wheat, multigrain, rye, and oat bread.
wheat – 2 items.
multigrain – 2 items.
rye – 2 items.
oat bread – 2 items.
2 + 2 + 2 + 2 = 8.
so the number of possible outcomes is 8.

Question 9.
Generalize How does the number of possible outcomes of a single event help you determine the total number of possible outcomes of a compound event?

Answer:
A compound event is a combination of two or more events.

Explanation:
In the above-given question,
given that,
two heads are equal to one tail.
one head and one tail are equal to head and tail.
two tails are equal to one head.
so compound event is a combination of two or more events.

Question 10.
A new car can be purchased with a choice of four exterior colors (A, B, C, and D) and three interior colors (1, 2, and 3). Make an organized list of all the possible color combinations for the car.

Answer:
The organized list of all the possible color combinations for the car is 12.

Explanation:
In the above-given question,
given that,
A new car can be purchased with a choice of four exterior colors (A, B, C, and D).
and three interior colors (1, 2, and 3).
4 x 3 = 12.
1 – A, B, C, and D.
2 – B, C, D, and A.
3 – C, D, A, and B.
so the organized list of all the possible color combinations for the car is 12.

Question 11.
Two friends each plan to order a fruit drink at the diner. The available flavors are kiwi (K), lemon (L), and watermelon (W). Make a list to represent all the possible outcomes of the friends’ fruit drink order. Write each outcome in the format (Friend 1, Friend 2).

Answer:
Each outcome is in format 6.

Explanation:
In the above-given question,
given that,
Two friends each plan to order a fruit drink at the diner.
The available flavors are kiwi (K), lemon (L), and watermelon (W).
2 friends – 3 flavors.
3 x 2 = 6.
1 – K, L, and W.
2 – K, L, and W.
so the number of outcomes is 6.

Question 12.
Plastic souvenir cups come in three different sizes: small (S), medium (M), and large (L). The available colors are red (R), white (w), and blue (B). Make a list to represent all the possible combinations of the different cups based on size and color. Write each outcome in the format (Size, Color).

Answer:
The number of outcomes = 9.

Explanation:
In the above-given question,
given that,
Plastic souvenir cups come in three different sizes: small (S), medium (M), and large (L).
The available colors are red (R), white (w), and blue (B).
red – 3.
white – 3.
blue – 3.
3 + 3 + 3 = 9.
so the number of outcomes = 9.

Question 13.
Higher Order Thinking Heidi’s older sister needs to take either Chemistry (C), Geometry (G), or Physics (P) this year. She can take the class during any one of six periods (1 through 6). Is there more than one way to draw a tree diagram to model this situation? Explain.

Answer:
Yes, there is more than one way.

Explanation:
In the above-given question,
given that,
Heidi’s older sister needs to take either Chemistry (C), Geometry (G), or Physics (P) this year.
She can take the class during any one of six periods (1 through 6).
1 – 3.
2 – 3.
3 – 3.
4 – 3.
5 – 3.
6 – 3.
3 + 3 + 3 + 3 + 3 + 3 = 18.
so there is more than one way.

Assessment Practice
Question 14.
A fruit basket has 6 oranges, 4 apples and 2 pears in it. 5 people each select a piece of fruit and eat it. Which of the following outcomes could represent this selection?
☐ All 5 people eat an orange.
☐ 1 person eats an orange, 4 people eat an apple.
☐ 2 people eat an orange, 3 people eat a pear.
☐ 3 people eat an orange, 1 person eats an apple, 1 person eats a pear.
☐ All 5 people eat an apple.

Answer:
Options B, C, and D are correct.

Explanation:
In the above-given question,
given that,
A fruit basket has 6 oranges, 4 apples, and 2 pears in it.
5 people each select a piece of fruit and eat it.
6 + 4 + 2 = 12.
12/3 = 4.
so options B, C, and D are correct.

Question 15.
Royce has a collection of trading cards. 16 of his cards are baseball cards, 21 are football cards and 13 are basketball cards. He selects half of this collection and gives them to his friend. Which of the following represent possible outcomes of this selection?
☐ He gives his friend 6 baseball cards, 10 football cards and all of his basketball cards.
☐ He gives his friend 4 baseball cards and all of his football cards.
☐ He gives his friend only football cards.
☐ He gives his friend 8 baseball cards, 10 football cards and 7 basketball cards.
☐ He gives his friend all of his baseball and basketball cards.

Answer:
Options B, C, and D are correct.

Explanation:
In the above-given question,
given that,
Royce has a collection of trading cards.
16 of his cards are baseball cards, 21 are football cards and 13 are basketball cards.
16 + 21 + 13 = 50.
50/3 = 16.6.
so options B, C, and D are correct.

### Lesson 7.6 Find Probabilities of Compound Events

Solve & Discuss It!
Talia is playing a game in which she must choose Option 1 or Option 2 and then spin the game wheel, flip the coin, and roll the number cube labeled 1 through 6. For her to win a prize, all the conditions listed under the chosen option must occur. Which option should Talia choose? Explain.
I can… find the probability of a compound event.

Option 1
• The game wheel lands on S.
• The coin lands on tails.
• An even number is rolled.

Option 2
• The game wheel lands on Z.
• The coin lands on either side.
• The number 3 is rolled.

Answer:
Both the options are correct.

Explanation:
In the above-given question,
given that,
in option 1 the game wheel lands on s.
the coil lands on tails.
an even number is rolled.
in option 2 the game wheel lands on z.
the coin lands on either side.
the number 3 is rolled.
so both the options are correct.

Look for Relationships
How can you use what you know about sample spaces to choose the best option?

Focus on math practices
Make Sense and Persevere Suppose an Option 3 was added to the game, with the conditions that the game wheel lands on Q, the coin lands on either side, and an odd number is rolled. Should Talia change her choice to Option 3? Explain.

Essential Question
How can a model help find the probability of a compound event?

Try It!
The designer of Flip ‘n’ Spin creates a new game using a 5-section spinner, as shown. How does the new spinner change the probability of winning a prize?
Using the 5-section spinner, the probability of winning a prize is ________ .
It is _________ likely that a player will win a prize when using the 5-section spinner than when using the 4-section spinner.

Answer:
It is more likely that a player will win a prize when using the 5-section spinner than when using the 4-section spinner.

Explanation:
In the above-given question,
given that,
The designer of Flip ‘n’ Spin creates a new game using a 5-section spinner, as shown.
if it is a 4-section spinner then it gives n/4.
if it is a 5 section spinner then it gives n/5.
so it is more likely that a player will win a prize when using the 5-section spinner than when using the 4-section spinner.

Convince Me! What generalization can you make about the number of sections on the spinner and the probability of winning a prize while playing the Flip ‘n’ Spin game?

Try It!
Is it more likely that a coin flipped 3 times will land heads up exactly once, or will land heads up exactly 2 twice? Explain using probability.

Answer:
Yes, two heads give tails and two tails give one head.

Explanation:
In the above-given question,
given that,
two heads give one tail.
two tails give one head.
so two heads give tails and two tails give one head.

Try It!
Does Marc have a greater chance than Carly of winning the tickets to Carly? Explain using probability.
Answer:

KEY CONCEPT
The probability of a compound event can be represented by a ratio of the number of favorable outcomes to the total number of possible equally likely outcomes. You can use an organized list, a table, or a tree diagram to determine the number of favorable outcomes and the total number of possible outcomes.

Do You Understand?
Question 1.
Essential Question How can a model help find the probability of a compound event?

Answer:
The probability of a compound event can be represented by a ratio of the number of favorable outcomes to the total number of possible equally likely outcomes.

Explanation:
In the above-given question,
given that,
The probability of a compound event can be represented by a ratio of the number of favorable outcomes to the total number of possible equally likely outcomes.
we can organize an organized list, a table, or a tree diagram to determine the number of favorable outcomes and the total number of possible outcomes.

Question 2.
Generalize What do you know about the outcomes of a compound event displayed in an organized list, a table, or a tree diagram?

Answer:
The probability of a compound event can be represented by a ratio of the number of favorable outcomes to the total number of possible equally likely outcomes.

Explanation:
In the above-given question,
given that,
The probability of a compound event can be represented by a ratio of the number of favorable outcomes to the total number of possible equally likely outcomes.
we can organize an organized list, a table, or a tree diagram to determine the number of favorable outcomes and the total number of possible outcomes.

Question 3.
How does finding the probability of a compound event compare with finding the probability of a simple event?

Answer:
The probability of a compound event can be represented by a ratio of the number of favorable outcomes to the total number of possible equally likely outcomes.

Explanation:
In the above-given question,
given that,
The probability of a compound event can be represented by a ratio of the number of favorable outcomes to the total number of possible equally likely outcomes.
we can organize an organized list, a table, or a tree diagram to determine the number of favorable outcomes and the total number of possible outcomes.

Do You Know How?
Question 4.
One of three contestants will be randomly selected to win a prize. One of three different prizes will be randomly awarded to the contestant whose name is selected to win. The tree diagram shows all possible outcomes of this contest.

What is the probability that Whitney will win Prize 2?

Answer:
The probability that Whitney will win prize 2 is 3.

Explanation:
In the above-given question,
given that,
One of three contestants will be randomly selected to win a prize.
One of three different prizes will be randomly awarded to the contestant whose name is selected to win.
the contestants are Pedro, Whitney, and Bryan.
Pedro wins 3 prizes.
Whitney wins 3 prizes.
Bryan wins 3 prizes.
so the probability that Whitney will win prize 2 is 3.

Question 5.
The table shows all the possible outcomes for flipping a coin and spinning the pointer of a spinner with four equal-sized sections labeled 1 through 4.

a. What is the probability that the pointer will stop on 3 and the coin will land on heads?

Answer:
The probability that the pointer will stop on 3 and the coin will land on heads = 6.

Explanation:
In the above-given question,
given that,
there are two combinations.
they are heads and tails.
heads 1 = 1 + 1 = 2.
heads 2 = 2 + 2 = 4.
heads 3 = 3 + 3 = 6.
heads 4 = 4 + 4 = 8.
so the probability that the pointer will stop on 3 and the coin will land on heads = 6.

b. What is the probability that the pointer will stop on an odd number and the coin will land on heads?

Answer:
The probability that the pointer will stop on an odd number and the coin will land on heads = 3.

Explanation:
In the above-given question,
given that,
there are two combinations.
they are heads and tails.
heads 1 = 1 + 1 = 2.
heads 2 = 2 + 2 = 4.
heads 3 = 3 + 3 = 6.
heads 4 = 4 + 4 = 8.
Practice & Problem Solving

Multimedia Leveled Practice in 6 and 7, find the probability of each event.
Question 6.
A fair coin is tossed twice in succession. The sample space is shown, where H represents heads up and T represents tails up. Find the probability of getting exactly one tail.

There are _________ outcomes that have exactly one tail. There are _________ possible outcomes, which are equally likely.
P(exactly one tail) = __________, or _________ %

Answer:
P(exactly one tail) = 2 or 2%.

Explanation:
In the above-given question,
given that,
A fair coin is tossed twice in succession.
The sample space is shown, where H represents heads up and T represents tails up.
toss1 and toss2.
in toss1 there are 2 heads, 2 tails.
p(exactly one tail) = 2 or 2%.

Question 7.
The tree diagram shows the sample space of two-digit numbers that can be created using the digits 2, 6, 7, and 9. What is the probability of choosing a number from the sample space that contains both 9 and 6?

There are _________ outcomes that include both 9 and 6. There are _________ possible outcomes, which are equally likely
P(9 and 6) = __________, or __________ %

Answer:
The probability of choosing a number from the sample space that contains both 9 and 6 = 4.

Explanation:
In the above-given question,
given that,
The tree diagram shows the sample space of two-digit numbers that can be created using the digits 2, 6, 7, and 9.
the probability of choosing a number from the sample space that contains both 9 and 6:
in 1st diagram, there is 1 combination.
in the 2nd diagram, there is 1 combination.
in the 3rd diagram, there is 1 combination.
in the 4th diagram, there is 1 combination.
so the probability of choosing a number from the sample space that contains both 9 and 6 = 4.

Question 8.
The table shows the possible outcomes of spinning the given spinner and flipping a fair coin. Find the probability of the coin landing heads up and the pointer landing on either 1, 2, or 4.

Answer:
The probability of the coin landing heads up and the pointer landing on 1.

Explanation:
In the above-given question,
given that,
The table shows the possible outcomes of spinning the given spinner and flipping a fair coin.
on coin 1 the pointer landing up.
so the probability of the coin landing heads up and the pointer landing on 1.

Question 9.
The organized list shows all the possible outcomes when three fair coins are flipped. The possible outcomes of each flip are heads (H) and tails (T). What is the probability that at least 2 fair coins land heads up when 3 are flipped?
Sample Space
HHT
HTH
HTT
THH
THT
ΤΤΗ
TTT

Answer:
HHT, HTH, THH.

Explanation:
In the above-given question,
given that,
The organized list shows all the possible outcomes when three fair coins are flipped.
The possible outcomes of each flip are heads (H) and tails (T).
HTH, HTH, and THH.

Question 10.
Look for Relationships Gary spins two game wheels at the carnival. He will win a prize if both of the wheels land on any red section. How does the chance of winning change if different game wheels are used with more sections that aren’t red?

Answer:

Question 11.
Model with Math Each week, a clothing store gives away a shirt to a lucky customer. The shirts vary by sleeve type (Long, Short, No Sleeve) and color (Gray, Blue, Pink). Draw a tree diagram to represent the sample space. What is the probability that the free shirt will have either long or short sleeves and be either pink or blue?

Answer:
The probability that the free shirt will have either long or short sleeves and be either pink, blue, and gray.

Explanation:
In the above-given question,
given that,
Each week, a clothing store gives away a shirt to a lucky customer.
The shirts vary by sleeve type (Long, Short, No Sleeve) and color (Gray, Blue, Pink).
the gray color shirt has a sleeve type and no sleeve.
the blue color shirt has a sleeve type and no sleeve.
the pink color shirt has a sleeve type and no sleeve.
so the probability that the free shirt will have either long or short sleeves and be either pink, blue, and gray.

Question 12.
Higher Order Thinking The table shows the sample space of picking a 2-character password using the letters Y, B, R, O, G, and P. If double letters are not allowed, what is the probability of choosing a password with no Y’s? With no O’s? Is one probability greater than the other? Explain.

Answer:
The probability of choosing a password with no y’s = 20.

Explanation:
In the above-given question,
given that,
The table shows the sample space of picking a 2-character password using the letters Y, B, R, O, G, and P.
there is a number of possibilities of choosing a password with no y’s.
there is a number of possibilities of choosing a password with no o’s.
so the probability of choosing a password with no y’s = 20.

Assessment Practice
Question 13.
A single number cube is rolled twice.
PART A
Determine the number of possible outcomes. Explain how you know you have found all the possible outcomes.

PART B
Find the probability of rolling two numbers that have a sum equal to 10.
Answer:

### Lesson 7.7 Simulate Compound Events

Solve & Discuss It!
Jillian lands the beanbag on the board in about half of her attempts in a beanbag toss game. How can she predict the number of times she will get the beanbag in the hole in her next 5 attempts using a coin toss?
I can… simulate a compound event to approximate its probability.

Make Sense and Persevere
How can you use what you know about the theoretical probability of landing heads-up or tails-up?

Focus on math practices
Use Appropriate Tools When might it be useful to model a scenario with a coin or other tool?

Essential Question
How can you use simulations to determine the probability of events?

Try It!
There is a 50% chance that a volleyball team will win any one of its four remaining games this year. A spinner with 2 equal sections numbered 1 (win) and 2 (loss) is used to simulate the probability that the team will win exactly two of its last four games. The results of the simulation are shown below.

1221 1121 2211 2121 2221 2212 1122 1111 1222 1112
Out of 10 trials, there are _________ favorable outcomes. Based on the simulation, the probability that the team will win exactly 2 of its last 4 games is ____________.

Answer:
Out of 10 trials, there are 10 favorable outcomes.

Explanation:
In the above-given question,
given that,
There is a 50% chance that a volleyball team will win any one of its four remaining games this year.
A spinner with 2 equal sections numbered 1 (win) and 2 (loss).
1111 is the favorable outcome.
so out of 10 trials, there are 10 favorable outcomes.

Convince Me! Does the probability that the team will win two games change when “exactly” is replaced with “at least”? Explain.

Try It!
In a tennis tournament, 25% of Sarah’s serves were aces. Design a simulation to predict how many aces you expect Sarah to serve out of 50 serves.

Answer:
The number of aces we expect Sarah to serve out of 50 serves = 2%.

Explanation:
In the above-given question,
given that,
In a tennis tournament, 25% of Sarah’s serves were aces.
50/25 = 2.
so the number of aces we expect Sarah to serve out of 50 serves = 2%.

KEY CONCEPT
A simulation is a model of a real-world situation that can be used to predict results or outcomes when the actual event is difficult to perform or record.
A simulation uses a tool, such as a spinner, number cube, coin, or random number generator, for which outcomes have the same probabilities as the actual event.
A greater number of trials will usually give results that are closer to the theoretical probability of the actual event.

Do You Understand?
Question 1.
Essential Question How can you use simulations to determine the probability of events?

Answer:
The simulation is a model of a real-world situation that can be used to predict results or outcomes when the actual event is difficult to perform or record.

Explanation:
In the above-given question,
given that,
A simulation uses a tool, such as a spinner, number cube, coin, or random number generator, for which outcomes have the same probabilities as the actual event.
A greater number of trials will usually give results that are closer to the theoretical probability of the actual event.
so simulation is a model of a real-world situation that can be used to predict results or outcomes when the actual event is difficult to perform or record.

Question 2.
Look for Relationships What is the connection between the tool used to simulate an event and the probability of the actual event?

Answer:
The simulation is a model of a real-world situation that can be used to predict results or outcomes when the actual event is difficult to perform or record.

Explanation:
In the above-given question,
given that,
A simulation uses a tool, such as a spinner, number cube, coin, or random number generator, for which outcomes have the same probabilities as the actual event.
A greater number of trials will usually give results that are closer to the theoretical probability of the actual event.
so simulation is a model of a real-world situation that can be used to predict results or outcomes when the actual event is difficult to perform or record.

Question 3.
Why are the results of simulations usually close to the probabilities of their related events?

Answer:
The simulation is a model of a real-world situation that can be used to predict results or outcomes when the actual event is difficult to perform or record.

Explanation:
In the above-given question,
given that,
A simulation uses a tool, such as a spinner, number cube, coin, or random number generator, for which outcomes have the same probabilities as the actual event.
A greater number of trials will usually give results that are closer to the theoretical probability of the actual event.
so simulation is a model of a real-world situation that can be used to predict results or outcomes when the actual event is difficult to perform or record.

Do You Know How?
Question 4.
Carl hits the target 50% of the time he throws a ball at it. Carl uses a coin to simulate his next three pitches. He assigns H for a hit and T for a miss. The results of 12 trials are shown below.
HHT HTH TTH HTT THT THH
HHT HTT HTH HTT TTH THT
Based on the results, what is the probability that Carl will hit the target with exactly two of his next three throws?

Answer:
The probability that carl will hit the target with exactly two of his next three throws = TTH and HHT.

Explanation:
In the above-given question,
given that,
Carl hits the target 50% of the time he throws a ball at it.
Carl uses a coin to simulate his next three pitches.
He assigns H for a hit and T for a miss.
the missing trials are TTH and HHT.

Question 5.
On average, Margo scores a goal for her field hockey team every 2 out of 3 shots. Margo uses a number cube to simulate her next three shots. She assigns 1 to 4 as “goals” and 5 and 6 as “missed shots.” Why does this assignment of numbers on the number cube make it a valid simulation?

Answer:
Yes, it is a valid simulation.

Explanation:
In the above-given question,
given that,
On average, Margo scores a goal for her field hockey team every 2 out of 3 shots.
Margo uses a number cube to simulate her next three shots.
She assigns 1 to 4 as “goals” and 5 and 6 as “missed shots.
so it is a valid simulation.

Practice & Problem Solving

Leveled Practice In 6 and 7, estimate the probability for each event.
Question 6.
Molly makes 70% of her free throws. The random numbers below represent 20 trials of a simulation of two free throws, using the numbers 0 through 9.

Let the numbers from to represent successful free throw.
Let the number
Let the numbers from to represent a missed free throw.
Based on the simulated results, the probability that Molly makes both free throws is

Answer:
The numbers from 3 to 21 represent a successful free throw.
The numbers from 27 to 93 represent missed free throws.

Explanation:
In the above-given question,
given that,
Molly makes 70% of her free throws.
The random numbers below represent 20 trials of a simulation of two free throws, using the numbers 0 through 9.
the numbers are 38, 38, 21, 50, 64, 71, 80, 87, 66, 92, 89, 42, 29, 89, 47, 98, 93, 90, 27, and 3.
so the numbers from 3 to 21 represent successful free throws.
the numbers from 27 to 93 represent missed free throws.

Question 7.
Survey results state that 80% of people enjoy going to the beach. The random numbers below represent 10 trials of a simulation of asking two people if they enjoy going to the beach, using the numbers 0 through 9 for their responses.

Let the numbers from to represent people who enjoy going to the beach.
Let the numbers from to represent people who do not enjoy going to the beach.
Based on the simulated results, the probability that exactly one of two people enjoys going to the beach is , or %.

Answer:
The probability that exactly one of two people enjoys going to the beach is 80%.

Explanation:
In the above-given question,
given that,
Survey results state that 80% of people enjoy going to the beach.
the numbers are 86, 53, 54, 07, 22, 65, 9, 56, 40, and 15.
the numbers from 65 to 86 represent people who enjoy going to the beach.
the numbers from 7 to 56 represent people who do not enjoy going to the beach.

Question 8.
In Stacia’s town, 60% of registered people vote regularly. A spinner with equal-sized sections numbered 0 to 9 can be used to represent those who do and do not vote.
a. What numbers can be assigned to represent those who do vote and those who do not vote?
b. Based on the simulated results below, what is the probability that at least one person out of three does not vote?

Answer:
The numbers 799 and 851 do not vote.
the numbers 117 to 768 do not vote.

Explanation:
In the above-given question,
given that,
In Stacia’s town, 60% of registered people vote regularly.
A spinner with equal-sized sections numbered 0 to 9 can be used to represent those who do and do not vote.
the numbers are 380, 799, 331, 205, 851, 182, 117, 768, 715 and 410.
the numbers 799 and 851 do not vote.
the numbers 117 to 768 do not vote.

Question 9.
Inspection of items at a company shows that an item has a 50% chance of being defective. A spinner with equal-sized sections numbered 0 to 9 can be used to simulate the event that the next 2 items inspected are defective.

a. How would you assign numbers to represent the defective and non-defective items?
b. Based on the simulated results below, what is the probability that the next 2 items are defective?

Answer:

Question 10.
Julie used a number cube to simulate a flower seed sprouting, for which the success rate is 50%. She used even numbers to represent success and odd numbers to represent failure. The results of 8 trials that simulate the sprouting of five seeds are shown below.

Based on the simulated results, what is the probability that none of the next five flower seeds will sprout successfully?

Answer:
The next five flower seeds will sprout successfully is 43631, 25143, 25643, and 64133.

Explanation:
In the above-given question,
given that,
Julie used a number cube to simulate a flower seed sprouting, for which the success rate is 50%.
She used even numbers to represent success and odd numbers to represent failure.
the given numbers are 31534, 35635, 43631, 35633, 25143, 25643, 64133, and 53113.
so the next five flower seeds will sprout successfully is 43631, 25143, 25643, 64133.

Question 11.
Construct Arguments How is the difference between the simulated probability and the theoretical probability of an actual event related to the number of simulated trials conducted?
Answer:

Question 12.
Higher Order Thinking Suppose Arun has an 80% chance of winning a game. For a simulation, the numbers 0 to 7 represent winning, and the numbers 8 and 9 represent losing. Write three different trial results that show 5 wins in a row out of 6 games played.

Answer:
The numbers are 1, 2, 3, 4, 5, and 6.

Explanation:
In the above-given question,
given that,
Arun has an 80% chance of winning a game.
For a simulation, the numbers 0 to 7 represent winning, and the numbers 8 and 9 represent losing.
3, 4, 5, 6, 7.
1, 2, 3, 4, 5.
2, 3, 4, 5, 6.
so the numbers are 1, 2, 3, 4, 5, and 6.

Assessment Practice
Question 13.
About 50% of the people surveyed in a certain county work for a small business. A random number generator was used to simulate the results of the next four people surveyed.
The numbers 0 to 4 represent people who work for a small business, and the numbers 5 to 9 represent people who do not work for a small business.

Answer:
The numbers who work for a small business are 0501, 0403, 3074, 2235, 0803, 3750, 1288, 3154.
the numbers who do not work for a small business are 6411, 7582, 7383, 5250, 7694, 9225, 7121, 7596, 8223, 8121, and 7652.

Explanation:
In the above-given question,
given that,
About 50% of the people surveyed in a certain county work for a small business.
A random number generator was used to simulate the results of the next four people surveyed.
The numbers 0 to 4 represent people who work for a small business, and the numbers 5 to 9 represent people who do not work for a small business.
so the numbers who work for a small business are 0501, 0403, 3074, 2235, 0803, 3750, 1288, 3154.
the numbers who do not work for a small business are 6411, 7582, 7383, 5250, 7694, 9225, 7121, 7596, 8223, 8121, and 7652.

PART A

Based on the simulated results shown above, what is the probability that at least one of the next four people surveyed works for a small business?

Answer:
The people who surveyed works for a small business = 5250.

Explanation:
In the above-given question,
given that,
About 50% of the people surveyed in a certain county work for a small business.
A random number generator was used to simulate the results of the next four people surveyed.
The numbers 0 to 4 represent people who work for a small business, and the numbers 5 to 9 represent people who do not work for a small business.
so the people who surveyed works for a small business = 5250.

PART B
How would the design of the simulation change if the percent of people who work for a small business was 70%?

Answer:
The number of people who work for a small business was 70% is 6411, 7582, 7383, 5250, 7694, 9225, 7121, 7596, 8223, 8121, and 7652.

Explanation:
In the above-given question,
given that,
About 70% of the people surveyed in a certain county work for a small business.
A random number generator was used to simulate the results of the next four people surveyed.
The numbers 0 to 4 represent people who work for a small business, and the numbers 5 to 9 represent people who do not work for a small business.
so the people who work for a small business were 6411, 7582, 7383, 5250, 7694, 9225, 7121, 7596, 8223, 8121, and 7652.

### Topic 7 REVIEW

Topic Essential Question

How can you investigate chance processes and develop, use, and evaluate probability models?

Vocabulary Review

Complete each definition, and then provide an example of each vocabulary word.

Answer:
The ratio of the number of times an event occurs to the total number of trials is the probability.
the set of all possible outcomes is the sample space.
a model of a real-world situation that is used to find probabilities is a(n) is simulation.
a single outcome or a group of outcomes is a(n) is an elementary event.

Explanation:
In the above-given question,
given that,
The ratio of the number of times an event occurs to the total number of trials is the probability.
for example:
p(red) = 7/12.
the set of all possible outcomes is the sample space.
for example:
if the set is finite = {H, T}.
a model of a real-world situation that is used to find probabilities is a(n) is simulation.
for example:
dice.
a single outcome or a group of outcomes is a(n) is an elementary event.
for example:
sample space for a pair of dice.

Use Vocabulary in Writing
A restaurant serves either skim milk or whole milk in glasses that are either small, medium, or large. Use vocabulary words to explain how you could determine all the possible outcomes of milk choices at the restaurant. Use vocabulary words in your explanation.

Concepts and Skills Review

Lesson 7.1 Understand Likelihood and Probability

Quick Review
The probability of an event describes the likelihood an event will occur. The likelihood of an event ranges from impossible to certain, but more common descriptions are likely or unlikely. An event is a single outcome or group of outcomes. An outcome is a possible result of an action. Probability can be represented as a fraction, as a decimal, or as a percent. Something is fair if there is an equal chance for each outcome to occur.

Practice

Question 1.
Use Luke’s 10-sided solid from the example. Describe an event that is certain and one that is impossible using this solid.

Answer:
The event describes the likelihood an event will occur.

Explanation:
In the above-given question,
given that,
Luke’s 10-sided solid from the example.
the event describes the likelihood an event will occur.

Question 2.
A spinner with 8 equal-sized sections is used for a game. Based on the descriptions below, is the spinner fair? Explain. The probability the pointer will land on yellow is 1 out of 4.
The probability the pointer will land on blue is 2 out of 8.
The probabilities that the pointer will land on green or red are both 25%.

Answer:
Yes, the spinner is fair.

Explanation:
In the above-given question,
given that,
A spinner with 8 equal-sized sections is used for a game.
the probability the pointer will land on blue is 2 out of 8.
the probability the pointer will land on yellow is 1 out of 4.
2/8 = 1/4.
so the spinner is fair.

Lesson 7.2 Understand Theoretical Probability

Quick Review

The theoretical probability of an event can be found if the possible outcomes are known and are all equally likely. Theoretical probability can be used to make predictions.

Practice

Question 1.
Using the example, fifteen people take out a slip of paper from the bucket without looking and record the results before replacing the slip back into the bucket. How many times is a slip labeled “5” expected to be drawn?

Answer:
The number of times is a slip labeled 5 is 3.

Explanation:
In the above-given question,
given that,
fifteen people take out a slip of paper from the bucket without looking.
15/3 = 5.
so the number of times is a slip labeled 5 is 3.

Lesson 7.3 Understand Experimental Probability

Quick Review
The experimental probability or relative frequency is based on actual results from an experiment and may differ from the theoretical probability of an event occurring. This discrepancy decreases as the number of trials of an experiment increases. You can use experimental probability and proportional reasoning to make predictions.

Practice

Question 1.
Jaylon and Paula spin the pointer 30 times and get the results shown in the table.

What is the theoretical probability of the pointer landing on the number 2?
Based on the results in the table, how does the experimental probability of the pointer landing on 2 compare to the theoretical probability?

Answer:
The theoretical probability of the pointer landing on the number 2 is 13.
the experimental probability of the pointer landing on the number 2 is 13.

Explanation:
In the above-given question,
given that,
Jaylon and Paula spin the pointer 30 times and get the results shown in the table.
Jaylon gets in 1st spin 6.
2nd spin 8, 3rd spin 9, and 4th spin 7.
Paula gets in 1st spin 8, 2nd spin is 5, 3rd spin is 7, and 4th spin is 10.

Question 2.
Based on the results in the table, about how many times should Jaylon and Paula expect the pointer to land on 4 out of a total of 130 spins? Explain your answer.

Answer:
The pointer to land on 4 out of a total of 130 is 0.30.

Explanation:
In the above-given question,
given that,
Jaylon and Paula expect the pointer to land on 4 out of a total of 130 spins.
4/130 = 0.30.
so the pointer to land on 4 out of a total of 130 is 0.30.

Lesson 7.4 Use Probability Models

Quick Review
A probability model consists of a sample space, or all possible outcomes of an action, and a list of events within the sample space with the probability of each. The sum of the probabilities in the model is 1. A probability model can be used to make conclusions about probabilities of events or to make estimates or predictions.

Practice

Question 1.
Abe has a different spinner. He also wants to develop a probability model.
How will his probability model be the same as, and how will it differ from, Jenna’s model?

Question 2.
Walter has a different spinner.

What is the probability that the pointer will land on a color that is not red?
Answer:

Question 3.
What is the sample space of Walter’s spinner?
Answer:

Question 4.
Walter will spin the pointer 50 times. About how many times will the pointer land on each color?
Answer:

Lesson 7.5 Determine Outcomes of Compound Events

Quick Review
A compound event is a combination of two or more events. An organized list, table, or tree diagram can be used to represent the sample space of a compound event.

Practice

Question 1.
A basket contains a red, a yellow, and a green apple. A second basket contains an orange, a lemon, and a peach. Use an organized list to show all the outcomes in the sample space.

Answer:
The organized list to show all the outcomes in the sample space is 1.

Explanation:
In the above-given question,
given that,
A basket contains a red, a yellow, and a green apple.
A second basket contains an orange, a lemon, and a peach.
p(3/3) = 1.
so the organized list to show all the outcomes in the sample space is 1.

Question 2.
Simon is playing a game with letter tiles. He has 5 tiles remaining and will spell a new word by placing two tiles-first a consonant and then a vowel-in front of a Y already on the board. Complete the table below to describe all combinations of tiles that Simon can use to spell a new word.

Lesson 7.6 Find Probabilities of Compound Events

Quick Review

The probability of a compound event can be represented by a ratio of the favorable outcomes to all possible outcomes. The probability can be calculated using an organized list, a table, or a tree diagram.

Practice

Question 1.
One set of cards has a beach, a road, a desert, a mountain, and an island. A second set of cards has a car, a truck, and a van. Complete the table below to find the probability of randomly drawing a mountain card and a truck card.

Answer:
Beach = 1, 2, and 3.
Road = 4, 5, and 6.
Desert = 7, 8, and 9.
Mountain = 10, 11, and 12.
Island = 13, 14, and 15.

Explanation:
In the above-given question,
given that,
One set of cards has a beach, a road, a desert, a mountain, and an island.
The second set of cards has a car, a truck, and a van.
Beach = 1, 2, and 3.
Road = 4, 5, and 6.
Desert = 7, 8, and 9.
Mountain = 10, 11, and 12.
Island = 13, 14, and 15.

Lesson 7.7 Simulate Compound Events

Quick Review

An actual event is sometimes difficult to perform or record. A simulation can be used to model the outcomes of a real-world event. Based on simulated results, you can approximate the probability and predict the future outcomes of an event.

Practice

Question 1.
Felix’s favorite cereal includes 1 of 3 different prizes inside each box. The chance of getting each prize is equally likely. Felix conducts a simulation to see what his chances are of collecting all 3 prizes if he buys 5 boxes over time. Each section of the spinner represents the possible prizes in a single box.

Based on the simulation, what is the probability that Felix will collect all three prizes?

Answer:
The spinner represents the possible prizes in a single box = GBGBG, BGGYG, BGBYB, GYBYY, and YYGYG.

Explanation:
In the above-given question,
given that,
Felix’s favorite cereal includes 1 of 3 different prizes inside each box.
The chance of getting each prize is equally likely.
Felix conducts a simulation to see what his chances are of collecting all 3 prizes if he buys 5 boxes over time.
The spinner represents the possible prizes in a single box = GBGBG, BGGYG, BGBYB, GYBYY, and YYGYG.

Question 2.
Reece is playing a carnival game in which he must guess under which of 2 cups a ball is hidden. To simulate the results of this game, he flips a coin with heads up (H) representing wins and tails up (T) representing losses. Based on the simulation below, what is the probability that Reece will win at least 2 of his next 4 games?

Answer:
The probability of representing the win of games is p(2) are HHTH and HHHT.

Explanation:
In the above-given question,
given that,
Reece is playing a carnival game in which he must guess under which of 2 cups a ball is hidden.
to simulate the results of this game, he flips a coin with heads up(H) representing wins and tails up(T) representing losses.
HH HT and HHTH represent the win of games.
so the probability of representing the win of games is p(2).

### Topic 7 Fluency Practice

Hidden Clue

For each ordered pair, one coordinate is given. Find the second coordinate by determining the sale price after the percent markup or markdown. Then locate and label the corresponding point on the graph. Draw line segments to connect the points in alphabetical order. Use the completed picture to help you answer the riddle below.

A (13, 25% markup on $4). 13, B (30% markdown on$10, 1) , 1
C (20% markdown on $1.25, 5) , 5 D (6, 40% markdown on$5.10) 6,
E (35% markup on $4, 9) , 9 F (4, 50% markdown on$18.50) 4,
G (60% markup on $2.50, 11) , 11 H (6, 25% markdown on$15) 6,
I (60% markup on $4, 13) , 13 J (8, 30% markdown on$18) 8,
K (50% markup on $5.30, 11) , 11 L (10,45% markdown on$20) 10,
M (35% markup on $7.60, 9) , 9 N (8, 30% markdown on$13) 8,

Answer:
A.13, 1.
B. 30, 1.
C. 0.25, 5.
D. 2.04, 6.
E. 1.4, 9.
F. 9.25, 4.
G. 1.5, 11.
H. 3.75, 6.
I. 2.4, 13.
J. 5.4, 8.
K. 2.65, 11.
L. 9, 10.
M. 2.66, 9.
N. 3.9, 8.

Explanation:
In the above-given question,
given that,
25/100 = 0.25 x 4 = 1.
30/100 = 0.3 x 10 = 3.
20/100 = 0.2 x 1.25 = 0.25.
40/100 = 0.4 x 5.10 = 2.04.
35/100 = 0.35 x 4 = 1.4.
50/100 = 0.5 x 18.50 = 9.25.
60/100 = 0.6 x 2.50 = 1.5.
25/100 = 0.25 x 15 = 3.75.
60/100 = 0.6 x 4 = 2.4.
30/100 = 0.3 x 5.18 = 5.4.
50/100 = 0.5 x 5.30 = 2.65.
45/100 = 0.4 x 20 = 8.
35/100 = 0.35 x 7.60 = 2.66.

## Envision Math Common Core 5th Grade Answers Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers

Essential Question: What are the standard procedures for estimating and finding products of multi-digit numbers?

enVision STEM Project: Water Usage
Do Research Use the Internet or other sources to find how much water is used for household activities like taking a shower or bath, using a dishwasher, hand washing dishes, and using a washing machine.
Journal: Write a Report Include what you found. Also in your report:

• Choose 3 of the activities. Estimate how many times each activity is done each week in your household.
• Estimate the weekly water usage for each activity. Organize your results in a table.
• Make up and solve multiplication problems based on your data.

Review What You Know

Vocabulary

Choose the best term from the box. Write it on the blank.
• multiple
• equation
• exponent
• power
• factor
• product

Question 1.
The answer to a multiplication problem is the ____.

Answer:
The answer to a multiplication problem is the product.

Explanation:
In the above-given question,
given that,
the answer to a multiplication problem is called the product.
for example:
15 x 2 = 30.
15 is called multiplicand.
2 is the multiplier.
30 is the product.
So the answer to a multiplication problem is called the product.

Question 2.
A number sentence that shows two expressions with the same value is a(n) _____

Answer:
A number sentence that shows two expressions with the same value is an equation.

Explanation:
In the above-given question,
given that,
A number sentence that shows two expressions with the same value is an equation.
for example:
4 + 8 = 12.
5 + 6 = 12.
so the number sentence that shows two expressions with the same value is an equation.

Queen 3.
A(n) ___ tells the number of times the base is used as a(n) ___.

Answer:
A(n) tells the number of times the base is used as an exponent.

Explanation:
In the above-given question,
given that,
A(n) ___ tells the number of times the base is used as a(n).
for example:
5².
where 2 is the exponent.
5 is base.

Question 4.
50 is a(n) ____ of 10 because 5 × 10 = 50.

Answer:
50 is a(n) base of 10 because 5 x 10 = 50.

Explanation:
In the above-given question,
given that,
50 is a(n) base of 10 because 5 x 10 = 50.
for example:
5 x a (n) = 10.
a(n) = 10 x 5.
a(n) = 50.

Operations

Find each sum or difference.

Question 5.
9,007 + 3,128

Answer:
9007 + 3128 = 12,135.

Explanation:
In the above-given question,
given that,
the two numbers are 9007 and 3128.
add the two numbers.
9007 + 3128 = 12,135.

Question 6.
7,904 – 3,199

Answer:
7904 – 3199 = 4705.

Explanation:
In the above-given question,
given that,
the two numbers are 7904 and 3199.
subtract the two numbers.
7904 – 3199 = 4705.

Question 7.
27,924 – 13,868

Answer:
27924 – 13,868 = 14,056.

Explanation:
In the above-given question,
given that,
the two numbers are 27924 and 13868.
subtract the two numbers.
27924 – 13868 =14,056.

Question 8.
9.27 + 3.128

Answer:
9.27 + 3.128 = 12.398.

Explanation:
In the above-given question,
given that,
the two numbers are 9.27 and 3.128.
add the two numbers.
9.27 + 3.128 = 12.398.

Question 9.
119.04 – 86.5

Answer:
119.04 – 86.5 = 32.54.

Explanation:
In the above-given question,
given that,
the two numbers are 119.04 and 86.5.
subtract the two numbers.
119.04 – 86.5 = 32.54.

Question 10.
165.2 – 133.18

Answer:
165.2 – 133.18 = 32.02.

Explanation:
In the above-given question,
given that,
the two numbers are 165.2 and 133.18.
subtract the two numbers.
165.2 – 133.18 = 32.02.

Round Whole Numbers and Decimals

Round each number to the place of the underlined digit.

Question 11.
14.3

Answer:
14.3.

Explanation:
In the above-given question,
given that,
the number is 14.3.
the underlined digit is 4.
if the tenths place is greater than 5.
then add by 1.
if tenths place is less than 5.
then write the same number as zeros.
so the answer is 14.3.

Question 12.
385.7

Answer:
395.7.

Explanation:
In the above-given question,
given that,
the number is 385.7.
the underlined digit is 8.
if the tenths place is greater than 5.
then add by 1.
if tenths place is less than 5.
then write the same number as zeros.
so the answer is 395.7.

Question 13.
0.545

Answer:
0.500.

Explanation:
In the above-given question,
given that,
the number is 0.545.
the underlined digit is 5.
if the tenths place is greater than 5.
then add by 1.
if tenths place is less than 5.
then write the same number as zeros.
so the answer is 0.500.

Question 14.
496.533

Answer:
497.533.

Explanation:
In the above-given question,
given that,
the number is 496.533.
the underlined digit is 6.
if the tenths place is greater than 5.
then add by 1.
if tenths place is less than 5.
then write the same number as zeros.
so the answer is 496.533.

Question 15.
496.353

Answer:
496.000.

Explanation:
In the above-given question,
given that,
the number is 496.353.
the underlined digit is 6.
if the tenths place is greater than 5.
then add by 1.
if tenths place is less than 5.
then write the same number as zeros.
so the answer is 496.000.

Question 16.
1,857.205

Answer:
1857.215.

Explanation:
In the above-given question,
given that,
the number is 1857.205.
the underlined digit is 0.
if the tenths place is greater than 5.
then add by 1.
if tenths place is less than 5.
then write the same number as zeros.
so the answer is 1857.205.

Compare Decimals

Question 17.
Write the numbers in order from least to greatest. 8.062 8.26 8.026 8.6

Answer:
The numbers in order from least to greatest is 8.026, 8.062, 8.26, and 8.6.

Explanation:
In the above-given question,
given that,
the numbers are 8.062, 8.26, 8.026, and 8.6.
the numbers in order from least to greatest is
8.026, 8.062, 8.26, and 8.6.

Question 18.
Write the numbers in order from greatest to least. 0.115 0.15 0.005 0.5

Answer:
The numbers in order from greatest to least are 0.005, 0.115, 0.15, and 0.5.

Explanation:
In the above-given question,
given that,
the numbers are 0.115, 0.15, 0.005, and 0.5.
the numbers in order from greatest to least are
0.115, 0.15, 0.005, and 0.5.

pick a Project

PROJECT ЗА
What puts the bounce in a bouncy ball?
Project: Make a Business Plan

PROJECT 3B
How can you build a fort?
Project: Build a Model Fort

PROJECT 3C
How many people can a ferry carry?
Project: Design a Prototype Ferry

3-ACT MATH PREVIEW

Math Modeling

Morning Commute

Before watching the video, think:
Train conductors don’t wear this kind of hat anymore. Even paper tickets are less common now that some train lines use an app to purchase tickets. What are some other ways we have updated transportation as part of our modern society? All aboard!

### Lesson 3.1 Multiply Greater Numbers by Powers of 10

Activity

At Izzy’s Party Store, party invitations come in packages of 8. How many invitations are in 10 packages? 100 packages? 1,000 packages? Solve this problem any way you choose.

Answer:
The number of invitations is in 10 packages = 80.
the number of invitations is in 100 packages = 800.
the number of invitations is in 1000 packages = 8000.

Explanation:
In the above-given question,
given that,
At Izzy’s Party Store, party invitations come in packages of 8.
8 x 10 = 80.
8 x 100 = 800.
8 x 1000 = 8000.
so the number of invitations is in 10 packages = 80.
the number of invitations is in 100 packages = 800.
the number of invitations is in 1000 packages = 8000.

You can use appropriate tools. Place-value blocks are useful for picturing problems that involve powers of 10.

Look Back! What patterns do you notice in your work above?

Visual Learning Bridge

Essential Question
How Can You Use Patterns and Mental Math to Question Multiply a Whole Number by a Power of 10?

A.
The value of each place in a number is 10 times the value of the place to the right. The place-value chart shows this relationship for the number 4. Look for patterns.

10 times greater than 4
10 times greater than 40
10 times greater than 400
10 times greater than 4,000
10 times greater than 40,000

B.
Find 32 × 10,000 by using place-value relationships.
Multiply 32 by 1; 10; 100; 1,000; and 10,000.
32 × 1 = 32 ones = 32
32 × 10 = 32 tens = 320
32 × 100 = 32 hundreds = 3,200
32 × 1,000 = 32 thousands = 32,000
32 × 10,000 = 32 ten thousands = 320,000
Pattern
Pattern The product ends with the same number of zeros as the power of 10.

C.
Instead of using the standard form, write each power of 10 using exponents.
32 × 1 = 32 × 100 = 32
32 × 10 = 32 × 101 = 320
32 × 100 = 32 × 102 = 3,200
32 × 1,000 = 32 × 103 = 32,000
32 × 10,000 = 32 × 104 = 320,000

Pattern
The exponent tells how many additional zeros the product will end with.

Convince Me! Critique Reasoning Nellie says that 60 × 1,000 is 6,000 because there are three zeros in 1,000. Kara says that 60 × 1,000 = 60,000. Whose thinking is correct? Explain.

Answer:
Kara says is correct.

Explanation:
In the above-given question,
given that,
Nellie says that 60 × 1,000 is 6,000.
Kara says that 60 × 1,000 = 60,000.
so kara says is correct.

Guided Practice

Do You Understand?

Question 1.
How many zeros will there be in the product of 39 × 1,000? How many zeros will there be in the product of 50 × 1,000?

Answer:
There are 3 zeros in the 39000.
there are 4 zeros in the 50000.

Explanation:
In the above-given question,
given that,
39 x 1000 = 39000.
50 x 1000 = 50000.
so there are 3 zeros in the 39000.
there are 4 zeros in the 50000.

Question 2.
Explain how to find the product of 90 × 104.

Answer:
The product of 90 x 104

Explanation:
In the above-given question,
given that,
90 x 104
90 x 10 x 10 x 10 x 10.
90 x 100 x 100.
90 x 10000.
900000.

Do You Know How?

In 3-5, use reasoning to fill in the missing numbers.

Question 3.
60 × 1 = ____
60 × 100 = ____
60 × 10,000 = ____

Answer:
60 x 1 = 60.
60 x 100 = 6000.
60 x 1000 = 60000.

Explanation:
In the above-given question,
given that,
60 x 1 = 60.
60 x 100 = 6000.
60 x 1000 = 60000.

Question 4.
13 × ___ = 13,000

Answer:
13 x 1000 = 13000.

Explanation:
In the above-given question,
given that,
13 x 10 x 10 x 10.
13 x 1000 = 13000.

Question 5.
___ × 104 = 100,000

Answer:
10 × 104 = 100,000.

Explanation:
In the above-given question,
given that,
10 x 1 =10.
104 = 10 x 10 x 10 x 10 =10000.
10 x 10000 = 100,000.

Independent Practice

Leveled Practice In 6-13, find each product.

Question 6.
89 × 1
89 × 10
89 × 100
89 × 1,000
89 × 10,000

Answer:
89 x 1 = 89.
89 x 10 = 890.
89 x 100 = 8900.
89 x 1000 = 89000.
89 x 10,000 = 890,000.

Explanation:
In the above-given question,
given that,
89 x 1 = 89.
89 x 10 = 890.
89 x 100 = 8900.
89 x 1000 = 89000.
89 x 10,000 = 890,000.

Question 7.
30 × 1
30 × 10
30 × 100
30 × 1,000
30 × 10,000

Answer:
30 × 1 = 30.
30 × 10 = 300.
30 × 100 = 3000.
30 × 1,000 = 30,000.
30 × 10,000 = 300,000.

Explanation:
In the above-given question,
given that,
30 × 1 = 30.
30 × 10 = 300.
30 × 100 = 3000.
30 × 1,000 = 30,000.
30 × 10,000 = 300,000.

Question 8.
41 × 1
41 × 101
41 × 102
41 × 103
41 × 104

Answer:
41 × 1 = 41.
41 × 101
41 × 102
41 × 103
41 × 104

Explanation:
In the above-given question,
given that,
41 × 1 = 41.
41 × 101= 4100.
41 × 102 = 42000.
41 × 103= 43000.
41 × 104= 430000.

Question 9.
90 × 1
90 × 101
90 × 102
90 × 103
90 × 104

Answer:
90 x 1 = 90.
90 × 101
90 × 102
90 × 103
90 × 104

Explanation:
In the above-given question,
given that,
90 x 1 = 90.
90 × 101
90 × 102
90 × 103
90 × 104

Question 10.
4 × 103

Answer:
4 x 103 = 4000.

Explanation:
In the above-given question,
given that,
4 x 103.
4 x 10 x 10 x 10.
4 x 1000.
4000.

Question 11.
85 × 100

Answer:
85 x 100 = 8500.

Explanation:
In the above-given question,
given that,
85 x 100.
8500.

Question 12.
16 × 102

Answer:
16 x 10 x 10 = 1600.

Explanation:
In the above-given question,
given that,
16 x 102.
16 x 10 x 10.
1600.

Question 13.
103 × 38

Answer:
10 x 10 x 10 x 38 = 38000.

Explanation:
In the above-given question,
given that,
103 × 38.
10 x 10 x 10 x 38.
100 x 10 x 38.
38 x 1000.
38000.

In 14-19, use reasoning to fill in the missing numbers.

Question 14.
52 × 10- = 520,000

Answer:
52 x 10 4 = 520,000.

Explanation:
In the above-given question,
given that,
52 x 10 x 10 x 10 x 10.
52 x 100 x 100.
52 x 10000.
520,000.

Question 15.
68,637 = 101 × ___

Answer:
10 x 68637 = 686370.

Explanation:
In the above-given question,
given that,
68637 x 10.
686370.

Question 16.
___ = 382 × 104

Answer:
382 x 10000 = 3820000.

Explanation:
In the above-given question,
given that,
382 x 10 x 10 x 10 x 10.
382 x 104.
382 x 100 x 100.
3820000.

Question 17.
___ = 103 × 80

Answer:
80 x 103 = 80000.

Explanation:
In the above-given question,
given that,
80 x 103.
80 x 10 x 10 x 10.
80 x 1000.
80000.

Question 18.
10 × 374 = 37,400

Answer:
10 x 374 x 10 = 37400.

Explanation:
In the above-given question,
given that,
10 x 374 x 10.
100 x 374.
37400.

Question 19.
500,000 = 50 × 10-

Answer:
50 x 10000 = 50,000.

Explanation:
In the above-given question,
given that,
50 x 10 x 10 x 10 x 10.
50 x 100 x 100.
50 x 10000.
50,000.

Problem Solving

Question 20.
At a football championship game, the home team gave a football to each of the first 100 fans who arrived at the stadium. Each football cost the team $28. How much did the team pay for the footballs it gave away? Answer: The team pay for the footballs it gave away =$2800.

Explanation:
In the above-given question,
given that,
At a football championship game,
the home team gave a football to each of the first 100 fans who arrived at the stadium.
Each football cost the team $28. 28 x 100 = 2800. so the team pay for the football it gave away =$2800.

Question 21.
Construct Arguments Without multiplying, tell which expression is greater, 93 × 103 or 11 × 104? How do you know?

Answer:
The expression 93 x 103 is greater.

Explanation:
In the above-given question,
given that,
the two expressions are 93 × 103 or 11 × 104.
93 x 10 x 10 x 10.
93 x 1000.
93000.
11 x 10 x 10 x 10 x 10.
11 x 10000.
110000.

Question 22.
A truck is carrying 102 bushels of onions, 101 bushels of peaches, and 103 bushels of corn. What is the total weight of the crops?

Answer:
The total weight of the crops = 76,200.

Explanation:
In the above-given question,
given that,
A truck is carrying 102 bushels of onions.
101 bushels of peaches, and 103 bushels of corn.
57 x 100 = 5700.
50 x 10 = 500.
70 x 10 x 10 x 10 = 70000.
5700 + 500 + 70000 = 76,200.
so the total weight of the crops = 76200.

Question 23.
Norman bought a 16-pound bag of charcoal for $7.89 and a 10.4-pound bag of charcoal for$5.69. What was the total weight of the two bags of charcoal?

Answer:
The total weight of the two bags of charcoal = $185.416. Explanation: In the above-given question, given that, Norman bought a 16-pound bag of charcoal for$7.89.
10.4-pound bag of charcoal for $5.69. 16 x 7.89 = 126.24. 10.4 x 5.69 = 59.176. 126.24 + 59.176 = 185.416. so the total weight of the two bags of charcoal =$185.416.

Question 24.
Higher Order Thinking There are 2,000 pounds in 1 ton. In the United States, the weight limit for a truck and its cargo is 40 tons. How many pounds is that? How did you find the answer?

Answer:
The number of pounds = 80,000.

Explanation:
In the above-given question,
given that,
There are 2,000 pounds in 1 ton.
In the United States, the weight limit for a truck and its cargo is 40 tons.
2000 x 40 = 80000.
so the number of pounds = 80,000.

Assessment Practice

Question 25.
Which is equivalent to multiplying a number by 104?
A. multiplying by 40
B. multiplying by 100
C. multiplying by 1,000
D. multiplying by 10,000

Answer:
The number is equivalent to multiplying a number by 10000.

Explanation:
In the above-given question,
given that,
multiplying by 10,000.
10 x 10 x 10 x 10 = 10,000.
so the number equivalent to multiplying a number by 10000.

Question 26.
Select the statements that are equivalent to multiplying 20 × 104.
Add 10 to 20 four times.
Multiply 20 by 10 four times.
Multiply 10 by 20 four times.
Multiply 20 by 10,000.
Multiply 20 by 100,000.

Answer:
Option B is correct.

Explanation:
In the above-given question,
given that,
20 x 104.
20 x 10 x 10 x 10 x 10.
20 x 100 x 100.
200000.
so option B is correct.

### Lesson 3.2 Estimate Products

Activity

Solve & Share

Answer:

A school club wants to buy shirts for each of its 38 members. Each shirt costs $23. About how much money will all the shirts cost? Solve this problem any way you choose. Answer: The much money will all the shirts cost =$874.

Explanation:
In the above-given question,
given that,
A school club wants to buy shirts for each of its 38 members.
Each shirt costs $23. 38 x 23 = 874. so the much money will all the shirts cost =$874.

Are you asked for an exact answer or an estimate?

Look Back! Construct Arguments How can you use number sense to tell that the exact answer has to be greater than $600? Explain how you know. Visual Learning Bridge Essential Question How Can You Estimate Products? You can use rounding to estimate. A. A store needs at least$75,000 in sales per month to make a profit. If the store is open every day in March and sales average $525 per day, will the store make a profit in March? B. Use Rounding to Estimate$525 rounds to $500. 31 rounds to 30. Find 30 × 500. 30 × 500 = 15,000 You know that 3 × 5 = 15. C. Both numbers used to estimate were less than the actual numbers, so 15,000 is an underestimate. The store will actually have more than$15,000 worth of sales.
So, the store will make a profit in March.

Convince Me! Critique Reasoning A different store needs to make at least $20,000 to make a profit in March. They average$685 a day for the month. James used rounding and estimation to say, “$685 is almost$700. $700 × 30 days is$21,000. I think it is going to be a close call!” What do you think?

Answer:
$685 x 30 = 20,550. Explanation: In the above-given question, given that, A different store needs to make at least$20,000 to make a profit in March.
They average $685 a day for the month. James used rounding and estimation to say, “$685 is almost $700.$700 x 30 = $21000.$685 x 30 = 20,550.
so they can make the profit.

Another example
Estimate 24 × 398.
25 and 4 are compatible numbers because their product is easy to compute mentally.
25 × 4 = 100
25 × 40 = 1,000
25 × 400 = 10,000
So, 10,000 is a good estimate for 24 × 398.
You can also use compatible numbers to estimate.

Both numbers used to estimate were greater than the actual numbers.
So, 10,000 is an overestimate.

Guided Practice

Do You Understand?

Question 1.
Number Sense Each egg carton holds one dozen eggs. Michael’s chicken farm fills 121 egg cartons. He thinks that there were over 1,500 eggs. Is he correct? Use an estimate to find out.

Answer:
Yes, the estimation was correct.

Explanation:
In the above-given question,
given that,
Each egg carton holds one dozen eggs.
1 dozen = 12.
Michael’s chicken farm fills 121 egg cartons.
121 is near to 125.
125 x 12 = 1500.
so the estimation was correct.

Do You Know How?

In 2-5, estimate. Then, tell if your estimate is an overestimate or underestimate.

Question 2.
29 × 688

Answer:
29 x 688 = 19,952.

Explanation:
In the above-given question,
given that,
the two numbers are 29 and 688.
multiply the numbers.
29 x 688 = 19,952.

Question 3.
210 × 733

Answer:
210 x 733 = 153930.

Explanation:
In the above-given question,
given that,
the two numbers are 210 and 733.
multiply the numbers.
210 x 733 = 153930.

Question 4.
43 × 108

Answer:
43 x 108 = 4644.

Explanation:
In the above-given question,
given that,
the two numbers are 43 and 108.
multiply the numbers.
43 x 108 = 4644.

Question 5.
380 × 690

Answer:
380 x 690 = 262200.

Explanation:
In the above-given question,
given that,
the two numbers are 380 and 690.
multiply the numbers.
380 x 690 = 262200.

Independent Practice

Leveled Practice In 6-17, estimate each product.

Question 6.
180 × 586

Answer:
180 x 586 = 1,05,480.

Explanation:
In the above-given question,
given that,
the two numbers are 180 and 586.
multiply the numbers.
180 x 586 = 1,05,480.

Question 7.
300 × 118

Answer:
300 x 118 = 35400.

Explanation:
In the above-given question,
given that,
the two numbers are 300 and 118.
multiply the numbers.
300 x 118 = 35400.

Question 8.
19 × 513

Answer:
19 x 513 = 9,747.

Explanation:
In the above-given question,
given that,
the two numbers are 19 and 513.
multiply the numbers.
19 x 513 = 9,747.

Question 9.
38 × 249

Answer:
38 x 249 = 9462.

Explanation:
In the above-given question,
given that,
the two numbers are 38 and 249.
multiply the numbers.
38 x 249 = 9462.

Question 10.
11 × 803

Answer:
11 x 803 = 8833.

Explanation:
In the above-given question,
given that,
the two numbers are 11 and 803.
multiply the numbers.
11 x 803 = 8833.

Question 11.
44 × 212

Answer:
44 x 212 = 9328.

Explanation:
In the above-given question,
given that,
the two numbers are 44 and 212.
multiply the numbers.
44 x 212 = 9328.

Question 2.
790 × 397

Answer:
790 x 397 = 313630.

Explanation:
In the above-given question,
given that,
the two numbers are 790 and 397.
multiply the numbers.
790 x 397 = 313630.

Question 13.
42 × 598

Answer:
42 x 598 = 25,116.

Explanation:
In the above-given question,
given that,
the two numbers are 42 and 598.
multiply the numbers.
42 x 598 = 25,116.

Question 14.
25 × 191

Answer:
25 x 191 = 4775.

Explanation:
In the above-given question,
given that,
the two numbers are 25 and 191.
multiply the numbers.
25 x 191 = 4775.

Question 15.
408 × 676

Answer:
408 x 676 = 275808.

Explanation:
In the above-given question,
given that,
the two numbers are 408 and 676.
multiply the numbers.
408 x 676 = 275808.

Question 16.
290 × 12

Answer:
290 x 12 = 3,480.

Explanation:
In the above-given question,
given that,
the two numbers are 290 and 12.
multiply the numbers.
290 x 12 = 3,480.

Question 17.
854 × 733

Answer:
854 x 733 = 6,25,982.

Explanation:
In the above-given question,
given that,
the two numbers are 854 and 733.
multiply the numbers.
854 x 733 = 6,25,982.

Problem Solving

Question 18.
Reasoning Estimate 530 × 375. Is the estimated product closer to 150,000 or 200,000? Explain.

Answer:
The estimated product is closer to 200,000.

Explanation:
In the above-given question,
given that,
530 x 375 = 198,750.
198750 is equal to 200,000.

Question 19.
Vocabulary Is 500 an underestimate or overestimate for the product of 12 and 53?

Answer:
500 is an underestimate for the product 12 and 53.

Explanation:
In the above-given question,
given that,
12 x 53 = 636.
10 x 50 = 500.
500 is an underestimate for the product 12 and 53.

Question 20.
Samuel needs to estimate the product of 23 × 495. Explain two different methods Samuel can use to estimate.

Answer:
23 x 495 = 11,385.
25 x 500 = 12500.

Explanation:
In the above-given question,
given that,
the product of 23 and 495.
23 x 495 = 11,385.
25 x 500 = 12500.

Question 21.
Rebekah said that 103 is 30 because 10 + 10 + 10 = 30. Do you agree? Explain.

Answer:
No, I do not agree with it.

Explanation:
In the above-given question,
given that,
Rebekah said that 103 is 30 because 10 + 10 + 10 = 30.
10 + 10 + 10 = 30.
30 is no equal to 103.
so I do not agree with it.

Question 22.
Higher Order Thinking Abby counts 12 large boxes and 18 small boxes of pencils in the supply cabinet. Each large box contains 144 pencils. Each small box contains 24 pencils. Estimate the total number of pencils. Is your estimate an overestimate or an underestimate? Explain why it might be better to have an underestimate rather than an overestimate.

Answer:
The total number of pencils = 2160.

Explanation:
In the above-given question,
given that,
Abby counts 12 large boxes and 18 small boxes of pencils in the supply cabinet.
Each large box contains 144 pencils.
Each small box contains 24 pencils.
144 x 12 =1728.
24  x 18 = 432.
1728 + 432 = 2160.
so the total number of pencils = 2160.

Question 23.
Susan used rounding to estimate 24 × 413 and found 20 × 400. Jeremy used compatible numbers and found 25 × 400. Whose method gives an estimate closer to the actual product? Explain.

Is your answer reasonable?

Answer:
Jeremy used compatible numbers and found 25 x 400 = 10000.

Explanation:
In the above-given question,
given that,
Susan used rounding to estimate 24 × 413 and found 20 × 400.
Jeremy used compatible numbers and found 25 × 400.
24 x 413 = 9912.
20 x 400 = 8000.
25 x 400 = 10000.
so jeremy used compatible numbers and found 25 x 400 = 10000.

Assessment Practice

Question 24.
Lance has 102 packages of sports cards. Each package has 28 cards. Use rounding to estimate. About how many cards does Lance have?
A. 2,000
B. 2,500
C. 3,000
D. 3,500

Answer:
The number of cards does Lance has = 3000.

Explanation:
In the above-given question,
given that,
Lance has 102 packages of sports cards.
Each package has 28 cards.
102 x 28 = 2856.
2856 is near to 3000.
so the number of cards does Lance has = 3000.

Question 25.
Which does NOT show a reasonable estimate of 24 338?
A. 6,000
B. 7,000
C. 7,500
D. 10,000

Answer:
The reasonable estimate is 10,000.

Explanation:
In the above-given question,
given that,
the two numbers are 24 and 338.
338 x 24 = 8,112.
8112 is near to 10,000.
so the reasonable estimate is 10,000.

### Lesson 3.3 Multiply by 1-Digit Numbers

Activity

Solve & Share

Suppose a school ordered 7 boxes of books. There are 25 books in each box. How can you use paper and pencil to find how many books were ordered? How can you check if your answer is reasonable? Solve these problems using any strategy you choose.

Answer:
The number of books that were ordered = 175.

Explanation:
In the above-given question,
given that,
Suppose a school ordered 7 boxes of books.
There are 25 books in each box.
7 x 25 = 175.
so the number of books that were ordered = 175.

You can make sense and persevere. Formulating a plan can help you solve problems. Show your work!

Look Back! Without finding the exact answer, how do you know that the answer to the problem above is less than 210?

Visual Learning Bridge

Glossary

Essential Question
What Is a Common Way to Essential Question Record Multiplication?

A.
Ms. Stockton ordered 6 boxes of T-shirts with the school name on them. Each 20 box contains 26 T-shirts. How many T-shirts did Ms. Stockton order?

You can multiply using partial products. You can write and add the partial products in any order.

B.
One Way to Record Multiplication

C.
Another Way to Record Multiplication
You can multiply each place value in order, beginning with the ones. Regroup if needed. Add any regrouped values to each place value.
Step 1: Multiply by the ones.

Step 2: Multiply by the tens.

Mrs. Stockton ordered 156 T-shirts.

Convince Me! Critique Reasoning A student did the calculation at the right. What did this student do wrong? What is the correct answer?

Another example!
Find 4 × 156.

Guided Practice

Do You Understand?

Question 1.
Use place value to explain each step in finding 3 × 2,746.

Answer:
The product is 8238.

Explanation:
In the above-given question,
given that,
the numbers are 3 and 2746.
3 x 2746 = 8238.
6 x 3 ones = 18; 18 = 1 ten and 8 ones.
4 x 3 tens = 12 tens; 12 tens + 1 ten = 13 tens = 1 hundred 3 tens.
7 x 3 hundreds = 21 hundreds + 1 hundred = 22 hundreds; 2 thousands 2 hundreds.
2 x 3 thousands = 6 thousands + 2 thousands; 8 thousands.
so the product is 8238.

Do You Know How?

For 2-5, find each product. Estimate to check if your answer is reasonable.

Question 2.

Answer:
23 x 4 = 92.

Explanation:
In the above-given question,
given that,
the two numbers are 23 and 4.
multiply the numbers.
23 x 4 = 92.

Question 3.

Answer:
378 x 2 = 756.

Explanation:
In the above-given question,
given that,
the two numbers are 378 and 2.
multiply the numbers.
378 x 2 = 756.

Question 4.

Answer:
157 x 5 = 785.

Explanation:
In the above-given question,
given that,
the two numbers are 157 and 5.
multiply the numbers.
157 x 5 = 785.

Question 5.

Answer:
1746 x 3 = 5238.

Explanation:
In the above-given question,
given that,
the two numbers are 1746 and 3.
multiply the numbers.
1746 x 3 = 5238.

Independent Practice

For 6-13, find each product. Estimate to check if your answer is reasonable.

Question 6.

Answer:
519 x 4 = 2076.

Explanation:
In the above-given question,
given that,
the two numbers are 519 and 4.
multiply the numbers.
519 x 4 = 2076.

Question 7.

Answer:
28 x 3 = 84.

Explanation:
In the above-given question,
given that,
the two numbers are 28 and 3.
multiply the numbers.
28 x 3 = 84.

Question 8.

Answer:
72 x 5 = 360.

Explanation:
In the above-given question,
given that,
the two numbers are 72 and 5.
multiply the numbers.
72 x 5 = 360.

Question 9.

Answer:
138 x 5 = 690.

Explanation:
In the above-given question,
given that,
the two numbers are 138 and 5.
multiply the numbers.
138 x 5 = 690.

Question 10.

Answer:
27 x 3 = 81.

Explanation:
In the above-given question,
given that,
the two numbers are 27 and 3.
multiply the numbers.
27 x 3 = 81.

Question 11.

Answer:
123 x 9 = 1107.

Explanation:
In the above-given question,
given that,
the two numbers are 123 and 9.
multiply the numbers.
123 x 9 = 1107.

Question 12.

Answer:
1445 x 5 = 7225.

Explanation:
In the above-given question,
given that,
the two numbers are 1445 and 5.
multiply the numbers.
1445 x 5 = 7225.

Question 13.

Answer:
2204 x 6 = 13224.

Explanation:
In the above-given question,
given that,
the two numbers are 2204 and 6.
multiply the numbers.
2204 x 6 = 13224.

Problem Solving

For 14-16, use the information in the pictures below to find each mass.

Question 14.
Elephant Seal

Answer:
The mass of Elephant Seal = 3480 kg.

Explanation:
In the above-given question,
given that,
the weight of elephants is 8 times as of zebra.
the weight of zebra is 435 kg.
435 x 8 = 3480 kg.

Question 15.
Sports Car

Answer:
The weight of the sports car = 1740 kg.

Explanation:
In the above-given question,
given that,
the weight of the sports car is 4 times as of zebra.
the weight of zebra is 435 kg.
435 x 4 =1740.
so the weight of the sports car = 1740 kg.

Question 16.
Bison

Answer:
The weight of the Bison = 870 kg.

Explanation:
In the above-given question,
given that,
the weight of the sports car is 2 times as of zebra.
the weight of zebra is 435 kg.
435 x 2 =870.
so the weight of the sports Bison = 870 kg.

Question 17.
Model with Math Last year, Anthony’s grandmother gave him 33 silver coins and 16 gold coins to start a coin collection. Now Anthony has six times as many coins in his collection. How many coins does Anthony have in his collection? Complete the bar diagram to show your work.

Answer:
The number of coins does Anthony has in his collection = 294.

Explanation:
In the above-given question,
given that,
Last year, Anthony’s grandmother gave him 33 silver coins and 16 gold coins to start a coin collection.
Now Anthony has six times as many coins in his collection.
33 x 6 = 198.
16 x 6 = 96.
198 + 96 = 294.
so the number of coins does Anthony have in his collection = 294.

Question 18.
Vocabulary Use Distributive or Commutative to complete the definition.
According to the ____ Property of Multiplication, factors can be multiplied in any order and the product remains the same.

Answer:
By using the commutative property, factors can be multiplied in any order.

Explanation:
In the above-given question,
given that,
By using the commutative property, factors can be multiplied in any order.
for example:
2 + 3 + 5.
5 + 5 = 10.
so the product remains the same.

Question 19.
Higher Order Thinking Do you think you could use a multiplication algorithm to multiply a 4-digit number by a 1-digit number? Explain.

Answer:
Yes, we can use a 4-digit number by a 1-digit number.

Explanation:
In the above-given question,
given that,
we can use a 4-digit number by a 1-digit number.
for example:
1234 x 1 = 1234.
so we can multiply a 4-digit number by a 1-digit number.

Assessment Practice

Question 20.
Find the product.

Answer:
768 x 8 = 6114.

Explanation:
In the above-given question,
given that,
the two numbers are 768 and 8.
multiply the numbers.
768 x 8 = 6114.

Question 21.
Find the product.

Answer:
1945 x 3 = 5835.

Explanation:
In the above-given question,
given that,
the two numbers are 1945 and 3.
multiply the numbers.
1945 x 3 = 5835.

### Lesson 3.4 Multiply 2-Digit by 2-Digit Numbers

Solve & Share

Ms. Silva has 12 weeks to train for a race. Over the course of one week, she plans to run 15 miles. If she continues this training, how many miles will Ms. Silva run before the race? Solve this problem using any strategy you choose.

Answer:
The number of miles will Ms. Silva run before the race = 180 miles.

Explanation:
In the above-given question,
given that,
Ms. Silva has 12 weeks to train for a race.
Over the course of one week, she plans to run 15 miles.
12 x 15 = 180.
so the number of miles will Ms. Silva run before the race = 180 miles.

You can use partial products to help make sense of and solve the problem. Show your work in the space below!

Look Back! Critique Reasoning Dwayne estimated 60 miles as an answer to the above problem. Is this estimate reasonable? If not, what mistake do you think Dwayne made?

Visual Learning Bridge

Essential Question What Is a Common Way to Record Multiplication?

A.
A ferry carried 37 cars per trip on the weekend. If the ferry made 11 trips on Saturday and 13 trips on Sunday, how many cars did it carry on the weekend?

You can add to find 24 trips were made on Saturday and Sunday. So the ferry carried 37 × 24 cars on the weekend.

B.
Use Partial Products
Use the area model to find the partial products for 24 × 37.

The ferry carried 888 cars on the weekend.

C.
Use the Standard Algorithm
Step 1: Multiply by the ones.

Step 2: Multiply by the tens.

The ferry carried 888 cars.

Convince Me! Make Sense and Persevere What are ways you can estimate to check the reasonableness of the answer?

Guided Practice

Do You Understand?

Question 1.
Janet said that the standard algorithm is just a shortcut for partial products. Do you agree? Explain.

Answer:
Yes, I will agree.

Explanation:
In the above-given question,
given that,
Janet said that the standard algorithm is just a shortcut for partial products.
for example:
37 x 24 = 148 is the standard algorithm.
37 x 24 = 888 is the partial products.
so i will agree.

Do You Know How?

For 2, use an algorithm or partial products to find the product. Estimate to check if your answer is reasonable.

Question 2.

Answer:
41 x 23 = 943.

Explanation:
In the above-given question,
given that,
the two numbers are 41 and 23.
40 + 1 = 43.
20 + 3 = 23.
20 x 40 = 800.
20 x 1 = 20.
800 + 20 = 820.
3 x 40 = 120.
3 x 1 = 3.
120 + 3 = 123.
820 + 123 = 943.

Independent Practice

Leveled Practice For 3-14, use an algorithm or partial products to find the product. Use and draw area models as needed.

Use estimation to check if your answers are reasonable.

Question 3.

Answer:
16 x 22 = 352.

Explanation:
In the above-given question,
given that,
the two numbers are 16 and 22.
10 + 6 = 16.
20 + 2 = 22.
10 x 20 = 200.
20 x 6 = 120.
200 + 120 = 320.
2 x 10 = 20.
2 x 6 = 12.
20 + 12 = 32.
320 + 32 = 352.

Question 4.

Answer:
15 x 16 = 240.

Explanation:
In the above-given question,
given that,
the two numbers are 16 and 15.
10 + 5 = 15.
10 + 6 = 16.
10 x 10 = 100.
10 x 5 = 50.
100 + 50 = 150.
6 x 10 = 60.
6 x 5 = 30.
60 + 30 = 90.
150 + 90 = 240.

Question 5.

Answer:
27 x 12 = 324.

Explanation:
In the above-given question,
given that,
the two numbers are 27 and 12.
20 + 7 = 27.
10 + 2 = 12.
10 x 20 = 200.
10 x 7 = 70.
200 + 70 = 270.
2 x 20 = 40.
2 x 7 = 14.
40 + 14 = 54.
270 + 54 = 324.

Question 6.

Answer:
18 x 15 = 270.

Explanation:
In the above-given question,
given that,
the two numbers are 18 and 15.
10 + 8 = 18.
10 + 5 = 15.
10 x 10 = 100.
10 x 8 = 80.
100 + 80 = 180.
5 x 10 = 50.
5 x 8 = 40.
50 + 40 = 90.
180 + 90 = 270.

Question 7.
53 × 17

Answer:
53 x 17 = 901.

Explanation:
In the above-given question,
given that,
the two numbers are 53 and 17.
multiply the numbers.
53 x 17 = 901.

Question 8.
81 × 46

Answer:
81 x 46 = 901.

Explanation:
In the above-given question,
given that,
the two numbers are 81 and 46.
81 x 46 = 901.

Question 9.
15 × 16

Answer:
15 x 16 = 240.

Explanation:
In the above-given question,
given that,
the two numbers are 15 and 16.
15 x 16 = 240.

Question 10.
17 × 21

Answer:
17 x 21 = 357.
Explanation:
In the above-given question,
given that,
the two numbers are 17 and 21.
17 x 21 = 357.

Question 11.
12 × 22

Answer:
12 x 22 = 264.
Explanation:
In the above-given question,
given that,
the two numbers are 12 and 22.
12 x 22 = 264.

Question 12.
38 × 41

Answer:
38 x 41 = 1558.

Explanation:
In the above-given question,
given that,
the two numbers are 38 and 41.
38 x 41 = 1558.

Question 13.
42 × 52

Answer:
42 x 52 = 2184.
Explanation:
In the above-given question,
given that,
the two numbers are 42 and 52.
42 x 52 = 2184.

Question 14.
38 × 19

Answer:
38 x 19 = 722.
Explanation:
In the above-given question,
given that,
the two numbers are 38 and 19.
38 x 19 = 722.

Problem Solving

Question 15.
Number Sense The Queen Mary 2’s height above water is about the same as a 14-story building. What is the Queen Mary 2’s height above water?

Answer:
The Queen Mary 2’s height above water = 168 feet.

Explanation:
In the above-given question,
given that,
The Queen Mary 2’s height above water is about the same as a 14-story building.
14 x 12 = 168.
so the Queen Mary 2’s height above water = 168 feet.

Question 16.
Model with Math Write the multiplication equation illustrated by the array drawn on the grid. Find the partial products. Then calculate the final product.

Answer:
The partial products are 18 and 15.
the final product is 270.

Explanation:
In the above-given question,
given that,
the two numbers are 18 and 15.
10 + 8 = 18.
10 + 5 = 15.
10 x 10 = 100.
10 x 8 = 80.
100 + 80 = 180.
5 x 10 = 50.
5 x 8 = 40.
50 + 40 = 90.
180 + 90 = 270.

Question 17.
Higher Order Thinking An elevator can carry 15 adults or 20 children at one time. During the course of a day, the elevator carries a full passenger load 52 times. If all the passengers were children, how many more people would the elevator carry than if all the passengers were adults?

Answer:
The more people would the elevator carry than if all the passengers were adults = 1040.

Explanation:
In the above-given question,
given that,
An elevator can carry 15 adults or 20 children at one time.
During the course of a day, the elevator carries a full passenger load 52 times.
52 x 20 = 1040.
so the more people would the elevator carry than if all the passengers were adults = 1040.

Assessment Practice

Question 18.
Ten years ago, Melissa planted a tree in her backyard. She has taken a photo of the tree every week so she can see how it has grown as time passed. How many photos of the tree does Melissa now have?
A. 62 photos
B. 120 photos
C. 520 photos
D. 620 photos
There are 52 weeks in one year.

Answer:
The number of photos of the tree does Melissa now has = 520 photos.

Explanation:
In the above-given question,
given that,
Ten years ago, Melissa planted a tree in her backyard.
She has taken a photo of the tree every week so she can see how it has grown as time passed.
there are 52 weeks in one year.
52 x 10 = 520.
so the number of photos of the tree does Melissa now have = 520 photos.

Question 19.
Mr. Morris bought sketchpads for 24 of his students. Each pad contained 50 sheets. How many sheets of paper were in all the pads?
A. 1,000 sheets
B. 1,200 sheets
C. 1,400 sheets
D. 1,600 sheets

Answer:
The number of sheets of paper was in all the pads = 1200 sheets.

Explanation:
In the above-given question,
given that,
Mr. Morris bought sketchpads for 24 of his students.
Each pad contained 50 sheets.
24 x 50 = 1200.
so the number of sheets of paper were in all the pads = 1200 sheets.

### Lesson 3.5 Multiply 3-Digit by 2-Digit Numbers

Activity

Solve & Share

A local charity collected 163 cans of food each day for 14 days. How many cans did they collect in all? Explain how you found your answer.

You can use what you know about multiplying 2-digit numbers by 2-digit numbers to help solve the problem.

Answer:
The number of cans did they collect in all = 2282 cans.

Explanation:
In the above-given question,
given that,
A local charity collected 163 cans of food each day for 14 days.
163 x 14 = 2282.
so the number of cans did they collect in all = 2282 cans.

Look Back! Make Sense and Persevere How can you check that your answer is reasonable?

Visual Learning Bridge

Essential Question How Do You Multiply 3-Digit Numbers by 2-Digit Numbers?

A.
Last month a bakery sold 389 boxes of bagels. How many bagels did the store sell last month? Find 12 × 389.

You can show all partial products or you can use the standard algorithm.

B.
Step 1
To use the Standard Algorithm, first multiply by the ones. Regroup as needed.

2 × 9 ones = 18 ones or 1 ten and 8 ones
2 × 8 tens =16 tens
16 tens + 1 ten = 17 tens
17 tens = 1 hundred 7 tens
2 × 3 hundreds = 6 hundreds
6 hundreds + 1 hundred = 7 hundreds

C.
Step 2
Multiply by the tens.
Regroup as needed.

10 × 9 ones = 90 ones
10 × 8 tens = 80 tens,
or 8 hundred 10 × 3 hundred = 30 hundred, or 3 thousand

D.
Step 3
Add to get the final product.

The store sold 4,668 bagels last month.

Convince Me! Construct Arguments Is 300 10 a good estimate for the number of bagels sold at the bakery? Explain.

Guided Practice

Do You Understand?

Question 1.
A theater can seat 540 people at one time. How many tickets are sold if the theater sells out every seat for one 30-day month?

Answer:
The number of tickets is sold if the theater sells out every seat for one 30-day month = 16200.

Explanation:
In the above-given question,
given that,
A theater can seat 540 people at one time.
540 x 30 = 16200.
so the number of tickets are sold if the theater sells out every seat for one 30-day month = 16200.

Question 2.
Number Sense Is 500 30 a good estimate for the number of tickets sold at the theater in one month? Explain.

Answer:
Yes, it is a good estimate for the number of tickets sold at the theater in one month.

Explanation:
In the above-given question,
given that,
A theater can seat 540 people at one time.
540 is equal to 500.
500 x 30 = 15000.
so it is a good estimate for the number of tickets sold at the theater in one month.

Do You Know How?

In 3-6, find each product. Estimate to check that your answer is reasonable.

Question 3.

Answer:
236 x 46 = 10856.

Explanation:
In the above-given question,
given that,
the two numbers are 236 and 46.
multiply the numbers.
236 x 46 = 10856.

Question 4.

Answer:
61 x 25 = 5185.

Explanation:
In the above-given question,
given that,
the two numbers are 61 and 25.
multiply the numbers.
61 x 25 = 5185.

Question 5.

Answer:
951 x 62 = 58962.

Explanation:
In the above-given question,
given that,
the two numbers are 951 and 62.
multiply the numbers.
951 x 62 = 58962.

Question 6.

Answer:
185 x 5 = 925.

Explanation:
In the above-given question,
given that,
the two numbers are 185 and 5.
multiply the numbers.
185 x 5 = 925.

Independent Practice

Leveled Practice In 7-18, find each product. Estimate to check that your answer is reasonable.

Question 7.

Answer:
51 x 10 = 510.

Explanation:
In the above-given question,
given that,
the two numbers are 51 and 10.
multiply the numbers.
51 x 10 = 510.

Question 8.

Answer:
892 x 18 = 16056.

Explanation:
In the above-given question,
given that,
the two numbers are 892 and 18.
multiply the numbers.
892 x 18 = 16056.

Question 9.

Answer:
946 x 33 = 31218.

Explanation:
In the above-given question,
given that,
the two numbers are 946 and 33.
multiply the numbers.
946 x 33 = 31218.

Question 10.

Answer:
735 x 41 = 30135.

Explanation:
In the above-given question,
given that,
the two numbers are 735 and 41.
multiply the numbers.
735 x 41 = 30135.

Question 11.

Answer:
100 x 25 = 2500.

Explanation:
In the above-given question,
given that,
the two numbers are 100 and 25.
multiply the numbers.
100 x 25 = 2500.

Question 12.

Answer:
81 x 11 = 891.

Explanation:
In the above-given question,
given that,
the two numbers are 81 and 11.
multiply the numbers.
81 x 11 = 891.

Question 13.

Answer:
106 x 7 = 742.

Explanation:
In the above-given question,
given that,
the two numbers are 106 and 7.
multiply the numbers.
106 x 7 = 742.

Question 14.

Answer:
90 x 59 = 5310.

Explanation:
In the above-given question,
given that,
the two numbers are 90 and 59.
multiply the numbers.
90 x 59 = 5310.

Question 15.

Answer:
360 x 18 = 6480.

Explanation:
In the above-given question,
given that,
the two numbers are 360 and 18.
multiply the numbers.
360 x 18 = 6480.

Question 16.

Answer:
222 x 75 = 16650.

Explanation:
In the above-given question,
given that,
the two numbers are 222 and 75.
multiply the numbers.
222 x 75 = 16650.

Question 17.

Answer:
481 x 35 = 16835.

Explanation:
In the above-given question,
given that,
the two numbers are 481 and 35.
multiply the numbers.
481 x 35 = 16835.

Question 18.

Answer:
659 x 17 = 11203.

Explanation:
In the above-given question,
given that,
the two numbers are 659 and 17.
multiply the numbers.
659 x 17 = 11203.

Problem Solving

Question 19.
enVision® STEM How many times does a rabbit’s heart beat in 1 hour?

Remember, there are 60 minutes in 1 hour.

Answer:
The number of times does a rabbit’s heartbeat in 1 hour = 212 beats.

Explanation:
In the above-given question,
given that,
1 hour = 60 minutes.
212 beats per minute.
so the number of times does a rabbit’s heartbeat in 1 hour = 212 beats.

Question 20.
In 1 hour, how many more times does a rabbit’s heart beat than a dog’s heart? Write an equation to show your work.

Answer:
The number of more times does a rabbit’s heartbeat than a dog’s heart = 112.

Explanation:
In the above-given question,
given that,
heart rate of the dog for a minute = 100.
heart rate of the rabbit for a minute = 212.
212 – 100 = 112.
so the number of more times does a rabbit’s heartbeat than a dog’s heart = 112.

Question 21.
Construct Arguments Is 3,198 a reasonable product for 727 × 44? Why or why not?

Answer:
Yes, it is a reasonable product.

Explanation:
In the above-given question,
given that,
the two numbers are 727 and 44.
multiply the two numbers.
727 x 44 = 31,988.
yes, it is a reasonable product.

Question 22.
Higher Order Thinking A garden store sells plants in flats. There are 6 plants in each tray. Each flat has 6 trays. The garden store sold 18 flats on Saturday and 21 flats on Sunday. How many plants did the garden store sell in all?

Answer:
The number of plants did the garden store sell in all = 234.

Explanation:
In the above-given question,
given that,
A garden store sells plants in flats.
There are 6 plants in each tray.
Each flat has 6 trays.
The garden store sold 18 flats on Saturday and 21 flats on Sunday.
18 x 6 = 108.
21 x 6 = 126.
108 + 126 = 234.
so the number of plants did the garden store sell in all = 234.

Assessment Practice

Question 23.
Tricia is building a rectangular patio. The patio will be 108 bricks wide and 19 bricks long. How many bricks does she need to build the patio?

Answer:
The number of bricks does she need to build the patio = 2052.

Explanation:
In the above-given question,
given that,
Tricia is building a rectangular patio.
The patio will be 108 bricks wide and 19 bricks long.
area of the rectangle = l x b.
where l = length and b = breadth.
108 x 19 = 2052.
so the number of bricks does she need to build the patio = 2052.

Question 24.
What is the product?

A. 1,560
B. 1,568
C. 4,268
D. 4,368

Answer:
312 x 14 = 4368.

Explanation:
In the above-given question,
given that,
the two numbers are 312 and 14.
multiply the two numbers.
312 x 14 = 4368.

### Lesson 3.6 Multiply Whole Numbers with Zeros

Activity

Solve & Share

A school district is replacing all of the desks in its classrooms. There are 103 classrooms and each classroom needs 24 new desks. How many desks will the school district need to buy? Solve this problem any way you choose!

Use what you know about multiplying 3-digit and 2-digit numbers. Show your work!

Answer:
The number of desks will the school district need to buy = 2472.

Explanation:
In the above-given question,
given that,
A school district is replacing all of the desks in its classrooms.
There are 103 classrooms and each classroom needs 24 new desks.
103 x 24 = 2472.
so the number of desks will the school district need to buy = 2472.

Look Back! Make Sense and Persevere What is a good estimate for the problem above? Explain.

Visual Learning Bridge

Essential Question
How Can You Multiply with Zeros?

A.
An antique steam train makes one sight-seeing tour each day. If every seat is filled for each trip, how many passengers can it carry for 31 tours?

The standard algorithm does not change when there is a zero in a factor.

B.
Step 1
Find 31 × 208.
Estimate:
30 × 200 = 6,000

C.
Step 2
Multiply by the ones.
Regroup if necessary.
Remember that multiplying with a zero gives a product of zero.

D.
Step 3
Multiply by the tens.
Regroup if necessary.
Add to get the final product.

The train can carry 6,448 passengers.

Convince Me! Model with Math Suppose the train fills an average of 102 seats for each tour. What is a reasonable estimate for the number of passengers that the train can carry in 28 tours? Write an equation to show your work.

Answer:
The number of passengers that the train can carry in 28 tours = 2856.

Explanation:
In the above-given question,
given that,
Suppose the train fills an average of 102 seats for each tour.
the two numbers are 102 and 28.
102 x 28 = 2856.
so the number of passengers that the train can carry in 28 tours = 2856.

Guided Practice

Do You Understand?

Question 1.
In an auditorium, there are 104 rows with 24 seats in each row. How many seats are available?

Answer:
The number of seats is available = 2496 seats.

Explanation:
In the above-given question,
given that,
there are 104 rows with 24 seats in each row.
104 x 24 = 2496.
so the number of seats are available = 2496 seats.

Question 2.
Why is it important to “estimate to check for reasonableness”?
Answer:

Do You Know How?

In 3-6, multiply to find the product. Estimate to check for reasonableness.

Question 3.

Answer:
205  x 23 = 4715.

Explanation:
In the above-given question,
given that,
the two numbers are 205 and 23.
multiply the numbers.
205 x 23 = 4715.

Question 4.

Answer:
108 x 34 = 3672.

Explanation:
In the above-given question,
given that,
the two numbers are 108 and 34.
multiply the numbers.
108 x 34 = 3672.

Question 5.

Answer:
410 x 44 = 18040.

Explanation:
In the above-given question,
given that,
the two numbers are 410 and 44.
multiply the numbers.
410 x 44 = 18040.

Question 6.

Answer:
302 x 30 = 9060.

Explanation:
In the above-given question,
given that,
the two numbers are 302 and 30.
multiply the numbers.
302 x 30 = 9060.

Independent Practice
Leveled Practice In 7-18, find each product. Estimate to check for reasonableness.

Question 7.

Answer:
302 x 17 = 5134.

Explanation:
In the above-given question,
given that,
the two numbers are 302 and 17.
multiply the numbers.
236 x 46 = 5134.

Question 8.

Answer:
608 x 23 = 13984.

Explanation:
In the above-given question,
given that,
the two numbers are 608 and 23.
multiply the numbers.
608 x 23 =13984.

Question 9.

Answer:
109 x 47 = 5123.

Explanation:
In the above-given question,
given that,
the two numbers are 109 and 47.
multiply the numbers.
109 x 47 = 5123.

Question 10.

Answer:
510 x 72 = 36864.

Explanation:
In the above-given question,
given that,
the two numbers are 510 and 72.
multiply the numbers.
510 x 72 = 36864.

Question 11.

Answer:
902 x 35 = 31570.

Explanation:
In the above-given question,
given that,
the two numbers are 902 and 35.
multiply the numbers.
902 x 35 = 31570.

Question 12.

Answer:
207 x 61 = 12627.

Explanation:
In the above-given question,
given that,
the two numbers are 207 and 61.
multiply the numbers.
207 x 61 = 12627.

Question 13.

Answer:
108 x 58 = 6264.

Explanation:
In the above-given question,
given that,
the two numbers are 108 and 58.
multiply the numbers.
108 x 58 = 6264.

Question 13.

Answer:
108 x 58 = 6264.

Explanation:
In the above-given question,
given that,
the two numbers are 108 and 58.
multiply the numbers.
108 x 58 = 6264.

Question 14.

Answer:
505 x 77 = 38885.

Explanation:
In the above-given question,
given that,
the two numbers are 505 and 77.
multiply the numbers.
505 x 77 = 38885.

Question 15.

Answer:
407 x 39 = 15873.

Explanation:
In the above-given question,
given that,
the two numbers are 407 and 39.
multiply the numbers.
407 x 39 = 15873.

Question 16.

Answer:
280 x 66 =18480.

Explanation:
In the above-given question,
given that,
the two numbers are 280 and 66.
multiply the numbers.
280 x 66 = 18480.

Question 17.

Answer:
105 x 24 =2520.

Explanation:
In the above-given question,
given that,
the two numbers are 105 and 24.
multiply the numbers.
105 x 24 =2520.

Question 18.

Answer:
360 x 48 = 17280.

Explanation:
In the above-given question,
given that,
the two numbers are 360 and 48.
multiply the numbers.
360 x 48 = 17280.

Problem Solving

Question 19.
There are 27 students in Mr. Mello’s class. Find the total number of pages the students read by the end of November.

Answer:
The total number of pages the students read by the end of November = 783 pages.

Explanation:
In the above-given question,
given that,
there are 27 students in Mr. Mello’s class.
in November there are 29 days.
29 x 27 = 783.
so the total number of pages the students read by the end of November = 783 pages.

Question 20.
Each student read 41 pages in December. How many total pages did the students read by the end of December?

Answer:
The number of total pages did the students read by the end of December = 1271 pages.

Explanation:
In the above-given question,
given that,
Each student read 41 pages in December.
in December there are 31 pages.
41 x 31 = 1271.
so the number of total pages did the students read by the end of December = 1271 pages.

Question 21.
Meredith says that 15.17 is greater than 15.8 because 17 is greater than 8. Do you agree? Explain your reasoning.

Answer:
No, I do not agree with it.

Explanation:
In the above-given question,
given that,
Meredith says that 15.17 is greater than 15.8 because
17 is greater than 8.
15.17 is less than 15.8.
so I do not agree with it.

Question 22.
Use Structure Trudy wants to multiply 66 × 606. She says that all she has to do is find 6 × 606 and then double that number. Explain why Trudy’s method will not give the correct answer. Then show how to find the correct product.

Answer:
Yes, Trudy’s method will not give the correct answer.

Explanation:
In the above-given question,
given that,
Trudy wants to multiply 66 × 606.
She says that all she has to do is find 6 × 606.
66 x 606 =
6 x 606 =
the two values are not equal.
so Trudy’s method will not give the correct answer.

Question 23.
Higher Order Thinking Maria needs a trombone for only 12 months. Renting the trombone costs $34 per month. She can buy the trombone for$495. Should she buy or rent the trombone? Explain. How much does she pay?

Answer:
Yes, she can rent the trombone.

Explanation:
In the above-given question,
given that,
Maria needs a trombone for only 12 months.
Renting the trombone costs $34 per month. She can buy the trombone for$495.
12 x $34 =$408.

Question 24.
Another music store rents trombones for $30 per month plus a yearly fee of$48. Which deal is better? Should Maria change her rental plan?

Answer:
Yes, maria change her rental plan.

Explanation:
In the above-given question,
given that,
Another music store rents trombones for $30 per month plus a yearly fee of$48.
30 x 48 = $1440. Assessment Practice Question 25. What is the product? Answer: 659 x 17 = 11203. Explanation: In the above-given question, given that, the two numbers are 659 and 17. multiply the numbers. 659 x 17 = 11203. ### Lesson 3.7 Practice Multiplying Multi-Digit Numbers Activity Solve & Share Which of the two car payment options will cost less for 1 year? How much less? Solve this problem any way you choose! Show all of your work You can use reasoning to connect mathematics to everyday life. Think about the situations multiplication describes. Answer: The Look Back! How can you estimate the total for the year when paying monthly? When paying quarterly? Visual Learning Bridge Essential Question How Can You Use Multiplication to Solve Problems? A. What is the yearly total for water, gas, and electric? What is the yearly total for cell phones? The standard algorithm for multiplying whole numbers involves breaking numbers apart using place value. B. What is the yearly total for water, gas, and electric? Find 4 × (760 + 510). Estimate: 4 × (760 + 510) is about 4 × 1,200 = 4,800. 4 × (760+ 510) = 4 × 1,270 The yearly total for water, gas, and electric is$5,080.

C.
What is the yearly total for cell phones?
Find 12 × 271.
Estimate:
12 × 271 is about 10 × 270 = 2,700.

The process for multiplying is the same regardless of the number of digits in 3,252 the factors.

The yearly total for cell phones is $3,252. Convince Me! Be Precise How are the processes for multiplying alike for the two calculations above? How are they different? Guided Practice Do You Understand? Question 1. Carlos saves 18 cents every day of the year. If there are 365 days this year, how many cents will he have saved by the end of the year? Write an equation that represents the problem. Then, solve the equation. Answer: The number of cents will he have saved by the end of the year = 6570. Explanation: In the above-given question, given that, Carlos saves 18 cents every day of the year. If there are 365 days this year. 365 x 18 = 6570. so the number of cents will he have saved by the end of the year = 6570. Question 2. Lila drives 129 kilometers round trip to work. How many kilometers does she drive in 31 days? Write an equation that represents the problem. Then solve the equation. Answer: The number of kilometers does she drive in 31 days = 3999 km. Explanation: In the above-given question, given that, Lila drives 129 kilometers round trip to work. 129 x 31 = 3999. so the number of kilometers does she drive in 31 days = 3999 km. Do You Know How? In 3-6, estimate each product. Then complete each calculation. Check that your answer is reasonable. Question 3. Answer: 134 x 11 = 1474. Explanation: In the above-given question, given that, the two numbers are 134 and 11. multiply the numbers. 134 x 11 = 1474. Question 4. Answer: 208 x 26 = 5408. Explanation: In the above-given question, given that, the two numbers are 208 and 26. multiply the numbers. 208 x 26 = 5408. Question 5. Answer: 428 x 35 = 14980. Explanation: In the above-given question, given that, the two numbers are 428 and 35. multiply the numbers. 428 x 35 = 14980. Question 6. Answer: 275 x 56 = 15400. Explanation: In the above-given question, given that, the two numbers are 275 and 56. multiply the numbers. 275 x 56 = 15400. Independent Practice Leveled Practice In 7-22, estimate and then compute each product. Check that your answer is reasonable. Question 7. Answer: 531 x 47 = 24,957. Explanation: In the above-given question, given that, the two numbers are 531 and 47. multiply the numbers. 531 x 47 = 24,957. Question 8. Answer: 759 x 68 = 51,612. Explanation: In the above-given question, given that, the two numbers are 759 and 68. multiply the numbers. 759 x 68 = 51,612. Question 9. Answer: 367 x 92 = 33,764. Explanation: In the above-given question, given that, the two numbers are 367 and 92. multiply the numbers. 367 x 92 = 33,764. Question 10. Answer: 817 x 45 = 36,765. Explanation: In the above-given question, given that, the two numbers are 817 and 45. multiply the numbers. 817 x 45 = 36,765. Question 11. Answer: 1206 x 77 = 92862. Explanation: In the above-given question, given that, the two numbers are 1206 and 77. multiply the numbers. 1206 x 77 = 92862. Question 12 Answer: 543 x 18 = 9774. Explanation: In the above-given question, given that, the two numbers are 543 and 18. multiply the numbers. 543 x 18 = 9774. Question 13. Answer: 908 x 62 = 56,296. Explanation: In the above-given question, given that, the two numbers are 908 and 62. multiply the numbers. 908 x 62 = 56,296. Question 14. Answer: 750 x 81 = 60,750. Explanation: In the above-given question, given that, the two numbers are 750 and 81. multiply the numbers. 750 x 81 = 60,750. Question 15. 6,755 × 9 Answer: 6755 x 9 = 60,795. Explanation: In the above-given question, given that, the two numbers are 6755 and 9. multiply the numbers. 6755 x 9 = 60,795. Question 16. 869 × 46 Answer: 869 x 46 = 39,974. Explanation: In the above-given question, given that, the two numbers are 869 and 46. multiply the numbers. 869 x 46 = 39,974. Question 17. 922 × 81 Answer: 922 x 81 = 74,682. Explanation: In the above-given question, given that, the two numbers are 922 and 81. multiply the numbers. 922 x 81 = 74,682. Question 18. 783 × 14 Answer: 783 x 14 = 10,962. Explanation: In the above-given question, given that, the two numbers are 783 and 14. multiply the numbers. 783 x 14 = 10,962. Question 19. 684 × 15 Answer: 684 x 15 = 10,260. Explanation: In the above-given question, given that, the two numbers are 684 and 15. multiply the numbers. 684 x 15 = 10,260. Question 20. 650 × 22 Answer: 650 x 22 = 14,300. Explanation: In the above-given question, given that, the two numbers are 650 and 22. multiply the numbers. 650 x 22 = 14,300. Question 21. 2,525 × 37 Answer: 2,525 x 37 = 93,425. Explanation: In the above-given question, given that, the two numbers are 2525 and 37. multiply the numbers. 2525 x 37 = 93,425. Question 22. 615 × 41 Answer: 615 x 41 = 25,215. Explanation: In the above-given question, given that, the two numbers are 615 and 41. multiply the numbers. 615 x 41 = 25,215. Problem Solving For 23 and 24, use the table. Question 23. Model with Math Jason frequently travels for work. This year he plans to make 15 trips to Chicago. What is the total cost for the airfare? Write an equation that represents the problem. Then, solve the equation. Answer: The total cost for the airfare =$7335.

Explanation:
In the above-given question,
given that,
Jason frequently travels for work.
the cost of the chicago is $489. This year he plans to make 15 trips to Chicago.$489 x 15 = 7335.
so the total cost of the airfare = $7335. Question 24. Which would cost more: 15 trips to Boston or 11 trips to New York? Explain. Answer: The trip would cost more. Explanation: In the above-given question, given that, the ticket cost of Boston is$178.
the ticket cost of new york is $225. 15 x 178 = 2670. 225 x 11 = 1958. so the trip would cost more. Question 25. A cook at a restaurant is planning her food order. She expects to use 115 pounds of potatoes each day for 12 days. How many pounds of potatoes will she order? Answer: The number of pounds of potatoes will she order = 1380 pounds. Explanation: In the above-given question, given that, A cook at a restaurant is planning her food order. She expects to use 115 pounds of potatoes each day for 12 days. 115 x 12 = 1380. so the number of pounds of potatoes will she order = 1380. Question 26. Higher Order Thinking Carolyn bought a gallon of paint that covers 250 square feet. She wants to paint a wall that is 16 feet wide and 12 feet high. Explain whether or not she will need more than one gallon of paint. Answer: Yes, she needs only one gallon of paint. Explanation: In the above-given question, given that, Carolyn bought a gallon of paint that covers 250 square feet. She wants to paint a wall that is 16 feet wide and 12 feet high. 16 x 12 = 192. 192 is less than 250. so she needs less than one gallon of paint. Assessment Practice Question 27. The product of the following expression is 7,453. What is the missing digit? A. 1 B. 2 C. 4 D. 7 Answer: Option B is the correct. Explanation: In the above-given question, given that, 257 x 29 = 7453. so option B is correct. Question 28. When you multiply a 3-digit number by a 2-digit number, what is the greatest number of digits the product can have? Answer: The greatest number of digits the product can have 4. Explanation: In the above-given question, given that, the three-digit number is 123. the two-digit number is 10. 123 x 10 = 1230. ### Lesson 3.8 Solve Word Problems Using Multiplication Activity Solve&S are Kevin’s family took 239 photos on their summer vacation. Marco and his family took 12 times as many photos on their vacation. How many photos did Marco’s family take? Solve this problem any way you choose. How can you use an equation to model the situation with math? Answer: The number of photos did Marco’s family take = 2629 photos. Explanation: In the above-given question, given that, Kevin’s family took 239 photos on their summer vacation. Marco and his family took 12 times as many photos on their vacation. 239 x 12 = 2868. 2868 – 239 = 2629. so the number of photos did Marco’s family take = 2629 photos. Look Back! How can you use estimation to tell if your answer is reasonable? Explain. Visual Learning Bridge Essential Question How Can You Use a Bar Diagram to Solve a Multiplication Problem? A. In 1980, a painting sold for$1,575. In 2015, the same painting sold for 5 times as much. What was the price of the painting in 2015?

You can draw a bar diagram and use a variable to find the new price of the painting.

B.
What am I asked to find?
The price of the painting in 2015.
Let p = the price of the painting in 2015.
Draw a bar diagram to represent the problem.

C.
Write and solve an equation using the variable.
$1,575 × 5 =p$1,575 × 5 = $7,875. So, p=$7,875.
In 2015, the painting sold for $7,875. You can use repeated addition or division to check your answer! Convince Me! Construct Arguments How can you use estimation to justify that the answer$7,875 is reasonable?

Guided Practice

Do You Understand?

Question 1.
Write a real-world problem that uses multiplication. Then, draw a bar diagram and write an equation to solve your problem.
Answer:

Do You Know How?

In 2, write and solve an equation.

Question 2.
Sharon’s Stationery Store has 1,219 boxes of cards. May’s Market has 3 times as many boxes of cards. How many boxes of cards does May’s Market have?

Answer:
The number of cards does May’s market have = 2.

Explanation:
In the above-given question,
given that,
Sharon’s Stationery Store has 1,219 boxes of cards.
May’s Market has 3 times as many boxes of cards.
3 x 1219 = 3657.
3657 – 1219 = 2438.
1219 x 2 = 2438.
so the number of cards does May’s market have = 2.

Independent Practice

In 3-5, draw a bar diagram to model the situation. Then, write and solve an equation.

Question 3.
There are 14 theaters at the mall. Each theater has 175 seats. How many seats are there in all?

Answer:
The number of seats is there in all = 2450.

Explanation:
In the above-given question,
given that,
There are 14 theaters at the mall.
Each theater has 175 seats.
14 x 175 = 2450.
so the number of seats are there in all = 2450.

Question 4.
Brad lives 12 times as far away from the ocean as Jennie. If Jennie lives 48 miles from the ocean, how many miles from the ocean does Brad live?

Answer:
The number of miles from the ocean does Brad live = 576 miles.

Explanation:
In the above-given question,
given that,
Brad lives 12 times as far away from the ocean as Jennie.
If Jennie lives 48 miles from the ocean.
48 x 12 = 576.
so the number of miles from the ocean does Brad live = 576 miles.

Question 5.
A hardware store ordered 13 packs of nails from a supplier. Each pack contains 155 nails. How many nails did the store order?

Answer:
The number of nails did the store order = 2015 nails.

Explanation:
In the above-given question,
given that,
A hardware store ordered 13 packs of nails from a supplier.
Each pack contains 155 nails.
13 x 155 = 2015 nails.
so the number of nails did the store order = 2015 nails.

Problem Solving

Question 6.
Algebra Sandi’s school has 1,030 students. Karla’s school has 3 times as many students as Sandi’s school. Write an equation to find s, the number of students in Karla’s school. Then, solve your equation.

Answer:
The number of students in Karla’s school = 3090.

Explanation:
In the above-given question,
given that,
Sandi’s school has 1,030 students.
Karla’s school has 3 times as many students as Sandi’s school.
1030 x 3 = 3090.
so the number of students in Karla’s school = 3090.

Question 7.
enVision® STEM Jupiter is about 5 times the distance Earth is from the Sun. Earth is about 93,000,000 miles from the Sun. About how far is Jupiter from the Sun?
Look for a relationship to help you solve this problem.

Answer:
The far is Jupiter from the sun = 46,50,00,000.

Explanation:
In the above-given question,
given that,
Jupiter is about 5 times the distance Earth is from the Sun.
Earth is about 93,000,000 miles from the sun.
93,000,000 x 5 = 46,50,00,000.
so the far is hupiter from the sun = 46,50,00,000.

Question 8.
Higher Order Thinking William travels only on Saturdays and Sundays and has flown 1,020 miles this month. Jason travels every weekday and has flown 1,200 miles this month. If each man travels about the same number of miles each day, who travels more miles per day for this month? Explain.

Answer:
Jason travels more miles per day for this month.

Explanation:
In the above-given question,
given that,
William travels only on Saturdays and Sundays and has flown 1,020 miles this month.
Jason travels every weekday and has flown 1,200 miles this month.
1200 is greater than 1020.
so Jason travels more miles per day for this month.

Question 9.
Make Sense and Persevere Hwong can fit 12 packets of coffee in a small box and 50 packets of coffee in a large box. He has 10 small boxes of coffee and would like to reorganize the packets into large boxes. How many large boxes could he fill? Explain.

Answer:
The number of large boxes could he fill = 6.

Explanation:
In the above-given question,
given that,
How can fit 12 packets of coffee in a small box and 50 packets of coffee in a large box.
He has 10 small boxes of coffee and would like to reorganize the packets into large boxes.
12 x 50 = 600.
10 x 6 = 600.
so the number of large boxes could he fill = 6.

Assessment Practice

Question 10.
Martin ran 108 miles last year. Katrina ran 13 times as many miles as Martin last year. How many miles did Katrina run last year?
A. 1,008 miles
B. 1,404 miles
C. 1,806 miles
D. 2,000 miles

Answer:
The number of miles did Katrina run last year = 1404 miles.

Explanation:
In the above-given question,
given that,
Martin ran 108 miles last year. Katrina ran 13 times as many miles as Martin last year.
108 x 13 = 1404.
so option B is the correct.

Question 11.
The Erie shoe factory makes 245 pairs of shoes a day. The Columbus shoe factory makes 34 times as many pairs of shoes a day as the Erie shoe factory. How many pairs of shoes does the Columbus shoe factory make a day?
A. 7,545 pairs of shoes
B. 8,010 pairs of shoes
C. 8,330 pairs of shoes
D. 8,750 pairs of shoes

Answer:
Option C is the correct answer.

Explanation:
In the above-given question,
given that,
The Erie shoe factory makes 245 pairs of shoes a day.
The Columbus shoe factory makes 34 times as many pairs of shoes a day as the Erie shoe factory.
245 x 34 = 8,330.
so the option C is the correct.

### Lesson 3.9 Critique Reasoning

Activity

Problem Solving

Solve & Share
A group of 44 students is planning a train trip to Washington, D.C. They held many fundraisers and raised $10,880. Nathan said, “We should have enough money to pay for the train tickets. There are about 50 students going on the trip and one round trip ticket costs about$200. That makes the total cost of the tickets less than $10,000.” Does Nathan’s reasoning make sense? Answer: The total cost of the tickets is less than$10,000.

Explanation:
In the above-given question,
given that,
A group of 44 students is planning a train trip to Washington, D.C.
They held many fundraisers and raised $10,880. Nathan said, “We should have enough money to pay for the train tickets. There are about 50 students going on the trip and one round trip ticket costs about$200.
50 x 200 = 10,000.
so the total cost of the ticket is less than $10000. Thinking Habits Be a good thinker! These questions can help you. • What questions can I ask to understand people’s thinking? • Are there mistakes in other people’s thinking? • Can I improve other people’s thinking? Look Back! Critique Reasoning What argument would you make to support Nathan’s estimate? Visual Learning Bridge Essential Question How Can You Critique Reasoning of Others? A. Ms. Lynch needs to ship 89 boxes. 47 boxes weigh 150 pounds each. Each of the other boxes weighs 210 pounds. Mia says that all 89 boxes can fit into one container. She reasons that 47 × 150 is less than 7,500 and 42 × 210 is a little more than 8,000, so the sum of their weights should be less than 15,400. What is Mia’s reasoning to support her estimate? Mia estimates the total weight of the lighter boxes and the total weight of the heavier boxes, then adds the two estimates. Here’s my thinking… B. How can I critique the reasoning of others? I can • ask questions for clarification. • decide if the strategy used makes sense. • look for flaws in estimates or calculations. C. Mia’s reasoning has flaws. She estimated that 42 × 210 is a little more than 8,000, but a better estimate is 9,000. She underestimated the products so her conclusion is not valid. The weight of the heavier boxes is 8,820 pounds. The weight of the lighter boxes is 7,050 pounds. The total weight is 15,870 pounds. The sum is greater than 15,400. Mia’s reasoning does not make sense. Convince Me! Critique Reasoning Raul states that one way to get the cargo under the weight limit is to remove two of the heavier boxes and one of the lighter boxes. How can you decide if Raul’s reasoning makes sense? Guided Practice Critique Reasoning A stadium has 58 sections of seats. There are 288 seats in each section. Mary estimated the total number of seats by multiplying 60 × 300. She concluded that the stadium has fewer than 18,000 seats. Question 1. What is Mary’s argument? How does she support it? Answer: Mary’s argument was correct. Explanation: In the above-given question, given that, A stadium has 58 sections of seats. There are 288 seats in each section. Mary estimated the total number of seats by multiplying 60 × 300. She concluded that the stadium has fewer than 18,000 seats. 58 x 288 = 16704. Question 2. Describe at least one thing you would do to critique Mary’s reasoning. Answer: Mary’s argument was correct. Explanation: In the above-given question, given that, A stadium has 58 sections of seats. There are 288 seats in each section. Mary estimated the total number of seats by multiplying 60 × 300. She concluded that the stadium has fewer than 18,000 seats. 58 x 288 = 16704. Question 3. Does Mary’s conclusion make sense? Explain. Answer: Mary’s argument was correct. Explanation: In the above-given question, given that, A stadium has 58 sections of seats. There are 288 seats in each section. Mary estimated the total number of seats by multiplying 60 × 300. She concluded that the stadium has fewer than 18,000 seats. 58 x 288 = 16704. Independent Practice Critique Reasoning An office manager has$10,000 to spend on new equipment. He planned to purchase 300 lamps for $72 each. He completed the calculations at the right and concluded that there would be plenty of money left to buy additional equipment. Question 4. What does the office manager do to support his thinking? Answer: Yes, he completed the calculations at the right and concluded that there would be plenty of money. Explanation: In the above-given question, given that, An office manager has$10,000 to spend on new equipment.
He planned to purchase 300 lamps for $72 each. 300 x 72 = 21,600. yes, he has enough money. Question 5. Describe how you could decide if the office manager’s calculation is reasonable. Answer: Yes, he completed the calculations at the right and concluded that there would be plenty of money. Explanation: In the above-given question, given that, An office manager has$10,000 to spend on new equipment.
He planned to purchase 300 lamps for $72 each. 300 x 72 = 21,600. yes, he has enough money. When you critique reasoning, you need to explain if the method used by another makes sense. Question 6. Does the office manager’s conclusion make sense? Explain. Answer: Yes, he completed the calculations at the right and concluded that there would be plenty of money. Explanation: In the above-given question, given that, An office manager has$10,000 to spend on new equipment.
He planned to purchase 300 lamps for $72 each. 300 x 72 = 21,600. yes, he has enough money. Problem Solving Performance Task Buying a Piano Over the summer Kathleen sold 1,092 jars of jam at outdoor markets. She made a$12 profit on each one. She wants to use the profits to buy the Ivory-5K piano. She said, “Since 1,000 × 12 = 12,000, and 1,092 is greater than 1,000, I know my profits add up to more than $12,000. So, I can buy the piano.” Question 7. Make Sense and Persevere Does it make sense for Kathleen to find an overestimate or an underestimate to decide if she has earned enough money? Why? Answer: Yes, she can make the Ivory panio. Explanation: In the above-given question, given that, Kathleen sold 1,092 jars of jam at outdoor markets. She made a$12 profit on each one.
1092 x 12 = 13,104.
She wants to use the profits to buy the Ivory-5K piano. She said, “Since 1,000 × 12 = 12,000, and 1,092 is greater than 1,000.
yes, she can make the Ivory patio.

Question 8.
Reasoning Should Kathleen use multiplication to estimate her total profits? Explain your reasoning.

Answer:
Yes, she can make the total profits.

Explanation:
In the above-given question,
given that,
Kathleen sold 1,092 jars of jam at outdoor markets.
She made a $12 profit on each one. 1092 x 12 = 13,104. She wants to use the profits to buy the Ivory-5K piano. She said, “Since 1,000 × 12 = 12,000, and 1,092 is greater than 1,000. yes, she can make the total profits. When you critique reasoning, ask questions to help understand someone’s thinking. Question 9. Be Precise Is Kathleen’s estimate appropriate? Is her calculation correct? Explain. Answer: Question 10. Critique Reasoning Explain whether Kathleen’s conclusion is logical. How did you decide? If it is not logical, what can you do to improve her reasoning? Answer: ### Topic 3 Fluency Practice Activity Follow the path Solve each problem. Then follow multiples of 10 to shade a path from START to FINISH. You can only move up, down, right, or left. Answer: The multiples of 10 are 1, 10, 20, 5, 2, 30, 40, 50, 60, 70, 80, 90, and 100. Explanation: In the above-given question, given that, 53 x 20 = 1060. 70 x 89 = 6230. 84 x 40 = 3360. 60 x 90 = 5400. 10 x 570 = 5700. 80 x 14 = 1120. 50 x 30 = 1500. 70 x 12 = 840. 100 x 100 = 10000. Topic 3 Vocabulary Review Glossary Word List • expression • multiple • overestimate • partial products • power • underestimate • variable For each of these terms, give an example and a non-example. Answer: power of 10 = 200. multiple of 10 x 10 = 100. an expression with a variable an underestimate of 532 x 11 = 5852. Explanation: In the above-given question, given that, power of 10 = 200. multiple of 10 x 10 = 100. an expression with a variable an underestimate of 532 x 11 = 5852. Write always, sometimes, or never. Question 5. The sum of partial products is equal to the final product. Answer: Always the sum of partial products is equal to the final product. Explanation: In the above-given question, given that, the sum of partial products is equal to the final product. for example: 12 x 10 = 120. 10 + 2 = 12. so Always the sum of partial products is equal to the final product. Question 6. A multiple of a number is the power of the number. Answer: Sometimes a multiple of a number is a power of the number. Explanation: In the above-given question, given that, multiple of a number is a power of the number. for example: 2 x 2 = 4. Question 7. An underestimate results from rounding each factor to a greater number. Answer: Always an underestimate results from rounding each factor to a greater number. Explanation: In the above-given question, given that, An underestimate results from rounding each factor to a greater number. for example: 12.5 round the number to tenth. 12.6. Question 8. A power of a number is a multiple of the number. Answer: Yes, the power of a number is a multiple of the number. Explanation: In the above-given question, given that, power of a number is a multiple of the number. for example: 2 x 2 = 4. the square root of 2 is 4. Write T for true or F for false. Question 9. 642 × 12 = 642 tens + 1,284 ones Answer: The expression is false. Explanation: In the above-given question, given that, 642 x 12 = 7704. 642 + 1284 = 1926. so the expression is false. Question 10. 41 × 106 = 41,000,000 Answer: The expression is true. Explanation: In the above-given question, given that, 41 x 106. 41 x 10 x 10 x 10 x 10 x 10 x 10. 41000000. so the expression is true. Question 11. 80 × 103 = 8,000 Answer: The expression is false. Explanation: In the above-given question, given that, 80 × 103. 80 x 10 x 10 x 10. 80 x 1000. 80,000. so the expression is false. Question 12. Suppose both factors in a multiplication problem are multiples of 10. Explain why the number of zeros in the product may be different than the total number of zeros in the factors. Include an example. Answer: Topic 3 Reteaching Set A pages 81-84 Find 65 × 103. Look at the exponent for the power of 10. Annex that number of zeros to the other factor to find the product. Remember to look at the number of zeros or the exponent for the power of 10. Question 1. 12 × 104 Answer: The number of zeros is 4. Explanation: In the above-given question, given that, 12 × 104. 12 x 10000. 120000. Question 2. 100 × 815 Answer: The number of zeros is 2. Explanation: In the above-given question, given that, 100 x 815. 81500. so the number of zeros is 2. Question 3. 102 × 39 Answer: The number of zeros is 3900. Explanation: In the above-given question, given that, 102 × 39 100 x 39 = 3900. Question 4. 6,471 × 101 Answer: The number of zeros is 64710. Explanation: In the above-given question, given that, 6471 x 10. 64710. Set B pages 85-88 Estimate 37 × 88. Step 1 Round both factors. 37 is about 40 and 88 is about 90. Step 2 Multiply the rounded factors. 40 × 90 = 3,600 Remember to either round the factors or use compatible numbers. Estimate each product. Question 1. 7 × 396 Answer: 7 x 400 = 2800. Explanation: In the above-given question, given that, the two numbers are 7 and 396. 396 is equal to 400. 7 x 400 = 2800. Question 2. 17 × 63 Answer: 17 x 63 = 1071. Explanation: In the above-given question, given that, the two numbers are 17 and 63. 17 x 63 = 1071. Question 3. 91 × 51 Answer: 90 x 50 = 4500. Explanation: In the above-given question, given that, the two numbers are 91 and 51. 91 is equal to 90. 51 is equal to 50. 90 x 50 = 4500. Question 4. 45 × 806 Answer: 45 x 806 = 36000. Explanation: In the above-given question, given that, the two numbers are 45 and 806. 806 is equal to 800. 45 x 800 = 36000. Set C pages 89-92 Think: 4 × 9 ones = 36; 36 is 3 tens 6 ones. 4 × 4 tens = 16 tens; 16 tens + 3 tens = 19 tens; 19 tens is 1 hundred 9 tens. 4 × 2 hundreds = 8 hundreds; 8 hundreds + 1 hundred = 9 hundreds Remember to keep track of the place values. Find each product. Question 1. 133 × 3 Answer: 133 x 3 = 399. Explanation: In the above-given question, given that, the two numbers are 133 and 3. multiply two numbers. 133 x 3 = 399. Question 2. 343 × 5 Answer: 343 x 5 = 1715. Explanation: In the above-given question, given that, the two numbers are 343 and 5. multiply two numbers. 343 x 5 = 1715. Question 3. 893 × 7 Answer: 893 x 7 = 6251. Explanation: In the above-given question, given that, the two numbers are 893 and 7. multiply two numbers. 893 x 7 = 6151. Question 4. 1,278 × 4 Answer: 1278 x 4 = 5112. Explanation: In the above-given question, given that, the two numbers are 1278 and 4. multiply two numbers. 1278 x 4 = 5112. Set D pages 93-96 Find 17 × 35. Remember that you can draw arrays or area models to represent multiplication. Find each product. Question 1. 21 × 13 Answer: 21 x 13 = 273. Explanation: In the above-given question, given that, the two numbers are 21 and 13. multiply two numbers. 21 x 13 = 273. Question 2. 34 × 52 Answer: 34 x 52 = 1768. Explanation: In the above-given question, given that, the two numbers are 34 and 52. multiply two numbers. 34 x 52 = 1768. Question 3. 89 × 27 Answer: 89 x 27 = 2403. Explanation: In the above-given question, given that, the two numbers are 89 and 27. multiply two numbers. 89 x 27 = 2403. Question 4. 78 × 47 Answer: 78 x 47 = 3666. Explanation: In the above-given question, given that, the two numbers are 78 and 47. multiply two numbers. 78 x 47 = 3666. Set E pages 97-100, 101-104, 105-108 Find 53 × 406. Estimate: 50 × 400 = 20,000 Remember to regroup if necessary. Estimate to check that your answer is reasonable. Find each product. Question 1. 54 × 9 Answer: 54 x 9 = 486. Explanation: In the above-given question, given that, the two numbers are 54 and 9. multiply two numbers. 54 x 9 = 486. Question 2. 76 × 59 Answer: 76 x 59 = 4484. Explanation: In the above-given question, given that, the two numbers are 76 and 59. multiply two numbers. 76 x 59 = 4484. Question 3. 47 × 302 Answer: 47 x 302 = 14194. Explanation: In the above-given question, given that, the two numbers are 47 and 302. multiply two numbers. 47 x 302 = 14194. Question 4. 32 × 871 Answer: 32 x 871 = 27,872. Explanation: In the above-given question, given that, the two numbers are 32 and 871. multiply two numbers. 32 x 871 = 27872. Question 5. Answer: 604 x 55 = 33,220. Explanation: In the above-given question, given that, the two numbers are 604 and 55. multiply two numbers. 604 x 55 = 33220. Question 6. Answer: 7133 x 4 = 28532. Explanation: In the above-given question, given that, the two numbers are 7133 and 4. multiply two numbers. 7133 x 4 = 28532. Set F pages 109-112 Draw a picture and write an equation. Solve. The length of James’s pool is 16 feet. The length of the pool at Wing Park is 4 times as long. How long is the pool at Wing Park? 16 × 4 = l l = 64 feet The length of Wing Park pool is 64 feet. Remember that pictures and equations can help you model and solve problems. Draw a picture and write an equation. Solve. Question 1. Alexandria has a collection of 34 dolls. A toy store has 15 times as many dolls as Alexandria. How many dolls are in the store? Answer: The number of dolls is in the store = 510. Explanation: In the above-given question, given that, Alexandria has a collection of 34 dolls. A toy store has 15 times as many dolls as Alexandria. 34 x 15 = 510. so the number of dolls are in the store = 510. Question 2. A store received a shipment of 37 TVs valued at$625 each. What is the total value of the shipment?

Answer:
The total value of the shipment = $23,125. Explanation: In the above-given question, given that, A store received a shipment of 37 TVS valued at$625 each.
37 x $625 = 23,125. so the total value of the shipment =$23,125.

Set G
pages 113-116
Think about these questions to help you critique the reasoning of others.

Thinking Habits
• What questions can I ask to understand other people’s thinking?
• Are there mistakes in other people’s thinking?

Remember you need to carefully consider all parts of an argument.

Sarah has 214 bags of beads. Each bag has enough beads for 22 bracelets. She estimates that since 200 × 20 = 4,000, there are enough beads for at least 4,000 bracelets.
Tell how you can critique Sarah’s reasoning.

### Topic 3 Assessment Practice

Question 1.
Dr. Peterson works 178 hours each month. How many hours does she work in a year?
A. 2,000
B. 2,136
C. 3,000
D. 2,200

Answer:
The number of hours does she work in a year = 2136.

Explanation:
In the above-given question,
given that,
Dr. Peterson works 178 hours each month.
1 year = 365 days.
1 week = 7 days.
12 x 178 = 2136.
so option B is the correct.

Question 2.
A banana contains 105 calories. Last week, Brendan and Lea ate a total of 14 bananas. How many calories does this represent?

Answer:
The number of calories does this represent = 1470 calories.

Explanation:
In the above-given question,
given that,
A banana contains 105 calories.
Last week, Brendan and Lea ate a total of 14 bananas.
105 x 14 = 1470 calories.
so the number of calories does this represent = 1470.

Question 3.
At a warehouse, 127 delivery trucks were loaded with 48 packages on each truck.
A. Estimate the total number of packages on the trucks. Write an equation to model your work.
B. Did you calculate an overestimate or an underestimate? Explain how you know.

Answer:
The total number of packages on the trucks = 6096 trucks.

Explanation:
In the above-given question,
given that,
At a warehouse, 127 delivery trucks were loaded with 48 packages on each truck.
127 x 48 = 6096.
so the total number of packages on the trucks = 6096.

Question 4.
Is the equation below correct? Explain.
5.6 × 103 = 560
A. The equation is incorrect. The product should have 3 zeros.
B. The equation is correct. The product should have 1 zero.
C. The equation is incorrect. The product should have 0 zeros.
D. The equation is incorrect. The product should have 2 zeros.

Answer:
Option A is correct.

Explanation:
In the above-given question,
given that,
5.6 × 103 = 560.
5.6 = 560.
560 x 1000 = 560000.
so option A is correct.

Question 5.
The latest mystery novel costs $24. The table shows the sales of this novel by a bookstore. A. What was the dollar amount of sales of the mystery novel on Saturday? Write an equation to model your work. B. What was the dollar amount of sales of the mystery novel on Friday? Write an equation to model your work. Answer: A. The dollar amount of sales of the mystery novel on Saturday = 2472. B. The dollar amount of sales of the mystery novel on Friday = 3768. Explanation: In the above-given question, given that, The latest mystery novel costs$24.
98 books were sold on Thursday.
103 books were sold on Friday.
157 books were sold on Saturday.
116 books were sold on Sunday.
103 x 24 = 2472.
157 x 24 = 3768.

Question 6.
There are 45 cans of mi×ed nuts. Each can has 338 nuts. Below is Mary’s work to find the total number of nuts. What is the missing number? Enter your answer in the box.

Answer:
The missing number is 5.

Explanation:
In the above-given question,
given that,
There are 45 cans of mi×ed nuts.
Each can have 338 nuts.
338 x 45 = 15210.
so the missing number is 5.

Question 7.
There are 36 large fish tanks at the zoo. Each tank holds 205 gallons of water. How many gallons of water would it take to fill all of the tanks?

Answer:
The number of gallons of water would it take to fill all of the tanks = 7380 gallons.

Explanation:
In the above-given question,
given that,
There are 36 large fish tanks at the zoo.
Each tank holds 205 gallons of water.
205 x 36 = 7380.
so the number of gallons of water would it take to fill all of the tanks = 7380 gallons.

Question 8.
Kai ordered 1,012 baseball cards. Sharon ordered 5 times as many cards as Kai. Write and solve an equation to find b, the number of baseball cards Sharon ordered.

Answer:
The number of baseball cards Sharon ordered = 5060 cards.

Explanation:
In the above-given question,
given that,
Kai ordered 1,012 baseball cards.
Sharon ordered 5 times as many cards as Kai.
1012 x 5 = 5060.
so the number of baseball cards Sharon ordered = 5060 cards.

Question 9.
Multiply

Answer:
289 x 16 = 4624.

Explanation:
In the above-given question,
given that,
the two numbers are 289 x 16.
multiply the two numbers.
289 x 16 = 4624.

Question 10.
Match each number on the left with an equivalent expression.

Answer:
12 x 100 = 1200.
120 = 12 x 10.
12 = 12 x 10.
12000 = 12 x 1000.

Explanation:
In the above-given question,
given that,
12 x 100 = 1200.
120 = 12 x 10.
12 = 12 x 10.

Question 11.
Select all the expressions that are equal to 3 × 103.
3 × 1,000
3 × 100
30 × 100
300 × 100
300 × 10

Answer:
3 x 1000, 30 x 100, 300 x 10.

Explanation:
In the above-given question,
given that,
3 x 1000 = 3000.
30 x 100 = 3000.
300 x 10 = 3000.

Question 12.
Rosanne has 142 songs on her MP3 player. Teresa has 11 times as many songs as Rosanne. How many songs does Teresa have?

Answer:
The number of songs does Teresa has = 1562 songs.

Explanation:
In the above-given question,
given that,
Rosanne has 142 songs on her MP3 player.
Teresa has 11 times as many songs as Rosanne.
142 x 11 = 1562 songs.
so the number of songs does Teresa has = 1562 songs.

### Topic 3 Performance Task

Baseball Apparel
Coach Sandberg wants to buy items for the baseball league. The league already has caps with the league logo on them, but the coach would like to offer the option of purchasing a T-shirt, sweatshirt, sweatpants, or jacket with the logo. Use the information in the table to answer the questions.

Question 1.
The players asked their families and friends if they want to buy T-shirts with the league logo. If 254 people want T-shirts, what would be the total cost? Write an equation to model your work.

Answer:
The total cost is $3556. Explanation: In the above-given question, given that, The players asked their families and friends if they want to buy T-shirts with the league logo. If 254 people want T-shirts, 254 x$14 = 3556.
so the total cost is $3556. Question 2. Coach Sandberg wants to order 127 sweatshirts. Part A Will the total cost of the sweatshirts be greater than or less than$3,000? Use estimation to decide. Explain your reasoning.
Part B
What is the total cost of 127 sweatshirts?

Answer:
The total cost of 127 sweatshirts = $4064. Explanation: In the above-given question, given that, the cost of sweatshirts =$32.
127 x $32 =$4064.
so the total cost of 127 sweatshirts = $4064. Question 3. Which would cost more, 32 T-shirts or 14 sweatshirts? How can you tell without multiplying? Answer: The two items cost the same. Explanation: In the above-given question, given that, the cost of the T-shirts =$14.
cost of  sweatshirts = $32. 32 x 14 =$448.
14 x 32 = $448. so the two items cost the same. Question 4. There are 18 × 101 players in the league. Part A The league raised$1,560 through fundraisers. Trenton estimates the cost of buying jackets for each player in the league. He concludes that the league has raised enough money. Do you agree with Trenton? Explain.

Answer:
Yes, I agree with it.

Explanation:
In the above-given question,
given that,
The league raised $1,560 through fundraisers. Trenton estimates the cost of buying jackets for each player in the league. 200 x 50 = 1000. so I agree with it. Part B How much would it cost to order sweatpants for each player? Write and solve an equation with a variable to show your work. Answer: The cost to order sweatpants for each player = Explanation: In the above-given question, given that,$24

Question 5.
Which costs more: 136 sweatpants or 103 sweatshirts? How much more?

Answer:
The more is 32.

Explanation:
In the above-given question,
given that,
136 x $24 =$3264.
103 x $32 = 3296. 3296 – 3264 = 32. Question 6. Coach Sandberg wants to order 115 jackets and 27 caps for$12 each.
Part A
Estimate the total cost for his order. Show your work.
Part B
What is his total cost? Compare your answer to your estimate.

Answer:
The total cost is $439. Explanation: In the above-given question, given that, Coach Sandberg wants to order 115 jackets and 27 caps for$12 each.
27 x 12 = $324.$324 + 115 = 439.

## Rate

### Topic 5 Essential Question

What are ratios and rates? How can you use ratios and rates to describe quantities and solve problems?

3-ACT MATH

Get in Line
It is hard to call it a freeway when you are stuck in the middle of a traffic jam. To keep vehicles moving on the freeway, some on-ramps have traffic signals. Controlling when cars enter the freeway is not only about reducing delays. It can decrease air pollution and collisions.
These ramp meters typically have alternating green and red lights. The time for one cycle depends on the time of day and the amount of traffic on the freeway. Think about this during the 3-Act Mathematical Modeling lesson.

### Topic 5 enVision STEM Project

Your Task: Get into Gear
Cyclists strive to achieve efficiency during continuous riding. But, which pairing of gears is the best or most efficient? And does the answer change depending on the terrain? You and your classmates will explore gear ratios and how they can affect pedaling and riding speeds.

### Topic 5 Get Ready!

Review What You Know!

Vocabulary
Choose the best term from the box to complete each definition.

• common factor
• common multiple
• equivalent fractions
• fraction

Question 1.
Fractions that name the same amount are called ___________

Answer:
Fractions that name the same amount are called equivalent fractions.

Explanation:
In the above-given question,
given that,
fractions that name the same amount are called equivalent fractions.
for example:
1/2 = 2/4.
2 x 1 = 2.
2 x 2 = 4.

Question 2.
The number 3 is a ___________ of 9 and 12.

Answer:
The number 3 is a factor of 9 and 12.

Explanation:
In the above-given question,
given that,
the number 3 is a factor of 9 and 12.
for example:
G.C.F of 9 and 12 = 3.
H.C.F of 9 and 12 = 3.

Question 3.
A number that can be used to describe a part of a set or a part of a whole is a(n) ___________

Answer:
A number that can be used to describe a part of a set or a part of a whole is a(n) common factor.

Explanation:
In the above-given question,
given that,
A number that can be used to describe a part of a set or a part of a whole is a(n) common factor.
for example:
G.C.F of 9 and 12 = 3.
H.C.F of 9 and 12 = 3.

Equivalent Fractions

Write two fractions equivalent to the given fraction.
Question 4.
$$\frac{3}{4}$$

Answer:
The two fractions are equivalent to the 3/4 = 9/12 and 27/36.

Explanation:
In the above-given question,
given that,
the fraction is 3/4.
the two fractions are equivalent to 3/4 is:
27/36 = 3/4.
9/12 = 3/4.

Question 5.
$$\frac{7}{8}$$

Answer:
The two fractions are equivalent to the 7/8 = 14/16 and 21/24.

Explanation:
In the above-given question,
given that,
the fraction is 7/8.
the two fractions are equivalent to 7/8 is:
14/16 = 7/8.
21/24 = 7/8.

Question 6.
$$\frac{12}{5}$$

Answer:
The two fractions are equivalent to the 12/5 = 24/10 and 84/60.

Explanation:
In the above-given question,
given that,
the fraction is 12/5.
the two fractions are equivalent to 12/5 is:
24/10 = 12/5.
84/60 = 12/5.

Question 7.
$$\frac{1}{2}$$

Answer:
The two fractions are equivalent to the 1/2 = 2/4 and 3/6.

Explanation:
In the above-given question,
given that,
the fraction is 1/2.
the two fractions are equivalent to 1/2 is:
2/4 = 1/2.
3/6 = 1/2.

Question 8.
$$\frac{8}{9}$$

Answer:
The two fractions are equivalent to the 8/9 = 16/18 and 24/27.

Explanation:
In the above-given question,
given that,
the fraction is 8/9.
the two fractions are equivalent to 8/9 is:
16/18 = 8/9.
24/27 = 8/9.

Question 9.
$$\frac{2}{3}$$

Answer:
The two fractions are equivalent to the 2/3 = 4/6 and 12/9.

Explanation:
In the above-given question,
given that,
the fraction is 2/3.
the two fractions are equivalent to 2/3 is:
4/6 = 2/3.
12/9 = 2/3.

Equation

Write an equation that represents the pattern in each table.
Question 10.

Answer:
The equation is y = 8x.

Explanation:
In the above-given question,
given that,
x contains numbers 2, 3, 4, 5, and 6.
y : 16, 24, 32, 40, and 48.
y = 8x.
16 = 8 x 2.
24 = 8 x 3.
32 = 8 x 4.
40 = 8 x 5.
48 = 8 x 6.

Question 11.

Answer:
The equation is y = 2x + 1.

Explanation:
In the above-given question,
given that,
x : 2, 4, 6, 8, and 10.
y : 5, 7, 9, 11, and 13.
y = 2x + 1.
5 = 2(2) + 1.
7 = 2(4) – 1.
9 = 2(6) – 3.

Units of Measure.

Choose the best unit of measure by writing inch, foot, yard, ounce, pound, ton, cup, quart, or gallon.
Question 12.
serving of trail mix

Answer:
The serving of a trail mix can be measured in cups.

Explanation:
In the above-given question,
given that,
Serving of a trail mix.
for example:
nutrition facts are also measured in cups.

Question 13.
height of a person

Answer:
The height of a person can be measured in feet.

Explanation:
In the above-given question,
given that,
the height of a person can be measured in feet.
for example:
the height of the short girl is 5 feet.
the height of the tall girl is 5.6 feet.

Question 14.
weight of a newborn kitten

Answer:
The weight of a newborn kitten can be measured in ounces.

Explanation:
In the above-given question,
given that,
the weight of a newborn kitten can be measured in ounces.
for example:
newborn kittens usually weigh about 3.5 ounces.
a healthy kitten should gain at least 10 grams per day.

Question 15.
gasoline

Answer:
Gasoline can be measured in cubic feet.

Explanation:
In the above-given question,
given that,
Gasoline can be measured in cubic feet.
for example:
gas is sometimes measured in cubic feet at a temperature of 60 degrees Fahrenheit and an atmospheric pressure of 14.7 pounds per square inch.

Measurement Conversions

Question 16.
Michael is 4 feet tall. Explain how Michael could find his height in inches. Then explain how he could find his height in yards.

Answer:
The height in inches = 48.
the height in yards = 12 yards.
Explanation:
In the above-given question,
given that,
Michael is 4 feet tall.
1 feet = 12 inches.
12 x 4 = 48 inches.
1 yard = 3 feets.
3 x 4 = 12 feets.

Language Development

A bag contains the following marbles:

Complete each math statement.
The following ratio statement reads, for every 1 red marble, there are ___2_____ yellow marbles.
A ratio that compares the yellow marbles to the green marbles is ___6_____ to ____4____.
3 : 4 is the ratio of red marbles to____green____ marbles.
The following ratio statement reads, for every 1 blue marble, there are 2 ___yellow_____ marbles.
$$\frac{3}{2}$$ represents the ____ratio____ of red marbles to blue marbles.
4 to 15 is the ratio of green marbles to the ___total_____ number of marbles.
In the ratio of yellow marbles to blue marbles, 6:2, the quantities 6 and 2 are called __ratio______
A __ratio______ compares one ___quantity_____ to another ___quantity_____.

Pick A Project

PROJECT 5A
What animal would you most like to have as a pet?
PROJECT: COMPARE COSTS OF PET FOODS

Answer:
The animal I would most like to have as a pet is the dog.

Explanation:
In the above-given question,
given that,
the animal I would most like to have as a pet is the dog.
for example:
the cost of the lams is $1.07. the cost of the Victor is$1.44.

PROJECT 5B
How fast do you think you can throw a baseball?
PROJECT: ANALYZE A SPORT STATISTIC

PROJECT 5C
What color would you want to paint a room?
PROJECT: EXPERIMENT WITH COMBINATIONS OF COLORS

Answer:
The color I would like to paint a room is cream color.

Explanation:
In the above-given question,
given that,
the color I would like to paint a room is cream color.
for example:
there are many different colors.
they are cream, red, pink, blue, and orange.

PROJECT 5D
If you could visit any U.S. National Park, which would it be?
PROJECT: PLAN A TOUR

Answer:
I could visit the Yellowstone National Park.

Explanation:
In the above-given question,
given that,
the Yellowstone national park wilderness and recreation area with active geysers like old faithful, plus canyons, rivers, and lakes.
so I could visit the Yellowstone national park.
the area of Yellowstone national park is 8,991 sq km.

### Lesson 5.1 Understand Ratios

Explore It!
A band just released an album that contains both pop songs and R&B (rhythm and blues) songs.

I can… use a ratio to describe the relationship between two quantities.

A. How can you describe the relationship between the number of pop songs and the number of R&B songs on the album?

Answer:
The relationship between the number of pop songs and the number of R&B songs is 3: 6.

Explanation:
In the above-given question,
given that,
there are 9 pop songs.
there are 6 R&B songs.
the relationship between the number of pop songs and the number of R&B songs is 9 and 6.
9 : 6 = 3 : 6.
3 x 3 = 9.
3 x 2 = 6.
so the relationship between the number of pop songs and the number of R&B songs is 3: 6.

B. How does the bar diagram represent the relationship between the number of pop songs and the number of R&B songs?

Answer:
The bar diagram represents the relationship between the number of pop songs and the number of R&B songs is 3: 6.

Explanation:
In the above-given question,
given that,
there are 9 pop songs.
there are 6 R&B songs.
the relationship between the number of pop songs and the number of R&B songs is 9 and 6.
9 : 6 = 3 : 6.
3 x 3 = 9.
3 x 2 = 6.
so the relationship between the number of pop songs and the number of R&B songs is 3: 6.

Focus on math practices
Reasoning Another album has 2 pop songs and 10 R&B songs. Draw a bar diagram that you could use to represent the relationship between the number of pop songs and the number of R&B songs.

Answer:
The relationship between the number of pop songs and the number of R&B songs = 1: 5.

Explanation:
In the above-given question,
given that,
Another album has 2 pop songs and 10 R&B songs.
2 : 10 = 1 : 5.
so the relationship between the number of pop songs and the number of R&B songs = 1: 5.

Essential Question
What is a mathematical way to compare quantities?

Try It!

What are three ways to write the ratio of the number of dogs to the total number of pets?

Answer:
The ratio of the number of dogs to the total number of pets = 3: 3.

Explanation:
In the above-given question,
given that,
the ratio of a number of dogs to the total number of pets is same.
for example:
3 : 3.
1: 1.

Convince Me! Is the ratio of dogs to cats the same as the ratio of cats to dogs? Explain.

Try It!

Chen’s friend Alisa can ride her bike 2 miles in 7 minutes. Use a bar diagram or a double number line diagram to find how long it would take Alisa to ride 10 miles if she rides at the same rate.

Answer:
The longer it would take Alisa to ride 10 miles = 70 minutes.

Explanation:
In the above-given question,
given that,
Chen’s friend Alisa can ride her bike for 2 miles in 7 minutes.
2 x 7 = 14 minutes.
10 x 7 = 70 minutes.
so the longer it would take Alisa to ride 10 miles = 70 minutes.

KEY CONCEPT
A ratio compares two quantities. A ratio can be written 3 ways: x to y, x:y, or Ratios can be represented using bar diagrams and double number line diagrams.

Do You Understand?
Question 1.
Essential Question What is a mathematical way to compare quantities?

Answer:
A ratio compares two quantities.
A ratio can be written in 3 ways: x to y, x: y, and x and y.

Explanation:
In the above-given question,
given that,
A ratio compares two quantities.
a ratio can be written in 3 ways are x to y.
x: y, and x/y.
so ratio compares two quantities.

Question 2.
Reasoning What are two different types of comparisons that a ratio can be used to make?

Answer:
The two different types of comparisons that a ratio can be used to make are x:y and x/y.

Explanation:
In the above-given question,
given that,
the ratio is x: y.
x to y.
x/y.
so the two different types of comparisons that a ratio can be used to make are x: y and x / y.

Question 3.
A science classroom has 5 turtles and 7 frogs. What is the ratio of frogs to total animals?

Answer:
The ratio of frogs to total animals is 7: 12.

Explanation:
In the above-given question,
given that,
A science classroom has 5 turtles and 7 frogs.
there are 7 frogs in the science classroom.
totally there are 12 animals.
frogs: animals.
7: 12.
so the ratio of frogs to total animals is 7: 12.

Question 4.
Tye is making trail mix with 3 cups of nuts for every 4 cups of granola. If Tye has 6 cups of nuts, how many cups of granola should he use?

Answer:
The number of cups of granola should use = 2: 3.

Explanation:
In the above-given question,
given that,
Tye is making trail mix with 3 cups of nuts for every 4 cups of granola.
6 : 3 = 2 : 3.
so the number of cups of granola should use = 2: 3.

Do You Know How?
In 5-7, use three different ways to write a ratio for each comparison.

A sixth-grade basketball team has 3 centers, 5 forwards, and 6 guards.
Question 5.
Forwards to guards

Answer:
Forwards to guards = 5: 6.

Explanation:
In the above-given question,
given that,
A sixth-grade basketball team has 3 centers, 5 forwards, and 6 guards.
Forwards to guards:
5: 6.

Question 6.
Centers to total players

Answer:
Centers to total players= 3: 14.

Explanation:
In the above-given question,
given that,
Centers to total players.
the total number of players = 14.
centers to total players = 3: 14.

Question 7.
Guards to centers

Answer:
Guards to centers = 6 : 3.

Explanation:
In the above-given question,
given that,
there are 6 Guards and 3 Guards.
6 : 3 = 2 : 3.
so Guards to centers = 6 : 3.

Question 8.
The ratio of blue cards to green cards is 2 to 5. There are 8 blue cards. Complete the diagram and explain how you can find the number of green cards.

Answer:
The number of blue cards to the number of green cards = 8: 11.

Explanation:
In the above-given question,
given that,
The ratio of blue cards to green cards is 2 to 5.
There are 8 blue cards.
the number of green cards is 11.
so the number of blue cards to the number of green cards = 8: 11.

Practice & Problem Solving

In 9-14, use the data to write a ratio for each comparison in three different ways.

A person’s blood type is denoted with the letters A, B, and O, and the symbols + and -. The blood type A+ is read as A positive. The blood type B- is read as B negative.

Question 9.
O+ donors to A+ donors

Answer:
O+ donors to A+ donors = 2 : 1.

Explanation:
In the above-given question,
given that,
A person’s blood type is denoted with the letters A, B, and O, and the symbols + and -.
O+ donors to A+ donors.
there are 90 O+ donors.
there are 45 A+ donors.
90 : 45 = 2 : 1.

Question 10.
AB-donors to AB+ donors

Answer:
AB- donors to AB+ donors = 2 : 3.

Explanation:
In the above-given question,
given that,
A person’s blood type is denoted with the letters A, B, and O, and the symbols+ and -.
AB- donors to AB+ donors.
there are 4 AB- donors.
there are 6 AB+ donors.
4 : 6 = 2 : 3.
so AB- donors to AB+ donors = 2 : 3.

Question 11.
B+ donors to total donors

Answer:
B+ donors to total donors = 20 : 195.

Explanation:
In the above-given question,
given that,
totally there are 195 donors.
B+ donors to total donors.
20: 195.

Question 12.
O- donors to A-donors

Answer:
O- donors to A- donors = 9 : 21.

Explanation:
In the above-given question,
given that,
O- donors to A- donors.
there are 9 O- donors.
there are 21 A- donors.
O- donors to A- donors = 9 : 21.

Question 13.
A+ and B+ donors to AB+ donors

Answer:
A+ and B+ donors to AB+ donors = 6 : 65.

Explanation:
In the above-given question,
given that,
A+ and B+ donors to AB+ donors.
there are A+ and B+ donors who are 65.
there are  6 AB+ donors.
A+ and B+ donors to AB+ donors = 6 : 65.

Question 14.
A- and B-donors to AB- donors

Answer:
A- and B- donors to AB- donors = 21: 4.

Explanation:
In the above-given question,
given that,
there are 21 A- donors.
there are 0 B- donors.
there are 4 AB- donors.
so A- and B- donors to AB- donors = 21: 4.

Question 15.
Which comparison does the ratio $$\frac{90}{9}$$ represent?

Answer:
The ratio 90/9 represent = 10 : 1.

Explanation:
In the above-given question,
given that,
the ratio 90/9 represents.
90 : 9 = 10: 1.
so the ratio represent = 10 : 1.

Question 16.
Which comparison does the ratio 20:21 represent?

Answer:
The comparison does the ratio 20:21 represent = B+ and A-.

Explanation:
In the above-given question,
given that,
there are 20 B+ donors.
there are 21 A- donors.
so the ratio 20:21 represent = B+ and A+.

Question 17.
Sam is packing gift boxes with fruit. For each apple, he packs 3 plums and 5 oranges. If he puts 3 apples in a box, how many plums and oranges will Sam put in the box? Draw a diagram to solve the problem.

Answer:
The number of plums and oranges will Sam put in the box is 9:15.

Explanation:
In the above-given question,
given that,
Sam is packing gift boxes with fruit.
For each apple, he packs 3 plums and 5 oranges.
1:3, 2:6, and 3:9.
1:5, 2:10, and 3:15.
so the number of plums and oranges will Sam put in the box is 9:15.

Question 18.
Write a ratio that compares the number of teal squares to the total number of squares in the quilt.

Answer:
The ratio that compares the number of teal squares to the total number of squares in the quilt = 1:3.

Explanation:
In the above-given question,
given that,
there are 18 teal squares and 6 squares.
6 : 18 = 1:3.
so the ratio that compares the number of teal squares to the total number of squares in the quilt = 1:3.

Question 19.
Reasoning Rita’s class has 14 girls and 16 boys. How does the ratio 14:30 describe Rita’s class?

Answer:
The ratio 14:30 describes there are 7 girls and 15 boys.

Explanation:
In the above-given question,
given that,
Rita’s class has 14 girls and 16 boys.
there are 14 girls and 16 boys.
there are 14 girls and 30 boys.
14: 30 = 7:15.
so there are 7 girls and 15 boys.

Question 20.
A math class surveyed students about their musical preferences and recorded the results in the table. Use the data to write a ratio for each comparison in three different ways.

a. Students who prefer classical to students who prefer techno

Answer:
The students who prefer classical to students who prefer techno is 1:3.

Explanation:
In the above-given question,
given that,
there are 4 classical students.
there are 12 techno students.
4 : 12 = 1:3.
so the students who prefer classical to students who prefer techno is 1:3.

b. Students who prefer hip-hop to total number of students surveyed

Answer:
Students who prefer hip-hop to the total number of students surveyed = 15:53.

Explanation:
In the above-given question,
given that,
there are 15 hip-hop students.
the total number of students is 53.
so the ratio is 15:53.

Question 21.
Construct Arguments Justin used blocks to model the following situation: A car dealership sells 7 cars for every 4 minivans it sells. How can Justin use his model to find the number of minivans the dealership sells if it sells 35 cars?

Answer:
The number of minivans the dealership sells if it sells 35 cars = 20 minivans.

Explanation:
In the above-given question,
given that,
A car dealership sells 7 cars for every 4 minivans it sells.
35/7 = 5.
4 x 5 = 20.
so the number of minivans the sealership sells if it sells 35 cars = 20 minivans.

Question 22.
Make Sense and Persevere The ratio of adult dogs to puppies at a dog beach in Florida on Monday was 3:2. There were 12 puppies there that day. On Tuesday, 15 adult dogs were at the dog beach. What is the difference between the number of adult dogs at the dog beach on Monday and Tuesday?

Answer:
The difference between the number of adult dogs at the dog beach on Monday and Tuesday = 1:3.

Explanation:
In the above-given question,
given that,
The ratio of adult dogs to puppies at a dog beach in Florida on Monday was 3:2.
There were 12 puppies there that day.
On Tuesday, 15 adult dogs were at the dog beach.
12:15 = 4:5.
4:5 – 3:2 = 1:3.
so the difference between the number of adult dogs at the dog beach on Monday and Tuesday = 1:3.

Question 23.
Higher Order Thinking At 9:30 A.M., Sean started filling a swimming pool. At 11:30 A.M., he had filled 1,800 gallons. At what time will the pool be full?

Answer:
At 2:30 P.M the pool will fill completely.

Explanation:
In the above-given question,
given that,
At 9:30 A.M., Sean started filling a swimming pool.
At 11:30 A.M., he had filled 1,800 gallons.
for 2 hours it will fill 1800 gallons.
1800 + 1800 = 3600.
3600 + 900 = 4500.
so at 2:30 P.M the pool will fill completely.

Assessment Practice

Question 24.
The diagram below represents the relationship between the number of students taking Spanish and the number of students taking French in a foreign language class.

What is the ratio of the number of students taking Spanish to the number of students taking French?
A. 8 : 3
B. 8 : 5
C. 8 : 8
D. 8 : 13

Answer:
The ratio of the number of students taking Spinach to the number of students taking French = 8: 5.

Explanation:
In the above-given question,
given that,
the number of Spanish students is 8.
the number of French students is 5.
the ratio is 8:5.
so the ratio of a number of students taking spinach to the number of students taking french = 8:5.

### Lesson 5.2 Generate Equivalent Ratios

Solve & Discuss It!
Sally used all of the paint shown below to make a certain tint of orange paint. How many pints of red paint should be mixed with 24 pints of yellow paint to make the same tint of orange?
I can… use multiplication and division to find equivalent ratios.

Look for Relationships
How can you use the relationship between the number of pints of yellow paint and the number of pints of red paint to answer the question?

Answer:
The ratio of yellow paint to the red paint is 4:3.

Explanation:
In the above-given question,
given that,
the number of yellow paint is 4.
the number of red paint is 3.
the relationship between the number of pints of yellow paint and the number of pints of red paint is 4:3.

Focus on math practices
Reasoning If Sally uses the same ratio of yellow paint to red paint, how many pints of yellow paint should she mix with 16 pints of red paint?

Essential Question
How can you find equivalent ratios?

Try It!

If you extend the table above, how would you find the next ratio of basketball players to soccer players?
Answer:

Convince Me! What is the relationship between the number of basketball players and the number of soccer players in each column in the table?

Try It!

Rashida uses 8 cups of tomatoes and 3 cups of onions to make salsa. How many cups of onions should Rashida use if she uses only 4 cups of tomatoes?

Answer:
Rashida use 1.5 cups of onions when she use 4 cups of tomatoes.

Explanation:
In the above-given question,
given that,
Rashida uses 8 cups of tomatoes and 3 cups of onions to make salsa.
for 1.5 cups of onions is used for 4 cups of tomatoes.
4 : 1.5.
so the ratio is 4:1.5.

Try It!

Which of the following ratios are equivalent to 16:20?
2:3, 4:5, 18:22, 20:25

Answer:
The ratio 4:5 equal to 16:20.

Explanation:
In the above-given question,
given that,
the ratios are 2:3, 4:5, 18:22, and 20:25.
16: 20 = 4:5.
so the ratio 4:5 equal to 16:20.

KEY CONCEPT
You can multiply or divide both terms of a ratio by the same nonzero number to find equivalent ratios.

Do You Understand?
Question 1.
Essential Question How can you find equivalent ratios?

Answer:
We can multiply or divide both terms of a ratio by the same nonzero number to find equivalent ratios.

Explanation:
In the above-given question,
given that,
multiply both terms by same non-zero number.
divide both terms by same non-zero number.
for example:
30 x 2 = 60.
40 x 2 = 80.

Question 2.
Critique Reasoning Deshawn says that the ratios 3:5 and 5:7 are equivalent ratios because by adding 2 to both terms of 3:5 you get 5:7. Is Deshawn correct? Explain.

Answer:
No, he was not correct.

Explanation:
In the above-given question,
given that,
Deshawn says that the ratios 3:5 and 5:7 are equivalent ratios because by adding 2 to both terms of 3:5 you get 5:7.
3:5 and 9:15.
so he was not correct.

Question 3.
What are two ways you can find an equivalent ratio for $$\frac{12}{16}$$?

Answer:
The equivalent ratio for 12/16 is 3:4.

Explanation:
In the above-given question,
given that,
the ratio is 12/16.
12: 16 = 3:4.
so the ratio is 3:4.

Question 4.
How can you show that the ratios 10:4 and 15:6 are equivalent?

Answer:
The ratios are not equal.

Explanation:
In the above-given question,
given that,
the ratios are 10:4 and 15:6.
10 : 4 = 15:6.
so the ratios are not equal.

Do You Know How?
Question 5.
Complete the table using multiplication to find ratios that are equivalent to 4:5.

Answer:
The ratios are 8:10, 12:15, and 16:20.

Explanation:
In the above-given question,
given that,
the ratio is 4:5.
8 : 10 = 4:5.
12:15 = 4:5.
16:20 = 4:5.

Question 6.
Complete the table using division to find ratios that are equivalent to 40:28.

Answer:
The ratios that are equivalent to 40:28 = 20:14 and 10:7.

Explanation:
In the above-given question,
given that,
the numbers are 40/28.
40/28 = 20/14.
40/28 = 10/7.
so the ratios that are equivalent to 40:28 = 20:14 and 10:7.

In 7-10, write an equivalent ratio for each given ratio.
Question 7.
$$\frac{12}{21}$$

Answer:
The equivalent ratio is

Explanation:
In the above-given question,
given that,
the ratio is 12/21.

Question 8.
1:3

Answer:
The equivalent ratio is 3:9.

Explanation:
In the above-given question,
given that,
the ratio is 1:3.
3:9 = 1:3.
3 x 1 = 3.
3 x 3 = 9.
so the equialent ratio is 3:9.

Question 9.
6 to 8

Answer:
The equivalent ratio is 3:4.

Explanation:
In the above-given question,
given that,
the ratio is 6 to 8.
6/8 = 3/4.
so the ratio is 3/4.
2 x 3 = 6.
2 x 4 = 8.

Question 10.
Pi (st) can be approximated using decimals as the ratio 3.14:1. Find 3 ratios equivalent to the ratio 3.14:1.

Answer:
The equivalent ratio is 22/7:1.

Explanation:
In the above-given question,
given that,
the ratio is 3.14:1.
22/7 = 3.14.
22/7:1 = 3.14:1.

Practice & Problem Solving

Question 11.
Eva is making French toast. How many ounces of milk should Eva use with 10 eggs?

Answer:
The recipe uses 10, 15, 20, 25, and 30.

Explanation:
In the above-given question,
given that,
for 2 eggs the recipe uses 5 ounces of milk.
for 4 eggs the recipe uses 10 ounces of milk.
for 6 eggs the recipe uses 15 ounces of milk.
for 8 eggs the recipe uses 20 ounces of milk.
for 10 eggs the recipe uses 25 ounces of milk.

In 12-15, write three ratios that are equivalent to the given ratio.
Question 12.
$$\frac{6}{7}$$

Answer:
The three ratios are 3/14 and 18/21.

Explanation:
In the above-given question,
given that,
the ratio is 6/7.
3/14 = 6/7.
18/21 = 6/7.
so the three ratios are 3/14 and 18/21.

Question 13.
$$\frac{9}{5}$$

Answer:
The three ratios are 27/15 and 18/10.

Explanation:
In the above-given question,
given that,
the ratio is 9/5.
18/10 = 9/5.
27/15 = 9/5.

Question 14.
8:14

Answer:
The three ratios are 16/28 and 24/42.

Explanation:
In the above-given question,
given that,
the ratio is 8/14.
16/28 = 8/14.
24/42 = 8/14.
so the three ratios are 16/28 and 24/42.

Question 15.
7:9

Answer:
The three ratios are 6/28 and 21/27.

Explanation:
In the above-given question,
given that,
the ratio is 7/9.
6/28 = 7/9.
21/27 = 7/9.

Question 16.
A teacher kept track of what students consumed at a school picnic. For three grades, the ratios of the amount of water consumed to the amount of fruit juice consumed were equivalent. Complete the table.

Answer:
The juice contains 21 and 28 gallons.

Explanation:
In the above-given question,
given that,
A teacher kept track of what students consumed at a school picnic.
For three grades, the ratios of the amount of water consumed to the amount of fruit juice consumed were equivalent.
for 6th grade 24 gallons of water contains the juice 28 gallons.
for 7th grade 18 gallons of water contains the juice 21 gallons.
so the juice contains 21 and 28 gallons.

Question 17.
The attendant at a parking lot compared the number of hybrid vehicles to the total number of vehicles in the lot during a weekend. The ratios for the three days were equivalent. Complete the table.

Answer:
The ratios for the three days were equivalent is 28 and 72.

Explanation:
In the above-given question,
given that,
The attendant at a parking lot compared the number of hybrid vehicles to the total number of vehicles in the lot during a weekend.
the ratios for the three days were equivalent.
28 and 72 are the equivalent ratio.

Question 18.
Shiloh is sharing jellybeans. The jar of jellybeans has the ratio shown. If Shiloh keeps the ratio the same and gives his friend 7 pink jellybeans, how many green jellybeans should he also share?

Answer:
The number of green jellybeans should he also share =

Explanation:
In the above-given question,
given that,
Shiloh is sharing jellybeans.
The jar of jellybeans has the ratio shown.
If Shiloh keeps the ratio the same and gives his friend 7 pink jellybeans.

Question 19.
Use Appropriate Tools Equivalent ratios can be found by extending pairs of rows or columns in a multiplication table. Write three ratios equivalent to $$\frac{2}{5}$$ using the multiplication table.

Answer:
The three ratios equivalent to 2/5 are 4/10 and 6/15.

Explanation:
In the above-given question,
given that,
the ratio is 2/5.
4/10 = 2/5.
6/15 = 2/5.
so the three ratios equivalent to 4/10 and 6/15.

Question 20.
If 5 mi ≈ 8 km, about how many miles would be equal to 50 km? Explain.

Answer:
The number of miles is equal to 31.069 miles.

Explanation:
In the above-given question,
given that,
5 miles ≈ 8 km.
50 km is equal to 31.069 miles.
so 31.069 miles is equal to 50 km.

Question 21.
Vocabulary How is the word term defined when used to describe a ratio relationship? How is the word term defined in the context of an expression?

Answer:
The term is one of the two numbers in the ratio a to b.
where a is the first term and b is the second term.

Explanation:
In the above-given question,
given that,
The term is one of the two numbers in the ratio a to b.
where a is the first term and b is the second term.
it is also used to indicate each one of the 4 numbers in a proportion.
if a = c.
then a,b,c, and d are the terms of the proportion.

Question 22.
Higher Order Thinking Three sisters are saving for a special vacation in Orlando, Florida. The ratio of Ada’s savings to Ellie’s savings is 7:3, and the ratio of Ellie’s savings to Jasmine’s savings is 3:4. Together all three girls have saved $56. How much has each girl saved? Complete the table. Explain how the table can be used to solve the problem. Answer: The Ada’s savings are$14 and $28. Ellie’s savings are$3, $9, and$12.
Jasmine’s savings are $8 and$12.

Explanation:
In the above-given question,
given that,
Three sisters are saving for a special vacation in Orlando, Florida.
The ratio of Ada’s savings to Ellie’s savings is 7:3, and the ratio of Ellie’s savings to Jasmine’s savings is 3:4.
Together all three girls have saved $56. 7 x 2 = 14, 7 x 4 = 28. 3 x 1 = 3, 3 x 3 = 9, and 3 x 4 = 12. 4 x 2 = 8, 4 x 3 = 12, and 4 x 4 = 16. Assessment Practice Question 23. Corey is making key lime pies for the school fair. For every 3 egg yolks, he uses 2 tablespoons of key lime zest. PART A Complete the table to find equivalent ratios. Answer: The egg yolks are 6, 9, and 12. Explanation: In the above-given question, given that, Corey is making key lime pies for the school fair. For every 3 egg yolks, he uses 2 tablespoons of key lime zest. the equivalent ratios are 6, 9, and 12. so the egg yolks are 6, 9, and 12. PART B How can you use the table to find how many egg yolks are needed for 8 tablespoons of lime zest? Answer: The number of egg yolks are 12 needed for 8 tablespoons of lime zest. Explanation: In the above-given question, given that, Corey is making key lime pies for the school fair. For every 3 egg yolks, he uses 2 tablespoons of key lime zest. 3 x 1 = 3, 3 x 2 = 6, 3 x 3 = 9, and 3 x 4 =12. so the number of egg yolks are 12 needed for 8 tablespoons of lime zest. Question 24. Which ratios can be represented by Pi (t)? Select all that apply. ☐ Diameter : Circumference ☐ Circumference : Diameter ☐ Circumference : Radius ☐ Radius : Circumference ☐ Circumference : Twice the radius Answer: Options A and B are correct. Explanation: In the above-given question, given that, diameter and circumference are represented. so options A and B are correct. ### Lesson 5.3 Compare Ratios Solve & Discuss It! Scott is making a snack mix using almonds and raisins. For every 2 cups of almonds in the snack mix, there are 3 cups of raisins. Ariel is making a snack mix that has 3 cups of almonds for every 5 cups of sunflower seeds. If Scott and Ariel each use 6 cups of almonds to make a batch of snack mix, who will make a larger batch? I can… compare ratios to solve problems. Model with Math How can you use ratio tables to represent Scott’s and Ariel’s snack mixes? Focus on math practices Look for Relationships Scott and Ariel want to make as much snack mix as possible, but no more than 25 cups of mix. If they can use only full cups of ingredients, who can make more mix without going over? Essential Question How can you compare ratios to solve a problem? Try It! Marlon had 6 hits in 15 at bats. How does Marlon’s hits to at bats ratio compare to Adrian’s? Answer: The ratio is 2:5. Explanation: In the above-given question, given that, Marlon had 6 hits in 15 at bats. 6 : 15 = 2:5. so the ratio is 2:5. Convince Me! Based on their hits to at bats ratios, who would you expect to have more hits in a game, Marlon or Dustin? Explain. Try It! Tank 3 has a ratio of 3 guppies for every 4 angelfish. Complete the ratio table to find the number of angelfish in Tank 3 with 12 guppies. Using the information in Example 2 and the table at the right, which tank with guppies has more fish? Answer: The number of Guppies is 6, 9, and 12. the number of Angelfish is 8, 12, and 16. Explanation: In the above-given question, given that, Tank 3 has a ratio of 3 guppies for every 4 angelfish. 3 x 1 = 3. 3 x 2 = 6. 3 x 3 = 9. 3 x 4 = 12. 4 x 1 = 4. 4 x 2 = 8. 4 x 3 = 12. 4 x 4 = 16. KEY CONCEPT You can use ratio tables to compare ratios when one of the corresponding terms is the same. Do You Understand? Question 1. Essential Question How can you compare ratios to solve a problem? Answer: We can use ratio tables to compare ratios when one of the corresponding terms is the same. Explanation: In the above-given question, given that, we can use ratio tables to compare ratios when one of the corresponding terms is the same. for example: 5 x 2 = 10. 5 x 3 = 15. 5 x 4 = 20. Question 2. In Example 1, how many hits would Adrian have in 50 at bats? Explain. Answer: Question 3. Reasoning During the first week of a summer camp, 2 out of 3 campers were boys. During the second week, 3 out of 5 campers were boys. There were 15 total campers each week. During which week were there more boy campers? Explain. Answer: In the second week, there are more boy campers. Explanation: In the above-given question, given that, During the first week of a summer camp, 2 out of 3 campers were boys. During the second week, 3 out of 5 campers were boys. There were 15 total campers each week. so in the second week, there are more boy campers. Do You Know How? Question 4. To make plaster, Kevin mixes 3 cups of water with 4 pounds of plaster powder. Complete the ratio table. How much water will Kevin mix with 20 pounds of powder? Answer: The amount of Kevin mix with 20 pounds of powder = 15 cups of water. Explanation: In the above-given question, given that, To make plaster, Kevin mixes 3 cups of water with 4 pounds of plaster powder. 3 x 2 = 6. 3 x 3 = 9. 3 x 4 = 12. 3 x 5 = 15. so the amount of Kevin mix with 20 pounds of powder = 15 cups of water. Question 5. Jenny makes plaster using a ratio of 4 cups of water to 5 pounds of plaster powder. Whose plaster recipe uses more water? Use the ratio table here and in Exercise 4 to compare. Answer: 16 cups of water to 20 pounds of powder. Explanation: In the above-given question, given that, Jenny makes plaster using a ratio of 4 cups of water to 5 pounds of plaster powder. 8 cups of water to 10 pounds of powder. 12 cups of water to 15 pounds of powder. 16 cups of water to 20 pounds of powder. Question 6. Kevin and Jenny each use 12 cups of water to make plaster. Who will make more plaster? Explain. Answer: Kevin makes more plaster than Jenny. Explanation: In the above-given question, given that, Kevin and Jenny each use 12 cups of water to make plaster. Kevin uses 12 cups of water to make 20 pounds of plaster. Jenny uses 12 cups of water to make 15 pounds of plaster. Practice & Problem Solving In 7-10, use the ratio table at the right. Question 7. Local radio station WMTH schedules 2 minutes of news for every 20 minutes of music. Complete the ratios shown in the table at the right. Answer: The ratios are 30/3, 40/4, 50/5, and 60/6. Explanation: In the above-given question, given that, 20 minutes of music is equal to 2 minutes of news. 20/2 = 10. 30/3 = 10. 40/4 = 10. 50/5 = 10. 60/6 = 10. so the ratios are 30/3, 40/4, 50/5, and 60/6. Question 8. What is the ratio of minutes of music to minutes of news? Answer: The ratio of minutes of music to minutes of news = 10:1. Explanation: In the above-given question, given that, 20 minutes of music is equal to 2 minutes of news. 20/2 = 10. 30/3 = 10. 40/4 = 10. 50/5 = 10. 60/6 = 10. so the ratios are 30/3, 40/4, 50/5, and 60/6. Question 9. Radio station WILM broadcasts 4 minutes of news for every 25 minutes of music. Which radio station broadcasts more news each hour? Answer: The radio station broadcasts more news each hour = Explanation: In the above-given question, given that, Radio station WILM broadcasts 4 minutes of news for every 25 minutes of music. Question 10. Which station will have to be on the air longer to broadcast 4 minutes of news? Explain. Answer: The station will have to be on the air longer to broadcast 4 minutes of news = 40 minutes of music. Explanation: In the above-given question, given that, 20 minutes of music is equal to 2 minutes of news. 20/2 = 10. 30/3 = 10. 40/4 = 10. 50/5 = 10. 60/6 = 10. so the station will have to be on the air longer to broadcast 4 minutes of news = 40 minutes of music. Question 11. Reasoning The ratio tables at the right show the comparison of books to games for sale at Bert’s Store and at Gloria’s Store. Complete the ratio tables. Which store has the greater ratio of books to games? Explain. Answer: The ratio of Bert’s store is 5/7, 6/8, 7/9, and 8/10. The ratio of Gloria’s store is 5/8, 6/9, 7/12, 8/15, and 9/18. Explanation: In the above-given question, given that, The ratio tables at the right show the comparison of books to games for sale at Bert’s Store and at Gloria’s Store. the ratios of Bert’s store are 5/7, 6/8, 7/9, and 8/10. the ratio of Gloria’s store is 5/8, 6/9, 7/12, 8/15, and 9/18. Question 12. The ratio of soy sauce to lime juice in a homemade salad dressing is 7:6. The ratio of soy sauce to lime juice in a store-bought dressing is 11:9. Which dressing has the greater ratio of soy sauce to lime juice? Answer: The ratio of Soy sauce to Lime juice is 7:6, 8:7, 9:8, 10:9, and 11:10. the ratio of Soy sauce to lime juice is 11:9, 12:10, 13:11, 14:12, and 15:13. Explanation: In the above-given question, given that, The ratio of soy sauce to lime juice in a homemade salad dressing is 7:6. The ratio of soy sauce to lime juice in a store-bought dressing is 11:9. the ratio of soy sauce to the Lime juice is 7:6, 8:7, 9:8, 10:9, and 11:10. the ratio of soy sauce to lime juice is 11:9, 12:10, 13:11, 14:12, and 15:13. Question 13. One bouquet of flowers has 3 milkweeds for every 5 tickseeds. Another bouquet has 4 tickseeds for every 5 canna lilies. If both bouquets have 20 tickseeds, which bouquet has more flowers? Answer: Bouquet 4 has more flowers. Explanation: In the above-given question, given that, One bouquet of flowers has 3 milkweeds for every 5 tickseeds. Another bouquet has 4 tickseeds for every 5 canna lilies. 3 x 5 =15. 4 x 6 = 24. 5 x 7 = 35. 6 x 8 = 48. 7 x 9 = 63. 4 x 5 = 20. 5 x 6 = 30. 6 x 7 = 42. 7 x 8 = 56. 8 x 9 = 72. Question 14. Higher Order Thinking Lauren can drive her car 320 miles on 10 gallons of gasoline. Melissa can drive her car 280 miles on 8 gallons of gasoline. Who can drive farther on 40 gallons of gasoline? Complete the ratio tables to justify your solution. Answer: Explanation: In the above-given question, given that, Lauren can drive her car 320 miles on 10 gallons of gasoline. Melissa can drive her car 280 miles on 8 gallons of gasoline. 320/10 and 280/8. Assessment Practice Question 15. Fran buys Florida cone seashells in packages that contain 9 purple-dyed Florida cone seashells for every 3 pink-dyed Florida cone seashells. Mia buys Florida cone seashells in packages with a ratio of 2 pink-dyed Florida cone seashells to 4 purple-dyed Florida cone seashells. PART A Complete the tables using the ratios given. Answer: The missing ratios of Fran’s shell packages are 18/6, 27/9, and 36/12. the missing ratios of Mia’s shell packages are 8/4, 12/6, and 16/8. Explanation: In the above-given question, given that, Fran buys Florida cone seashells in packages that contain 9 purple-dyed Florida cone seashells for every 3 pink-dyed Florida cone seashells. 6 x 3 = 18, 3 x 3 = 9, 9 x 3 = 27, and 12 x 3 = 36. 2 x 2 = 4, 4 x 2 = 8, 6 x 2 = 12, and 8 x 2 = 16. PART B If the girls each buy packages that contain 6 pinkdyed Florida cone seashells, how many purple-dyed Florida cone seashells would each have? Explain. Answer: The number of purple-dyed Florida cone seashells would each have = 18. Explanation: In the above-given question, given that, If the girls each buy packages that contain 6 pink dyed Florida cone seashells. 6 x 3 = 18. so the number of purple-dyed Florida cona seashells would each have = 18. ### Lesson 5.4 Represent and Graph Ratios Solve & Discuss It! For every 4 adults at the beach one afternoon, there were 3 children. How many children were at the beach if there were 8, 12, 16, or 20 adults at the beach? I can… solve ratio problems by using tables and graphs to show equivalent ratios. Model with Math How does the graph show the ratio? Answer: The graph shows the ratio y = x-1, y = x-2, y = x-4. Explanation: In the above-given question, given that, For every 4 adults at the beach one afternoon, there were 3 children. if there were 8 adults there were 6 children. if there were 12 adults there were 9 children. 4 x 1 = 4, 4 x 2 = 8, 4 x 3 = 12, and 4 x 4 = 16. 3 x 1 = 3, 3 x 2 = 6, 3 x 3 = 9, and 3 x 4 = 12. Focus on math practices Critique Reasoning There were 25 children and 15 adults at the beach. Emery said that there were 5 children for every 3 adults. Is he correct? Explain. Answer: Yes, Emery was correct. Explanation: In the above-given question, given that, There were 25 children and 15 adults at the beach. Emery said that there were 5 children for every 3 adults. 3 x 5 = 15. for 15 adults there were 25 children. so Emery was correct. Essential Question How can you use tables and graphs to show equivalent ratios? Try It! What are the coordinates of the point that represents the number of balloons you can buy for$6?

Answer:
The coordinates of the point that represents the number of balloons we can but for $6 is (6,0) and (0,6). Explanation: In the above-given question, given that, for example: for example: the points are (6, 0) and (0,6). Convince Me! How can you use the graph to find the cost of 15 balloons? Try It! Can you draw an object with a diameter of 10 inches and a circumference of 50 inches? Explain. Answer: Yes, we can draw a diameter of 10 inches and a circumference of 50 inches. Explanation: In the above-given question, given that, we can draw a diameter of 10 inches and a circumference of 50 inches. the circumference is the center of the circle. radius is half of the diameter. diameter is 50/2 = 25. so we can draw a diameter. KEY CONCEPT You can use ratio tables and graphs to show equivalent ratios. When ordered pairs representing equivalent ratios are graphed as points in the coordinate plane, they form a line. Do You Understand? Question 1. Essential Question How can you use tables and graphs to show equivalent ratios? Answer: When ordered pairs representing equivalent ratios are graphed as points in the coordinate plane, they form a line. Explanation: In the above-given question, given that, for example: for every 3 tennis rackets sold, 4 tennis balls are sold. for every 12 tennis rockets sold, 16 tennis balls are sold. so when ordered pairs representing equivalent ratios are graphed as points in the coordinate plane, they form a line. Question 2. Look for Relationships In Example 2, how could you use the graph to find the number of apples needed for 30 celery sticks? Answer: The number of apples needed for 30 celery sticks = 40 apples. Explanation: In the above-given question, given that, if there are 30 celery sticks, there would be 40 apples. 3 x 10 = 30. 4 x 10 = 40. so the number of apples needed for 30 celery sticks = 40 apples. Question 3. How could you use repeated addition to show ratios equivalent to 1:3 on a graph? Answer: The ratios forms a straight line. Explanation: In the above-given question, given that, for example: for every 3 tennis rackets sold, 4 tennis balls are sold. for every 12 tennis rockets sold, 16 tennis balls are sold. so when ordered pairs representing equivalent ratios are graphed as points in the coordinate plane, they form a line. Do You Know How? Question 4. Complete the table to show equivalent ratios representing a cost of$8 for every 3 boxes. Then write the pairs of values as points to be plotted on a coordinate plane.

Answer:
The points are (12, 32) and (15, 40).

Explanation:
In the above-given question,
given that,
3 x 2 = 6, 3 x 3 = 9, 3 x 4 = 12, and 3 x 5 = 15.
8 x 2 = 16, 8 x 3 = 24, 8 x 4 = 32, and 8 x 5 = 40.
so the coordinate points are (12, 32) and (15, 40).

Question 5.
Model with Math Plot the equivalent ratios (3, 4), (6, 8), and (9, 12) on the graph. Use the graph to find the number of nonfiction books purchased if 10 fiction books are purchased.

Answer:
The number of nonfiction books purchased if 10 fiction books are purchased = 14.

Explanation:
In the above-given question,
given that,
the equivalent ratios are (3, 4), (6, 8), (9, 12), and (10, 14).
the points form a straight line.
so the number of nonfiction books purchased if 10 fiction books are purchased = 14.

Practice & Problem Solving

Leveled Practice in 6 and 7, complete the table and graph the pairs of values.
Question 6.

Answer:
The points are (6, 9).

Explanation:
In the above-given question,
given that,
The points are (2,3) and (4,6).
2 x 2 = 4, 2 x 3 = 6.
3 x 2 = 6, 3 x 3 = 9.
so the points are (6, 9).

Question 7.

Answer:
The points are (50, 20).

Explanation:
In the above-given question,
given that,
the points are (5, 2) and (25, 10).
5 x 5 = 25, 5 x 10 = 50.
2 x 5 = 10, 2 x 10 = 20.
so the points are (50, 20).

Question 8.
A student runs 2 minutes for every 10 minutes she walks.
a. Complete the table. Graph the pairs of values.

Answer:
The points are (6, 30).

Explanation:
In the above-given question,
given that,
the running minutes are 2, 4.
the walking minutes are 10, 20.
2 x 2 = 4.
2 x 3 = 6.
10 x 2 = 20.
10 x 3 = 30.
so the points are (6, 30).

b. For how long would the student walk if she runs for 7 minutes?

Answer:
The student walks if she runs for 7 minutes = 35.

Explanation:
In the above-given question,
given that,
the running minutes are 2, 4.
the walking minutes are 10, 20.
2 x 2 = 4.
2 x 3 = 6.
10 x 2 = 20.
10 x 3 = 30.
so the student walks if she runs for 7 minutes = 35.

Question 9.
A car magazine reports the number of miles driven for different amounts of gas for three cars. Which car travels the farthest on 1 gallon of gas? Explain.

Answer:
Car A can travels the farthest on 1 gallon of gas.

Explanation:
In the above-given question,
given that,
Car A can travels for 1 gallon of gas the number of miles driven is 50.
Car B can travels for 1 gallon of gas the number of miles driven is 30.
Car C can travels for 1 gallon of gas the number of miles driven is 25.
So car A can travels the farthest on 1 gallon of gas.

Question 10.
Model with Math A bread recipe calls for 4 cups of white flour for every 5 cups of whole-wheat flour. Complete the table to show how many cups of whole-wheat flour are needed to mix with 16 cups of white flour. Then graph the pairs of values.

Answer:
The points of whole-wheat flour are 4, 8, 12, and 16.

Explanation:
In the above-given question,
given that,
A bread recipe calls for 4 cups of white flour for every 5 cups of whole-wheat flour.
4 x 1 = 4, 4 x 2 = 8, 4 x 3 = 12, and 4 x 4 = 16.
so the points of whole-wheat flour is 4, 8, 12, and 16.

Question 11.
The graph shows the relationship between the number of cups of sugar and the number of cups of flour in a key-lime bread recipe. What point on the graph represents the number of cups of sugar that would be used with 8 cups of flour?

Answer:
The number of cups of sugar that would be used with 8 cups of flour is 2.

Explanation:
In the above-given question,
given that,
the flour(c) is on the x-axis.
the sugar (c) is on the y-axis.
the points are (2, 0.5), (4, 1), (6, 1.5), (8, 2), (10, 2.5),  (12, 3), and (14, 3.5).
so the number of cups of sugar that would be used with 8 cups of flour is 2.

Question 12.
Higher Order Thinking Ishwar can read 5 pages in 15 minutes. Anne can read 15 pages in 1 hour. Explain how you could use a table or graph to find how much longer it would take Anne to read a 300-page book than Ishwar.

Answer:
The much longer it would take Anne to read a 300-page book than Ishwar = 20 hours.

Explanation:
In the above-given question,
given that,
Ishwar can read 5 pages in 15 minutes.
Anne can read 15 pages in 1 hour.
30 pages in 2 hours.
60 in 4 hours.
90 in 6 hours.
90 + 90 = 180 pages in 12 hours.
8 hours is 120 pages.
12 + 8 = 20 hours.

Assessment Practice

Question 13.
The measurements of a circular object are given in the ratio table.
PART A
Find the missing dimensions of other circular objects by completing the ratio table.

Answer:
The missing dimensions are 42 and 301.

Explanation:
In the above-given question,
given that,
The measurements of a circular object are given in the ratio table.
the missing dimensions are 42 and 301.

PART B
Graph the pairs of values.

Answer:
The points are (200, 200), (400, 400), (600, 600), and (800, 800).

Explanation:
In the above-given question,
given that,
the diameter is shown on the x-axis.
the circumference is shown on y-axis.
so the points are (200, 200), (400, 400), (600, 600), and (800, 800).

### Topic 5 Mid-Topic Checkpoint

Question 1.
Vocabulary How can a ratio be used to compare quantities? Lesson 5-1
Answer:

Question 2.
The circumference of the outside of a ring is 66 mm, and it has an outer diameter of 21 mm. If the circumference of the inside of the ring is 50 mm, what is the inner diameter of the ring? Lesson 5-4

Answer:
The inner diameter of the ring = 50 mm.

Explanation:
In the above-given question,
given that,
The circumference of the outside of a ring is 66 mm, and it has an outer diameter of 21 mm.
If the circumference of the inside of the ring is 50 mm.
the inner diameter of the ring = 50 mm.

Question 3.
During the breakfast service, the D-Town Diner sells 12 cups of coffee for every 10 glasses of orange juice. How many cups of coffee would the diner have sold if 40 glasses of orange juice had been sold? Complete the table with equivalent ratios. Lesson 5-2

Answer:
The ratios are (24, 20), (36, 30), and (48,40).

Explanation:
In the above-given question,
given that,
During the breakfast service, the D-Town Diner sells 12 cups of coffee for every 10 glasses of orange juice.
2 x 5 = 10, and 2 x 6 = 12.
3 x 10 = 30, and 3 x 12 = 36.
4 x 10 = 40, and 4 x 12 = 48.
so the ratios are (24, 20), (36, 30), and (48, 40).

Question 4.
The ratio of cows to chickens at Old McDonald’s Farm is 2:7. Select all the farms that have a greater ratio of cows to chickens than Old McDonald’s Farm. Lessons 5-3
☐ Red’s Farm: 3 cows for every 5 chickens
☐ Pasture Farm: 2 cows for every 9 chickens
☐ Cluck & Moo Farm: 1 cow for every 5 chickens
☐ C & C Farm: 3 cows for every 8 chickens
☐ T Family Farm: 1 cow for every 3 chickens

Answer:

Question 5.
A package of 3 notebooks costs $5. Complete the ratio table and graph the pairs of values. How much will 18 notebooks cost? Lesson 5-4 Answer: The cost of 18 notebooks is$30.

Explanation:
In the above-given question,
given that,
A package of 3 notebooks costs $5. 3 x 2 = 6, 3 x 3 = 9, 3 x 4 = 12, 3 x 5 =15, 3 x 6 = 18. 5 x 2 = 10, 5 x 3 = 15, 5 x 4 = 20, 5 x 5 = 25, 5 x 6 = 30. so the cost of 18 notebook is$30.

### Topic 5 Mid-Topic Performance Task

Hillsdale Orchard grows Fuji apples and Gala apples. There are 160 Fuji apple trees and 120 Gala apple trees in the orchard.

PART A
Hillsdale Orchard’s owners decide to plant 30 new Gala apple trees. Complete the ratio table to find the number of new Fuji apple trees the owners should plant if they want to maintain the same ratio of Fuji apple trees to Gala apple trees.

Answer:
The number of Fuji Apple trees to the Gala Apple trees is (40,30), (80, 60), and (160,120).

Explanation:
In the above-given question,
given that,
Hillsdale Orchard’s owners decide to plant 30 new Gala apple trees.
30 x 2 = 60.
60 x 2 = 120.
40 x 2 = 80.
80 x 2 = 160.
so the number of Fuji Apple trees to the Gala Apple trees is (40,30), (80, 60), and (160, 120).

PART B
Use the ratio table to complete a graph that shows the relationship between the number of Fuji apple trees and Gala apple trees at Hillsdale Orchard.

Answer:
Fuji Apple trees on the x-axis.
Gala Apple trees on the y-axis.

Explanation:
In the above-given question,
given that,
Fuji Apple trees on the x-axis.
Gala Apple trees on the y-axis.
the points are (80, 60), (160, 120), and (240, 180).

PART C
By the end of the next season, the owners of Hillsdale Orchard plan to have 240 Fuji apple trees. Explain how you could use the graph to find the total number of Fuji and Gala apple trees that Hillsdale Orchard will have if the owners achieve their goal.

Answer:
The total number of Fuji and Gala apple trees that Hillsdale Orchard is (240, 180).

Explanation:
In the above-given question,
given that,
the owners of Hillsdale Orchard plan to have 240 Fuji apple trees.
the points are (80, 60), (160, 120), and (240, 180).
so the total number of Fuji and Gala apple trees that Hillsdale Orchard is (240, 180).

### Lesson 5.5 Understand Rates and Unit Rates

Solve & Discuss It!
What is the cost of 10 bottles of fruit juice?

I can… solve problems involving rates.

Make Sense and Persevere
How can you use tables or diagrams to make sense of the quantities in the problem?

Focus on math practices
Critique Reasoning Monica says, “If 4 bottles cost $10, then 2 bottles cost$5, and 8 bottles cost $20. So 10 bottles cost$5 + $20.” Is Monica correct? Explain. Answer: No Monica was not correct. Explanation: In the above-given question, given that, If 4 bottles cost$10, then 2 bottles cost $5, and 8 bottles cost$20.
4 bottles cost $10. 8 bottles cost$20.
10 bottles cost $30. so Monica was not correct. Essential Question What are rates and unit rates? Try It! At the same rate, how long would it take the car to travel 60 kilometers? It will take the car _______ minutes to travel ________ kilometers. Answer: It will take the car 6 minutes to travel 60 kilometers. Explanation: In the above-given question, given that, 10 x 6 = 60. 3 x 6 = 18. it will take the car 6 minutes to travel 60 kilometers. Convince Me! Sal draws the double number line diagram at the right. He says it shows that at this rate the race car will travel 35 kilometers in 10.5 minutes. Critique Sal’s reasoning. Is he correct? Explain. Answer: Yes, he was correct. Explanation: In the above-given question, given that, Sal draws the double number line diagram at the right. He says it shows that at this rate the race car will travel 35 kilometers in 10.5 minutes. the points are (3, 10), (6, 20), (9, 30), (10.5, 35), and (12, 40). so he was correct. Try It! A recipe for scrambled eggs uses 2 tablespoons of milk for every 3 eggs. What are two unit rates that could represent the recipe? Answer: The two-unit rates that could represent the recipe = 2/3, 4/6, and 6/9. Explanation: In the above-given question, given that, A recipe for scrambled eggs uses 2 tablespoons of milk for every 3 eggs. 2 x 2 = 4, 2 x 3 = 6, 2 x 4 = 8. 3 x 2 = 6, 3 x 3 = 9, 3 x 4 = 12. so the two-unit rates that could represent the recipe = 2/3, 4/6, and 6/9. Try It! A canoeing club travels 78 miles in 3 days. How far could they travel in 5 days if they maintain the same speed? Answer: They can travel 130 miles in 5 days. Explanation: In the above-given question, given that, A canoeing club travels 78 miles in 3 days. 78 miles in 3 days. 78/3 = 26. 26 + 26 = 52. 78 + 52 = 130. so they can travel 130 miles in 5 days. KEY CONCEPT A rate compares quantities with unlike units of measure. $$\frac{\ 3.50}{7 \text { oranges }}$$ A unit rate compares a quantity to 1 unit of another quantity. $$\frac{\ 3.50}{7 \text { oranges }}=\frac{\ 0.50}{1 \text { orange }}$$ Do You Understand? Question 1. Essential Question What are rates and unit rates? Answer: A rate compares quantities with unlike units of measure. A unit rate compares a quantity to 1 unit of another quantity. Explanation: In the above-given question, given that, A rate compares quantities with unlike units of measure. A unit rate compares a quantity to 1 unit of another quantity. for example: 3.50/7 = 0.5. Question 2. Be Precise Use what you know about ratios to describe a rate. Answer: A rate compares quantities with unlike units of measure. A unit rate compares a quantity to 1 unit of another quantity. Explanation: In the above-given question, given that, A rate compares quantities with unlike units of measure. A unit rate compares a quantity to 1 unit of another quantity. for example: 3.50/7 = 0.5. Question 3. Reasoning A bathroom shower streams 5 gallons of water in 2 minutes. a. Find the unit rate for gallons per minute and describe it in words. Answer: The unit rate for gallons per minute is 2.5. Explanation: In the above-given question, given that, A bathroom shower streams 5 gallons of water in 2 minutes. 5/2 = 2.5. so the unit rate for gallons per minute. b. Find the unit rate for minutes per gallon and describe it in words. Answer: The unit rate for minutes per gallon = 2.5. Explanation: In the above-given question, given that, A bathroom shower streams 5 gallons of water in 2 minutes. 5/2 = 2.5. so the unit rate for gallons per minute. Do You Know How? In 4 and 5, find the value of n. Question 4. Answer: The number of hours for n = 12. Explanation: In the above-given question, given that, the number of hours is 4 for the number of miles = 45. 45 + 45 + 45 = 135. 4 + 4 + 4 = 12. so for the n hours the number of miles = 12. Question 5. Answer: The value of n is 4. Explanation: In the above-given question, given that, the cost in dollars are shown in the figure. the pounds are also shown. for 2 dollars the pounds count is 1. for 3 dollars the pounds count is 2. for 4 dollars the pounds count is 3. for 5 dollars the pounds count is 4. so the value of n is 4. Question 6. Jenny packaged 108 eggs in 9 cartons. Write this statement as a rate. Answer: Jenny packaged 108 eggs in 2 cartons. Explanation: In the above-given question, given that, Jenny packaged 18 eggs in 9 cartons. 18/9 = 2. so the Jenny packaged 108 eggs in 9 cartons is 2. Question 7. Anna Maria read 40 pages in 60 minutes. What is her unit rate in pages per minute? Answer: The unit rate in pages per minute = 0.6. Explanation: In the above-given question, given that, Anna Maria read 40 pages in 60 minutes. 40/60 = 4/6. 2/3 = 0.6. so the unit rate in pages per minute = 0.6. In 8 and 9, use the unit rates that you found in Exercise 3. Question 8. How many gallons of water does the shower stream in 6 minutes? Answer: The number of gallons of water does the shower stream in 6 minutes = 6.6. Explanation: In the above-given question, given that, 40/6 = 20/3. 20/3 = 6.6. so the number of gallons of water does the shower stream in 6 minutes = 6.6. Question 9. How long can someone shower to use only 10 gallons of water? Answer: The shower to use only 10 gallons of water = 0.5. Explanation: In the above-given question, given that, the long can someone shower to use only 10 gallons of water. 10/20 = 1/2 = 0.5. so the length can someone shower to use only 10 gallons of water = 0.5. Practice & Problem Solving In 10 and 11, write each statement as a rate. Question 10. Jan saw 9 full moons in 252 days. Answer: The rate is 0.03. Explanation: In the above-given question, given that, Jaw saw 9 full moons in 252 days. 9/252 = 0.03. so the rate is 0.03. Question 11. It took Hannah 38 minutes to run 8 laps. Answer: The rate is 4.75. Explanation: In the above-given question, given that, It took Hannah 38 minutes to run 8 laps. 38/8 = 4.75. so the rate is 4.75. In 12 and 13, find the value of x. Question 12. Answer: The value of x is 8. Explanation: In the above-given question, given that, the number of bowls is 2. the number of fish in 2 bowls is 16. the number of bowls is 6. the number of fish in 6 bowls is 48. 48/6 = 8. so the value of x is 8. Question 13. Answer: The value of x is 46. Explanation: In the above-given question, given that, the number of miles and the number of hours are given. the number of miles for 4 is 46. the number of miles for 8 is 92. the number of miles for 12 is 138. the number of miles for 16 is 184. In 14 and 15, find the unit rate. Question 14. Answer: The number of miles is 20. Explanation: In the above-given question, given that, 320 mi/16 gal. 320/16 / 16/16. 20/1 = 20. so the number of miles is 20. Question 15. Answer: The value of cm = 15. Explanation: In the above-given question, given that, 75 cm to 5 h. 75/5 / 5/5. 15 / 1 = 15. so the value of the cm = 15. In 16-19, complete each table. Question 16. Answer: The missing values are 2, 5, 5. Explanation: In the above-given question, given that, the minutes on the number of pages are given. 9 x 2 = 18. 1 x 2 = 2. 5 x 2 =10. 5 x 3 = 15. Question 17. Answer: The missing values are 62, 434, and 682. Explanation: In the above-given question, given that, 186/3 = 62. 62/1 = 62. 434/7 = 62. 682/11 = 62. so the missing values are 62, 434, and 682. Question 18. Answer: The missing values are 12.3, 61.5, and 10. Explanation: In the above-given question, given that, 12.3/1 = 12.3. 24.6/2 = 12.3. 61.5/5 =12.3. 123/10 = 12.3. so the missing values are 12.3, 61.5, and 10. Question 19. Answer: The missing values are 1, 75, 300. Explanation: In the above-given question, given that, the number of gallons and miles are given. 125/5 = 25. 25/1 = 25. 75/3 = 25. 300/12 = 25. so the missing values are 1, 75, and 300. Question 20. Which runner set the fastest pace? Explain. Answer: Allison runs at the fastest pace. Explanation: In the above-given question, given that, the runner Martha did 20 laps in 32 min. the runner Allison did 16 laps in 25 min. the runner Rachel did 17 laps in 27.2 min. speed = distance/time. Martha = 20/32. Martha = 0.625. Allison = 16/25. Allison = 0.64. Rachel = 17/27.2. Rachel = 0.625. Allison runs at the fastest pace. Question 21. Model with Math Over the summer, Alexis read 15 books in 12 weeks. The diagram below can be used to track her progress. If Alexis read at the same rate each week, how many books had she read in 4 weeks? In 8 weeks? Complete the diagram. Answer: The missing books are 5 and 10. Explanation: In the above-given question, given that, Over the summer, Alexis read 15 books in 12 weeks. Alexis read the 5 books in 4 weeks. the number of books Alexis read in 8 weeks = 10. 5 x 1 = 5. 5 x 2 = 10. 5 x 3 = 15. so the missing books are 5 and 10. Question 22. An elephant charges an object that is 0.35 kilometer away. How long will it take the elephant to reach the object? Answer: The distance will it take the elephant to reach the object = 0.5 km. Explanation: In the above-given question, given that, An elephant charges an object that is 0.35 kilometers away. Elephants can charge at speeds of 0.7 km per minute. 0.35/0.7 = 0.5. 7 x 5 = 35. so the distance will it take the elephant to reach the object = 0.5 km. Question 23. A machine takes 1 minute to fill 6 cartons of eggs. At this rate, how many minutes will it take to fill 420 cartons? Answer: The number of minutes will it take to fill 420 cartons = 70 min. Explanation: In the above-given question, given that, A machine takes 1 minute to fill 6 cartons of eggs. 70 x 6 = 420. 420/6 = 70. so the number of minutes will it take to fill 420 cartons = 70 min. Question 24. Higher Order Thinking How are the ratios $$\frac{24 \text { laps }}{1 \text { hour }}$$ and $$\frac{192 \text { laps }}{8 \text { hours }}$$ alike? How are they different? Answer: They are same. Explanation: In the above-given question, given that, the ratios 24, 192, and 8 are alike. 8 x 1 = 8. 8 x 3 = 24. 24 x 8 = 192. so they are same. Assessment Practice Question 25. A bakery sells 12 gourmet orange-zest cupcakes for$36.00. Select all the statements that are true.
☐ $$\frac{\ 3.00}{1 \text { cupcake }}$$ is a unit rate for the cost per 1 cupcake cupcake.
☐ $$\frac{36}{12}$$ represents the ratio of $36.00 for 12 cupcakes. ☐ Using the same rate, the bakery can sell 6 cupcakes for$20.00.
☐ Using the same rate, the bakery can sell 2 dozen cupcakes for $72.00. ☐ Using the same rate, it would cost$24.50 for 8 cupcakes.

Answer:
Option A is correct.

Explanation:
In the above-given question,
given that,
A bakery sells 12 gourmet orange-zest cupcakes for $36.00. 36/12 = 3. 12 x 3 = 36. so option A is correct. ### Lesson 5.6 Compare Unit Rates Solve & Discuss It! Rick and Nikki own remote-control cars. They use a stopwatch to record the speed of each car. Whose car is faster? I can… compare unit rates to solve problems. Be Precise Use precise numbers and units to describe and compare rates. Focus on math practices Make Sense and Persevere If each car maintains its rate of speed, how long will it take Rick’s car to travel 300 feet? How long will it take Nikki’s car to travel the same distance? Explain. Answer: The long will it take Nikki’s car to travel the same distance = 60 sec. Explanation: In the above-given question, given that, Rick and Nikki own remote-control cars. the distance in 30 sec is 150 feet. the distance will it take 300 feet in 60 sec. 300/150 = 2. 30 x 2 = 60. so the long will it take Nikki’s car to travel the same distance = 60 sec. Essential Question How can you use unit rates to make comparisons? Try It! Ashley is Austin’s older sister. She trains in the same pool and can swim 9 laps in 6 minutes. Is Ashley a faster swimmer than Austin? Ashley swims 1.5 laps per minute. Because ________ 1.4, Ashley is a _______ swimmer than Austin. Answer: Ashley swims faster than Austin. Explanation: In the above-given question, given that, Ashley is Austin’s older sister. She trains in the same pool and can swim 9 laps in 6 minutes. Ashley swims 1.5 laps per minute. 9/1.5 = 6. 6/1.4 = 4.2. so Ashley swims faster than Austin. Convince Me! How can you use the unit rate in minutes per lap to compare Ashley’s speed to Austin’s speed? Try It! Explain how to decide which is the better value, 4 greeting cards for$10 or 6 greeting cards for $14. Answer: The better value is 4 greeting cards for$10.

Explanation:
In the above-given question,
given that,
4 greeting cards for $10. 6 greeting cards for$14.
10/4 = 2.5.
14/6 = 2.3.
so the better value is 4 greeting cards for $10. KEY CONCEPT You can use unit rates to make comparisons.$8.50 per hour > $8.00 per hour $$\frac{32 \mathrm{~cm}}{1 \mathrm{sec}}$$ < $$\frac{45 \mathrm{~cm}}{1 \mathrm{sec}}$$ $$\frac{7 \text { laps }}{1 \mathrm{~min}}$$ < $$\frac{9 \text { laps }}{1 \mathrm{~min}}$$ 175 words per minute > 95 words per minute Do You Understand? Question 1. Essential Question How can you use unit rates to make comparisons? Answer: We can use unit rates to make comparisons. Explanation: In the above-given question, given that, You can use unit rates to make comparisons.$8.50 per hour > $8.00 per hour. 175 words per minute > 95 words per minute. Question 2. Critique Reasoning Paul says that a lower unit rate is a better value only if you can use all the items purchased to get the lower unit rate. Do you agree? Explain. Answer: Yes, i will agree. Explanation: In the above-given question, given that, Paul says that a lower unit rate is a better value only if you can use all the items purchased to get the lower unit rate. so I will agree. Question 3. Reasoning Car A travels 115 miles on 5 gallons of gas. Car B travels 126 miles on 6 gallons of gas. How can you find which car gets better gas mileage? Answer: The Car A gets better gas mileage. Explanation: In the above-given question, given that, Car A travels 115 miles on 5 gallons of gas. Car B travels 126 miles on 6 gallons of gas. 115/5 = 23. 126/6 = 21. so car A gets better gas mileage. Do You Know How? Question 4. Hakim’s car travels 600 feet in 20 seconds. Andre’s motorcycle travels 300 feet in 12 seconds. Which is faster, the car or the motorcycle? Explain. a. Find the unit rates. Answer: The car is faster. Explanation: In the above-given question, given that, Hakim’s car travels 600 feet in 20 seconds. Andre’s motorcycle travels 300 feet in 12 seconds. 300/12 = 25. 600/20 = 30. so the car is faster. b. Compare the unit rates. Answer: In 5 and 6, find each unit price. Question 5. 7 movie tickets for$56

Answer:
Each ticket cost $8. Explanation: In the above-given question, given that, 7 movie tickets for$56.
$56/7 = 8. so each ticket cost$8.

Question 6.
12 fluid ounces of shampoo for $2.76 Answer: 12 fluid ounces of shampoo for$2.76.

Explanation:
In the above-given question,
given that,
12 fluid ounces of shampoo for $2.76. 12/2.76 = 4.34. so 12 fluid ounces of shampoo for$2.76.

Question 7.
Which is the better value, 2 books for $15 or 6 books for$45? Explain.

Answer:
Both of them have better value.

Explanation:
In the above-given question,
given that,
2 books for $15. 6 books for$45.
15 + 15 + 15 = 45.
so both of them have the better value.

Practice & Problem Solving

Leveled Practice In 8 and 9, find each unit price.
Question 8.
9 pens for $3.60 $$\frac{\ 3.60 \div 9}{9 \div 9}=\frac{ }{1}$$ Answer: 9 pens for$3.60 = $2.5. Explanation: In the above-given question, given that, 9 pens for$3.60.
9/3.60 = 2.5.
so 9 pens for $3.60 =$2.5.

Question 9.
15 ounces of canned beans for $2.25 $$\frac{\ 2.25 \div}{15 \div}=\frac{}{}$$ Answer: 15 ounces of canned beans for$2.25 = 6.6.

Explanation:
In the above-given question,
given that,
15 ounces of canned beans for $2.25. 15/2.25 = 6.6. so 15 ounces of canned beans for$2.25 = 6.6.

In 10 and 11, determine which is the better value.
Question 10.
3 kilograms of charcoal for $7.95 or 5 kilograms of charcoal for$12.50

Answer:
7.95/3 = 2.65 and 12.50/5 = 2.5.

Explanation:
In the above-given question,
given that,
3 kilograms of charcoal for $7.95. 5 kilograms of charcoal for$12.50.
7.95/3 = 2.65.
12.50/5 = 2.5.

Question 11.
50 envelopes for $2.49 or 90 envelopes for$5.50

Answer:
50 envelopes for $2.49 or 90 envelopes for$5.50 is 16.36.

Explanation:
In the above-given question,
given that,
50 envelopes for $2.49 or 90 envelopes for$5.50.
50/2.49 = 20.08.
90/5.50 = 16.36.

In 12-15, compare the rates to find which is greater.
Question 12.
35 points in 20 minutes or 49 points in 35 minutes

Answer:
35 points in 20 minutes or 49 points in 35 minutes are 1.75 and 1.4.

Explanation:
In the above-given question,
given that,
35 points in 20 minutes or 49 points in 35 minutes.
35/20 = 1.75.
49/35 = 1.4.

Question 13.
12 laps in 8 minutes or 16 laps in 10 minutes

Answer:
16 laps in 10 minutes.

Explanation:
In the above-given question,
given that,
12 laps in 8 minutes or 16 laps in 10 minutes.
12/8 = 1.5.
16/10 = 1.6.
so 16 laps in 10 minutes.

Question 14.
45 strikeouts in 36 innings or 96 strikeouts in 80 innings

Answer:
45 strikeouts in 36 innings or 96 strikeouts in 80 innings = 1.2.

Explanation:
In the above-given question,
given that,
45 strikeouts in 36 innings or 96 strikeouts in 80 innings.
45/36 = 1.25.
96/80 = 1.2.

Question 15.
480 stickers on 6 sheets or 120 stickers on 2 sheets

Answer:
120 stickers on 2 sheets is 60.

Explanation:
In the above-given question,
given that,
480 stickers on 6 sheets or 120 stickers on 2 sheets.
480/6 = 80.
120/2 = 60.
so 120 stickers on 2 sheets is 60.

In 16-18, compare the rates to find which is the better value.

Question 16.
$27 for 4 large pizzas or$32 for 5 large pizzas

Answer:
The both values are same.

Explanation:
In the above-given question,
given that,
$27 for 4 large pizzas or$32 for 5 large pizzas.
$27/4 =$6.75.
$32/5 =$6.4.
so both values are same.

Question 17.
$30 for 100 flyers or$65 for 250 flyers

Answer:
$30 for 100 flyers = 3.3. Explanation: In the above-given question, given that,$30 for 100 flyers.
$65 for 250 flyers. 100/30 = 3.3. 250/100 = 2.5. Question 18. 36 pictures for$8 or 24 pictures for $5 Answer: 24 pictures for$5 = 4.8.

Explanation:
In the above-given question,
given that,
36 pictures for $8 or 24 pictures for$5.
36/8 = 4.5.
24/5 = 4.8.
so 24 pictures for $5 = 4.8. Question 19. Model with Math Katrina and Becca exchanged 270 text messages in 45 minutes. An equal number of texts was sent each minute. The girls can send 90 more text messages before they are charged additional fees. Complete the double number line diagram. At this rate, for how many more minutes can the girls exchange texts before they are charged extra? Answer: The missing values are 90/6, 180/6, and 360/6. Explanation: In the above-given question, given that, Katrina and Becca exchanged 270 text messages in 45 minutes. An equal number of texts was sent each minute. The girls can send 90 more text messages before they are charged additional fees. 90/6 = 15. 180/6 = 30. 360/6 = 60. so the missing values are 90/6, 180/6, and 360/6. Question 20. Reasoning Which container of milk would you buy? Explain. Answer: I will buy 1/2 gallon of milk for$2.29.

Explanation:
In the above-given question,
given that,
1/2 gallon of milk for $2.29. 1 gallon of milk for$3.99.
$2.29/0.5 = 4.58.$3.99/1 = 3.99.
I will buy 1/2 gallon of milk for $2.29. Question 21. Higher Order Thinking Amil and Abe rode in a bike-a-thon. Abe rode for 77 minutes at a faster rate per mile than Amil. Find Amil’s unit rate. Then explain how you could use it to find a possible unit rate for Abe. Answer: Abe’s rate is 4.5. Explanation: In the above-given question, given that, Amil and Abe rode in a bike-a-thon. Abe rode for 77 minutes at a faster rate per mile than Amil. 77/17 = 4.5. so Abe’s rate was 4.5. Assessment Practice Question 22. A food warehouse sells cans of soup in boxes. Bargain shoppers have four options. PART A Complete the table to find the unit price for each option. Answer: The unit price is 1.13, 1.17, 1.16, and 1.12. Explanation: In the above-given question, given that, A food warehouse sells cans of soup inboxes. 12 cans for$10.56 = 1.13.
16 cans for $13.60 = 1.17. 20 cans for$17.20 = 1.16.
24 cans for $21.36 = 1.12. PART B Compare the unit rates found in Part A and identify the best value. Answer: The best value is 1.17. Explanation: In the above-given question, given that, A food warehouse sells cans of soup inboxes. 12 cans for$10.56 = 1.13.
16 cans for $13.60 = 1.17. 20 cans for$17.20 = 1.16.
24 cans for $21.36 = 1.12. ### Lesson 5.7 Solve Unit Rate Problems Solve & Discuss It! Suppose you are traveling by train to visit a friend who lives 275 miles away. How long will the trip take? Moving at a constant speed, how long would it take the train to travel 385 miles? I can… use unit rates to solve problems. Model with Math How can you use what you know about unit rates to model and solve this problem? Focus on math practices Reasoning Suppose the train was traveling at a constant speed that is twice as fast as 55 miles per hour. How long would it take the train to go 275 miles? Explain. Answer: The long would it take the train to go 275 miles = 5. Explanation: In the above-given question, given that, Suppose you are traveling by train to visit a friend who lives 275 miles away. the train travels at a constant speed of 55 miles per hour. 275/55 = 5. so the long would it take the train to go 275 miles = 5. Essential Question How can you use unit rates to solve problems? Try It! At the same rate, how far would the jet fly in 75 minutes? The jet would fly ________ miles. Answer: The jet would fly 1 mile. Explanation: In the above-given question, given that, 15 x 5 = 75. 1 minute x 75 = 75. 75 miles/75 minutes = 1. Convince Me! How could you use the table from Example 1 to find how far the jet would fly in 75 minutes? Explain. Try It! Jarod paid$13.80 for 5 tickets to the game. At the same rate, how much would 3 tickets cost?

Answer:
The 3 tickets cost is $8.28. Explanation: In the above-given question, given that, Jarod paid$13.80 for 5 tickets to the game.
$13.80/5 =$2.76.
$2.76 +$2.76 + $2.76 =$8.28.
so the 3 tickets cost is $8.28. Try It! A submarine travels 19 miles in $$\frac{1}{2}$$ hour. Write an equation to find out how long it would take the submarine to travel 57 miles at the same rate. Then find the time. Answer: The submarine to travel 57 miles at the same rate = 3. Explanation: In the above-given question, given that, A submarine travels 19 miles in $$\frac{1}{2}$$ hour. 19 miles in 0.5 hours. 19 x 3 = 57. 57/19 = 3. so the submarine to travel 57 miles at the same rate = 3. KEY CONCEPT You can use ratio tables or unit rates to solve rate problems, including constant speed problems. Do You Understand? Question 1. Essential Question How can you use unit rates to solve problems? Answer: We can use ratio tables to solve rate problems including constant speed problems. Explanation: In the above-given question, given that, we can use ratio tables to solve rate problems including constant speed problems. Ant traveled 6 cm in 1.5 sec. ant traveled 3 cm in 12 sec. ant traveled 4.5 cm in 18 sec. Question 2. Construct Arguments An ostrich runs 6 miles in 12 minutes at a constant speed. Explain how you can use a unit rate to find how far the ostrich could run in 40 minutes. Answer: The far ostrich could run in 40 minutes = 8 miles. Explanation: In the above-given question, given that, An ostrich runs 6 miles in 12 minutes at a constant speed. 40/6 = 8. so the far ostrich could run in 40 minutes = 8 miles. Question 3. Bananas sell for$0.58 per pound. How could you write an equation to show the relationship between the total cost, c, and the number of pounds of bananas, p?

Answer:
C = $0.58p. Explanation: In the above-given question, given that, Bananas sell for$0.58 per pound.
the total cost is written as c.
the number of pounds is written as p.
c = $0.58p. Do You Know How? In 4 and 5, use unit rates to solve. Question 4. A football player runs 80 yards in 25 seconds. If he maintains the same rate of speed, how far could he run in 60 seconds? Answer: The far could he run in 60 seconds = 1.3. Explanation: In the above-given question, given that, A football player runs 80 yards in 25 seconds. 80/25 = 3.2. 80/60 = 1.3. so the far could he run in 60 seconds = 1.3. Question 5. On a family vacation, Amy’s dad drove the car at a constant speed and traveled 585 miles in 13 hours. At this rate, how long would it have taken the family to travel 810 miles? What was the car’s rate of speed? Answer: The rate of the car’s speed = 45 hours. Explanation: In the above-given question, given that, Amy’s dad drove the car at a constant speed and traveled 585 miles in 13 hours. 585/13 = 45. 45 x 13 = 585. so the rate of the car’s speed = 45 hours. Question 6. Look at Exercise 5. Write an equation to find the total distance, d, that Amy’s family traveled after t hours. Answer: Practice & Problem Solving Leveled Practice In 7-9, solve the rate problems. Question 7. A horse named Northern Dancer won the Kentucky Derby with a time of exactly 2 minutes. At this constant rate, how long would it take Northern Dancer to run the Belmont Stakes? It would take Northern Dancer ______ minutes to run the Belmont Stakes. Answer: It would take Northern dancer 2 minutes to run the Belmont stakes. Explanation: In the above-given question, given that, A horse named Northern Dancer won the Kentucky Derby with a time of exactly 2 minutes. 1.25/ 1.25/1 = 1 mile/1 minute. the equivalent rate is 1.5/2. so it would take Northern dancer 2 minutes to run the Belmont stakes. Question 8. If a cyclist rides at a constant rate of 24 miles per hour, how long would it take the cyclist to ride 156 miles? Answer: The long would it take the cyclist to ride 156 miles = 6.5 hours. Explanation: In the above-given question, given that, If a cyclist rides at a constant rate of 24 miles per hour. 156/24 = 6.5. so the long would it take the cyclist to ride 156 miles = 6.5 hours. Question 9. The price of an 8-minute phone call is$1.20. What is the price of a 17-minute phone call?

Answer:
The price of a 17-minute phone call = 0.07.

Explanation:
In the above-given question,
given that,
The price of an 8-minute phone call is $1.20.$1.20/8 = 0.15.
$1.20/17 = 0.07. so the price of a 17-minute phone call = 0.07. In 10 and 11, use the map at the right. The Garcia family is driving from Sacramento, California, to Key West, Florida. In 5 days, they have traveled 2,045 miles. At this rate, how long will it take them to travel from Sacramento to Key West? Question 10. How can you use rate reasoning to solve this problem? Explain. Answer: The rate will it take them to travel from Sacramento to Key West = 409. Explanation: In the above-given question, given that, The Garcia family is driving from Sacramento, California, to Key West, Florida. In 5 days, they have traveled 2,045 miles. 2045/5 = 409. so the rate will it take them to travel from Sacramento to Key West = 409. Question 11. Be Precise Show how to use numbers, units, and symbols precisely to solve the problem. Answer: Question 12. Vik wrote the equation 470 • h = 3,008, where h is the number of hours it took a plane flying at a constant speed of 470 miles per hour to travel 3,008 miles. Solve for h. Answer: The number of miles per hour to travel 3008 miles = 6.4. Explanation: In the above-given question, given that, Vik wrote the equation 470 • h = 3,008, 470. h = 3008. h = 3008/470. h = 6.4. so the number of miles per hour to travel 3008 miles = 6.4. Question 13. A nursery owner buys 7 panes of glass to fix some damage to his greenhouse. The 7 panes cost$15.05. Unfortunately, he breaks 2 more panes while repairing the damage. What is the cost of another 2 panes of glass?

Answer:
The cost of another 2 panes of glass = $7.5. Explanation: In the above-given question, given that, A nursery owner buys 7 panes of glass to fix some damage to his greenhouse. the 7 panes cost$15.05.
$15.05 / 7 =$2.15.
$15.05 / 2 = 7.525. so the cost of another 2 panes of glass =$7.5.

Question 14.
Cheyenne drew a circle with diameter 1 meter. She measured the circumference to estimate the value of Pi. Complete the table, and then write an equation to find the circumference, C, for a circle with diameter d.

Answer:
The circumference, C, for a circle with diameter d is 6.28, 9.42, and 12.56.

Explanation:
In the above-given question,
given that,
Cheyenne drew a circle with a diameter of 1 meter.
She measured the circumference to estimate the value of Pi.
3.14 x 1 = 3.14.
3.14 x 2 = 6.28.
3.14 x 3 = 9.42.
3.14 x 4 = 12.56.
so the circumference, c, for a circle with diameter d is 6.28, 9.42, and 12.56.

Question 15.
Jayden bought 70 feet of speaker wire for $18.20. He needs 30 more feet. If the unit price is the same, how much will Jayden pay for the extra 30 feet of wire? Explain. Answer: The much will Jayden pay for the extra 30 feet of wire = 0.60. Explanation: In the above-given question, given that, Jayden bought 70 feet of speaker wire for$18.20.
He needs 30 more feet.
$18.20/70 = 0.26.$18.20/30 = 0.60.
so the much will Jayden pay for the extra 30 feet of wire = 0.60.

Question 16.
Higher Order Thinking Sasha runs at a constant speed of 3.8 meters per second for $$\frac{1}{2}$$ hour. Then she walks at a constant rate of 1.5 meters per second for $$\frac{1}{2}$$ hour. How far did Sasha run and walk in 60 minutes?

Answer:
The far did Sasha run and walk in 60 minutes = 30.

Explanation:
In the above-given question,
given that,
Sasha runs at a constant speed of 3.8 meters per second for $$\frac{1}{2}$$ hour.
Then she walks at a constant rate of 1.5 meters per second for $$\frac{1}{2}$$ hour.
3.8/ 0.5 = 7.6.
1.5 / 0.5 = 30.
so the far did Sasha run and walk in 60 minutes = 30.

Assessment Practice

Question 17.
Suppose that a leatherback turtle swam 7.5 kilometers in 3 hours at a constant speed.
PART A
How many kilometers per hour did the turtle swim?

Answer:
The turtle swim per hour = 2.5 miles.

Explanation:
In the above-given question,
given that,
leatherback turtle swam 7.5 kilometers in 3 hours.
7.5/3 = 2.5.
so the turtle swim per hour = 2.5 miles.

PART B
At this rate, how long would it take the turtle to swim 10 kilometers?

Answer:
The long would it take the turtle to swim 10 kilometers = 0.75.

Explanation:
In the above-given question,
given that,
the long would it take the turtle to swim 10 kilometers.
7.5/10 = 0.75.
so the long would it take the turtle to swim 10 kilometers = 0.75.

3-ACT MATH

3-Act Mathematical Modeling: Get in Line

ACT 1
Question 1.
After watching the video, what is the first question that comes to mind?
Answer:

Question 2.
Write the Main Question you will answer.

Answer:

Question 3.
Construct Arguments Predict an answer to this Main Question. Explain your prediction.
Answer:

Question 4.
On the number line below, write a number that is too small to be the answer. Write a number that is too large.

Answer:

Question 5.
Plot your prediction on the same number line.
Answer:

ACT 2
Question 6.
What information in this situation would be helpful to know? How would you use that information?

Answer:

Question 7.
Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
Answer:

Question 8.
Model with Math Represent the situation using mathematics. Use your representation to answer the Main Question.
Answer:

Question 9.
What is your answer to the Main Question? Is it higher or lower than your prediction? Explain why.

Answer:

ACT 3
Question 10.
Write the answer you saw in the video.

Answer:

Question 11.
Reasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
Answer:

Question 12.
Make Sense and Persevere Would you change your model now that you know the answer? Explain.

Answer:

Reflect
Question 13.
Model with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?

Answer:

Question 14.
Generalize Will your model work on other lights? Explain your reasoning.
Answer:

SEQUEL
Question 15.
Use Structure Later that week, it took between 20 and 21 minutes to get through the same light. How many cars were in line?
Answer:

### Lesson 5.8 Ratio Reasoning: Convert Customary Units

Solve & Discuss It!
If 6.5 feet of snow were to fall in a 24-hour period, would the 1921 record be broken? There are 12 inches in 1 foot.
I can… use ratio reasoning to convert customary measurements.

Reasoning
Use the relationship between inches and feet to solve the problem.

Focus on math practices
Make Sense and Persevere How many feet of snow would need to fall in Silver Lake, Colorado, to break the 1921 24-hour snowfall record from 1921?

Answer:
The record from 1921 is 3.15.

Explanation:
In the above-given question,
given that,
the number of feet of snow would need to fall in Silver Lake, Colorado, to break the 1921 24-hour snowfall record from 1921.
75.8 inches in 24 hours.
75.8/24 = 3.15.
so the record from 1921 is 3.15.

Essential Question
How can you use ratios to convert customary units of measure?

Try It!

According to city regulations, how many feet wide is the maximum sidewalk width? Explain.
Answer:

Convince Me! What conversion factor would you use when converting 66 inches to feet? Explain.

Try It!

Brandon is making bread. His recipe says to use 21 tablespoons of sugar. How many teaspoons of sugar should he use?

Answer:
The number of teaspoons of sugar should be used = 21 tablespoons.

Explanation:
In the above-given question,
given that,
Brandon is making bread.
His recipe says to use 21 tablespoons of sugar.
so the number of teaspoons of sugar should be used = 21 tablespoons.

Try It!

How many pounds does the elephant weigh?

Answer:
The elephant weigh about 3.3 tons.

Explanation:
In the above-given question,
given that,
stella weighs approximately 3.3 tons.
so the weight of the elephant is 3.3 tons.

KEY CONCEPT
You can convert customary measures by finding an equivalent rate or by using dimensional analysis.

Use an equivalent rate.
1 mi = 5,280 ft
$$\frac{5,280 \mathrm{ft} \times 4.25}{1 \mathrm{mi} \times 4.25}=\frac{22,440 \mathrm{ft}}{4.25 \mathrm{mi}}$$

Use dimensional analysis.
4.25 m1 × $$\frac{5,280 \mathrm{ft}}{1 \mathrm{~m}}$$
= 4.25 × 5,280 ft
= 22,440 ft

Do You Understand?
Question 1.
Essential Question How can you use ratios to convert customary units of measure?

Answer:
We can find an equivalent rate by using dimensional analysis.

Explanation:
In the above-given question,
given that,
You can convert customary measures by finding an equivalent rate or by using dimensional analysis.
1 feet = 12 inches.
1 yard = 36 inches.
1 yard = 3 feet.
1 mile = 5280 feet.
1 mile = 1760 yards.
5280 – 1760 = 3520.
so we can find an equivalent rate by using dimensional analysis.

Question 2.
What is a conversion factor that relates miles to yards?

Answer:
The conversion factor that relates miles to yards is 5280 feet.

Explanation:
In the above-given question,
given that,
for example:
1 mile = 5280 feet.
1 mile = 1760 yards.
5280 – 1760 = 3520.
so the conversion factor that relates miles to yards is 5280 feet.

Question 3.
Construct Arguments Jenna used the conversion factor $$\frac{1 \mathrm{~T}}{2,000 \mathrm{lb}}$$ to convert 50 tons to pounds. Did she use the correct conversion factor? Explain.

Answer:
Yes, she uses the correct conversion factor.

Explanation:
In the above-given question,
given that,
Jenna used the conversion factor $$\frac{1 \mathrm{~T}}{2,000 \mathrm{lb}}$$ to convert 50 tons to pounds.
2000/50 = 40.
so she use the correct conversion factor.

Question 4.
How can you use the conversion rates of fluid ounces to cups, and cups to pints, to find the number of fluid ounces in a pint?
Answer:

Do You Know How?
Question 5.
Convert 27 inches to yards by finding an equivalent rate.

Answer:
The equivalent rate is 0.729.

Explanation:
In the above-given question,
given that,
1 inch = 0.027 yards.
27 x 0.027 = 0.729.
the equivalent rate is 0.729.

Question 6.
Use dimensional analysis to convert 1.8 pounds to ounces.

Answer:
1.8 pounds to ounces = 28.8 ounces.

Explanation:
In the above-given question,
given that,
1 pound = 16 ounces.
1.8 x 16 = 28.8 ounces.
so 1.8 pounds to ounces = 28.8 ounces.

Question 7.
Critique Reasoning Sam is tripling a recipe for an organic cleaning solution. The new recipe calls for 15 tsp of orange oil. To find how many tbsp this is, Sam converted this way:
Conversion factor: $$\frac{3 \text { tsp }}{1 \text { tbsp }}$$
$$15 t s p \times \frac{3 t s p}{1 \text { tbsp }}=\frac{45}{1} \text { tbsp }=45 \text { tbsp }$$
What error did Sam make?
Answer:

Practice & Problem Solving

In 8-13, complete each conversion.
Question 8.
5 pt = _______c

Answer:
5 pt = 10 cups.

Explanation:
In the above-given question,
given that,
5 us liquid cups = 10 us cups.
5 pt = 10 us cups.

Question 9.
2$$\frac{1}{2}$$gal = _________ qt

Answer:
2$$\frac{1}{2}$$gal = 10 qt.

Explanation:
In the above-given question,
given that,
a half of a gallon equal to 2 quarts.
2 + 2 + 2 + 2 + 2 = 10.
2$$\frac{1}{2}$$gal = 10 qt.

Question 10.
2,640 yd = ________ mi

Answer:
2640 yd = 1.5 miles.

Explanation:
In the above-given question,
given that,
1 yard = 0.000568 miles.
2640 yards = 1.5 miles.
so 2640 yds = 1.5 miles.

Question 11.
Convert 16 yards to feet. Use the conversion rate 3 feet = 1 yard.

Answer:
16 yards = 48 feet.

Explanation:
In the above-given question,
given that,
3 feet = 1 yard.
16 yards = 48 feet.
16 x 3 = 48.

Question 12.
Convert 10 pints to quarts. Use the conversion rate 1 quart = 2 pints.

Answer:
10 pints = 5 quarts.

Explanation:
In the above-given question,
given that,
1 quart = 2 pints.
10 us liquid pints = 5 us liquid quarts.
10 pints = 5 quarts.

Question 13.
Convert 12 ounces to pounds. Use the conversion rate 16 ounces = 1 pound.

Answer:
12 ounces = 0.75 pounds.

Explanation:
In the above-given question,
given that,
16 ounces = 1 pound.
12 ounces = 0.75 pounds.
so 12 ounces = 0.75 pounds.

Question 14.
Two neighbors in a rural area want to know the distance between their homes in miles. What should the neighbors use as a conversion factor to convert this distance to miles?

Answer:
The distance to miles is 0.8 miles.

Explanation:
In the above-given question,
given that,
Two neighbors in a rural area want to know the distance between their homes in miles.
4224 x 2 = 0.8 miles.
1 feet = 12 inches.
so the distance to miles is 0.8 miles.

Question 15.
A school custodian discovered a leak in a water pipe. The custodian found that 1,920 fluid ounces of water had leaked out. How many gallons of water is this? Use the conversion factor $$\frac{1 \text { gallon }}{128 \text { fluid ounces }}$$.

Answer:
The conversion factor is 15 gallons.

Explanation:
In the above-given question,
given that,
A school custodian discovered a leak in a water pipe.
The custodian found that 1,920 fluid ounces of water had leaked out.
1920/128 = 15.
1920 us fluid ounces = 15 us liquid gallons.

Question 16.
Critique Reasoning Two students, Stella and Vladimir, complete the conversion statement 12 feet 8 inches = __________ inches.
Stella stated that 12 feet 8 inches = 152 inches. Vladimir stated that 12 feet 8 inches = 9 inches.
Which student is incorrect? Explain.

Answer:
Vladimir is incorrect.

Explanation:
In the above-given question,
given that,
12 feet 8 inches = 152 inches.
Vladimir is incorrect.
Stella is correct.

Question 17.
The hole for a support post needs to be 6 feet deep. It is currently 1 foot 8 inches deep. How much deeper must the hole be? Use the conversion factor 

Answer:
1 ft 8 in is 50.8 cm.

Explanation:
In the above-given question,
given that,
The hole for a support post needs to be 6 feet deep.
It is currently 1 foot 8 inches deep.
1 ft 8 in is 50.8 cm.

In 18 and 19, use the recipe card.

Question 18.
Look for Relationships Cheryl has measured 3 cups of water. Is this enough water for Cheryl to make a double recipe of green slime for a class project? Explain.

Answer:

Explanation:
In the above-given question,
given that,
Cheryl has measured 3 cups of water.

Question 19.
There are 16 tablespoons in 1 cup. How many tablespoons of cornstarch would Cheryl need to make the green slime recipe 15 times?

Answer:
The tablespoons of cornstarch would Cheryl need to make = 92 cups.

Explanation:
In the above-given question,
given that,
There are 16 tablespoons in 1 cup.
tablespoons = 1 cup.
so the tablespoons of cornstarch would Cheryl need to make = 92 cups.

Question 20.
Make Sense and Persevere Len plans to run at least 3 miles each day to get ready for a cross-country race. One lap of the school track is 440 yards. If Len runs 10 laps each day, will he cover at least 3 miles? Explain.

Answer:
The Len runs 10 laps each day, will he cover at least 3 miles = 146.6.

Explanation:
In the above-given question,
given that,
Len plans to run at least 3 miles each day to get ready for a cross-country race.
One lap of the school track is 440 yards.
440/3 = 146.6.
so len runs 10 laps each day, will he cover at least 3 miles = 146.6.

Question 21.
Higher Order Thinking Hunter is splitting a quart of ice cream with 7 members of his family. If the quart is split evenly, how many cups will each family member get? Explain.

Answer:

Explanation:
In the above-given question,
given that,
Hunter is splitting a quart of ice cream with 7 members of his family.

Question 22.
A fully loaded and fueled space shuttle can weigh close to 4.5 million pounds at liftoff. What is this weight expressed in tons?

Answer:
The weight expressed in tons = 2250 us tons.

Explanation:
In the above-given question,
given that,
A fully loaded and fueled space shuttle can weigh close to 4.5 million pounds at liftoff.
weighs almost 4.5 million pounds.
the weight expressed in tons = 2250 us tons.

Assessment Practice

Question 23.
Select all the conversions that are true.

☐ 18 ft = 6 yd
☐ 18 yd = 6 ft
☐ 0.5 mi = 10,560 ft
☐ 0.5 mi = 2,640 ft
☐ $$\frac{1}{2}$$ mi = 880 yd

Answer:
18 ft = 6 yd.

Explanation:
In the above-given question,
given that,
1 ft = 12in.
1yd = 36 in.
6yd = 18 ft.
so 18 ft = 6 yd.

### Lesson 5.9 Ratio Reasoning: Convert Metric Units

Solve & Discuss It!
Sam needs to fill a 5-liter water jug for his team. If Sam uses the water bottle to fill the jug, how many times does he
need to fill the water bottle to fill the jug?

I can… use unit rates to convert metric measurements.

Reasoning
How many milliliters are in 5 liters?

Metric Units of Capacity
1,000 milliliters (ml) = 1 liter (L)
100 centiliters (CL) = 1 liter
10 deciliters (dL) = 1 liter
1 dekaliter (dal) = 10 liters
1 hectoliter (hL) = 100 liters
1 kiloliter (KL) = 1,000 liters

Focus on math practices
Be Precise How many liters of water does Sam’s water bottle hold when full?

Essential Question
How can you use ratios to convert metric units of measure?

Try It!

The middle of the skate ramp is 2.5 meters wide. Emelia and her father want to use a board that is 23.5 decimeters long. Is this board wide enough for them to use? Convert the decimeters to meters to explain.

Answer:
Yes, it is board-wide enough for them to use.

Explanation:
In the above-given question,
given that,
The middle of the skate ramp is 2.5 meters wide.
Emelia and her father want to use a board that is 23.5 decimeters long.
23.5 = 2.35 meters.
so it is board-wide enough for them to use.

Convince Me! How can you convert 2.5 meters to decimeters to determine whether the board is wide enough?

Try It!

To make violet paint, Iris mixes 0.25 liter of red paint, 0.25 liter of blue paint, and 4.5 centiliters of white paint. How many centiliters of paint are in the mixture?

Answer:
The number of centiliters of paint are in the mixture = 0.545 liters.

Explanation:
In the above-given question,
given that,
Iris mixes 0.25 liter of red paint, 0.25 liter of blue paint, and 4.5 centiliters of white paint.
4.5 centiliters = 0.045 litre.
0.045 + 0.25 + 0.25 = 0.545.
so the number of centiliters of paint are in the mixture = 0.545 liters.

KEY CONCEPT
You can convert metric measures by finding an equivalent rate or by using dimensional analysis.

Use an equivalent rate.
1 kg = 1,000 g
$$\frac{1 \mathrm{~kg} \times 1.4}{1,000 \mathrm{~g} \times 1.4}=\frac{1.4 \mathrm{~kg}}{1,400 \mathrm{~g}}$$

Use dimensional analysis.
1.4 kg is equivalent to 1,400 g.
$$1.4 \mathrm{~kg} \times \frac{1,000 \mathrm{~g}}{1 \mathrm{~kg}}$$
= 1.4 × 1,000 g
= 1,400 g

Do You Understand?
Question 1.
Essential Question How can you use ratios to convert metric units of measure?

Answer:
We can convert metric measures by finding an equivalent rate or by using dimensional analysis.

Explanation:
In the above-given question,
given that,
1 kg = 1000 g.
1.4 kilogram is equivalent to 1400 g.
1.4 x 1000 = 1400 g.

Question 2.
Be Precise How are the metric units kilometer and kilogram the same? How are they different?

Answer:
The metrics units kilometers and kilogram are the same.

Explanation:
In the above-given question,
given that,
1 kg = 1000 g.
1.4 kilogram is equivalent to 1400 g.
1.4 x 1000 = 1400 g.

Question 3.
Reasoning Which is greater, 250 m or 0.25 km? Justify your reasoning.

Answer:
Both 250m or 0.25 km are the same.

Explanation:
In the above-given question,
given that,
250 meters = 0.25 kilometers.
divide the length value by 1000.
so both 250m or 0.25 km are the same.

Question 4.
How can you find the conversion rate for milliliters to kiloliters?

Answer:
The conversion rate for milliliters to kiloliters by 1000.

Explanation:
In the above-given question,
given that,
for example:
we can divide the length value by 1000.
1 ml = 1000 l.

Do You Know How?
Question 5.
What is the conversion factor when converting from liters to milliliters?
Answer:

Question 6.
Use an equivalent rate to convert 35 centimeters to meters.

Answer:
The equivalent rate to convert 35 centimeters to meters = 3500.

Explanation:
In the above-given question,
given that,
1 meter = 100 cm.
35 x 100 = 3500.
so the equivalent rate to convert 35 centimeters to meters = 3500.

Question 7.
Critique Reasoning Maddy wants to know how many centigrams are in 0.75 gram. She converted 0.75 gram to its equivalent in centigrams as shown. Is her work correct? Explain.
$$\frac{10 \mathrm{cg} \times 0.75}{1 \mathrm{~g} \times 0.75}=\frac{7.5 \mathrm{cg}}{0.75 \mathrm{~g}}$$
Answer:

Question 8.
Look at Exercise 7. Use dimensional analysis to convert 0.75 gram to centigrams.
Answer:

Practice & Problem Solving

Leveled Practice In 9 and 10, complete each conversion using an equivalent rate.
Question 9.

Answer:
4m = 400 cm.

Explanation:
In the above-given question,
given that,
4m = 400 cm.
100 cm x 4/1m x 4m.
400cm/4m.

Question 10.

Answer:
800 ml = 0.8 l.

Explanation:
In the above-given question,
given that,
800 ml = 0.8 liter.
(1000 ml/ 8)/ (1l/ 1000).
800 ml/ 1000l.
0.8 l.

Leveled Practice In 11 and 12, complete each conversion using dimensional analysis.
Question 11.

Answer:
200 cl = 2 liters.

Explanation:
In the above-given question,
given that,
200 cl = 2 liters.
200 cl x 20000/100cl.
200/100 l.
2 l.

Question 12.

Answer:
2.5 kg = 2500 grams.

Explanation:
In the above-given question,
given that,
2.5 kg x 1000g/1 kg.
250000/100 = 2500.
2.5 kg = 2500 grams.

In 13 and 14, complete each conversion.
Question 13.
80 cm = _______ m

Answer:
80 cm = 0.8m.

Explanation:
In the above-given question,
given that,
1m = 100 cm.
0.8m = 80 cm.
8 = 800 cm.

Question 14.
2.1 g = ________ mg

Answer:
2.1 g = 2100 mg.

Explanation:
In the above-given question,
given that,
2.1 g = 2100 milligrams.
1 kg = 1000 g.
2.1 g = 2100 mg.

In 15-17, use the table showing the amount of liquid that Whitney drinks each day.

Question 15.
How many liters of water does Whitney drink each day?

Answer:
The number of liters of water does Whitney drinks each day = 1.5 l.

Explanation:
In the above-given question,
given that,
the amount of juice = 250 ml.
the amount of milk = 400 ml.
the amount of water = 1500 ml.
1 l = 1000 ml.
so the number of liters of water does Whitney drinks each day = 1.5 L.

Question 16.
What is the total amount of liquid, in liters, that Whitney drinks each day?

Answer:
The total amount of liquid in liters that Whitney drinks each day = 0.25 l.

Explanation:
In the above-given question,
given that,
the amount of juice = 250 ml.
the amount of milk = 400 ml.
the amount of water = 1500 ml.
1 l = 1000 ml.
so the total amount of liquid in liters that Whitney drinks each day = 0.25 L.

Question 17.
Troy drinks 1.8 L of water each day. How many more milliliters of water does Troy drink each day than Whitney?

Answer:
The more milliliters of water does Troy drink each day than Whitney = 300 ml.

Explanation:
In the above-given question,
given that,
Troy drinks 1.8 L of water each day.
1800 – 1500 = 300.
so the number of milliliters does Troy drink each day = 300 ml.

Question 18.
There are 10 millimeters in 1 centimeter, so about how many millimeters long is this dinosaur bone? Explain.

Answer:
The number of milliliters long is the dinosaur bone = 220 ml.

Explanation:
In the above-given question,
given that,
There are 10 millimeters in 1 centimeter.
22 x 10 = 220 ml.
so the number of milliliters long is the dinosaur bone = 220 ml.

Question 19.
Critique Reasoning Savannah says that 1 kilogram is equivalent to 1,000,000 milligrams. Is Savannah correct? Explain.

Answer:
Yes, Savannah was correct.

Explanation:
In the above-given question,
given that,
Savannah says that 1 kilogram is equivalent to 1,000,000 milligrams.
1 kg = 1000 g.
1 g = 1000 mg.
1000 x 1000 = 1000000 mg.
so Savannah was correct.

Question 20.
Model with Math Lucas hiked 14,300 meters through the Everglades in the morning. After lunch, he continued hiking. When he finished the hike, he had covered 31.5 kilometers in all. Write an equation that can be used to find how far Lucas hiked after lunch.

Answer:
Lucas hiked after lunch = 453.96.

Explanation:
In the above-given question,
given that,
Lucas hiked 14,300 meters through the Everglades in the morning.
After lunch, he continued hiking.
When he finished the hike, he had covered 31.5 kilometers in all.
14300/31.5 = 453.96.
so Lucas hiked after lunch = 453.96.

Question 21.
Tariq has a collection of 35 quarters that he wants to send to his cousin. What is the total weight of the quarters in kilograms?

Answer:
The total weight of the quarters in kilograms = 6.17.

Explanation:
In the above-given question,
given that,
Tariq has a collection of 35 quarters that he wants to send to his cousin.
One quarter weighs 5.67 grams.
35/5.67 = 6.17.
so the total weight of the quarters in kilograms = 6.17.

Question 22.
Higher Order Thinking Louis has a bag of 25 pen shells. Each pen shell is 18 centimeters long. What is the combined length of the pen shells in meters?

Answer:
The combined length of the pen shells in meters = 450.

Explanation:
In the above-given question,
given that,
Louis has a bag of 25 pen shells.
Each pen shell is 18 centimeters long.
25 x 18 = 450.
so the combined length of the pen shells in meters = 450.

Assessment Practice

Question 23.
Select all the conversions that are equivalent to the capacity of a 5.5-liter pitcher of lemonade.
☐ 0.0055 kL
☐ 55 mL
☐ 0.055 kL
☐ 550 mL
☐ 5,500 ml

Answer:
Option A is correct.

Explanation:
In the above-given question,
given that,
5.5 liter = 5.5 x 1000.
0.0055.
so option A is correct.

Question 24.
Select all the conversions that are equivalent to the mass of a 425-gram football.
☐ 42,000 mg
☐ 42,500 cg
☐ 450 dg
☐ 4.25 hg
☐ 00.425 kg

Answer:

Explanation:
In the above-given question,
given that,
425 grams.

### Lesson 5.10 Relate Customary and Metric Units

Explain It!
Gianna and her friends are in a relay race. They have a pail that holds 1 liter of water. They need to fill the 1-liter pail, run 50 yards, and dump the water into the large bucket until it overflows. Gianna says that as long as they do not spill any of the water, they will need 7 trips with the 1-liter pail before the large bucket overflows.

I can… convert between customary and metric units.

A. Which conversion factor could you use to determine whether Gianna is correct? Explain.

Answer:
1 gal = 4 qt.

Explanation:
In the above-given question,
given that,
Gianna and her friends are in a relay race.
They have a pail that holds 1 liter of water.
1 gal is equal to 4 qt.
They need to fill the 1-liter pail, run 50 yards, and dump the water into the large bucket until it overflows.
1 gal = 4 qt.

B. Critique Reasoning Gianna’s friend Linus says that you cannot be certain how many trips it will take because the conversion is approximate. Is Linus’s reasoning appropriate? Explain.

Answer:
Yes, Gianna’s was correct.

Explanation:
In the above-given question,
given that,
Gianna and her friends are in a relay race.
They have a pail that holds 1 liter of water.
1 gal is equal to 4 qt.
They need to fill the 1-liter pail, run 50 yards, and dump the water into the large bucket until it overflows.
1 gal = 4 qt.

C. Construct Arguments Is Gianna correct that 7 trips are needed before the bucket overflows? If not, how many trips will it take? Use the table to justify your answer.
Answer:

Focus on math practices

Construct Arguments Morgan says that 4 liters is less than 1 gallon. Construct an argument to show that Morgan is incorrect.

Essential Question
How can you use ratios to convert customary and metric units of measure?

Try It!

Jacob is building a robot named T3-X that is 75 inches tall. To the nearest tenth, how many centimeters tall is T3-X?

Answer:
T3-x is 75 cm tall.

Explanation:
In the above-given question,
given that,
Jacob is building a robot named T3-X that is 75 inches tall.
1 in = 12 cm.
75in ( 1 cm/1 in).
75 x 1 cm = 75 cm.
T3-x is 75 cm tall.

Convince Me! If you want to find the height of T3-X in meters, will you get the same answer if you convert inches to centimeters, and then centimeters to meters, as you would if you convert inches to feet, and then feet to meters? Explain.

Try It!

Find the length of a 100-yard football field in meters. Use 1 yard = 3 feet and 1 meter ≈ 3.28 feet. Round to the nearest tenth.

Answer:
1 yard = 3 feet.
1 meter = 3.28 feet.

Explanation:
In the above-given question,
given that,
the length of a 100-yard football field meters.
1 yard = 3 feet.
1 meter = 3.28 feet.

KEY CONCEPT
You can use what you know about converting within one measurement system to relate customary and metric units. You can convert measures with customary and metric units by finding an equivalent rate or using dimensional analysis.
Use an equivalent rate.
1 kg ≈ 2.20 lb
$$\frac{1 \mathrm{~kg} \times 5}{2.20 \mathrm{lb} \times 5}=\frac{5 \mathrm{~kg}}{11 \mathrm{lb}}$$
5 kg ≈ 11 lb

Use dimensional analysis.
$$5 \mathrm{~kg} \times \frac{2.20 \mathrm{lb}}{1 \mathrm{~kg}}$$
5 × 2.20 = 11 lb
5 kg ≈ 11 lb

Do You Understand?
Question 1.
Essential Question How can you use ratios to convert customary and metric units of measure?
Answer:

Question 2.
Reasoning When converting centimeters to inches, do you multiply or divide by 2.54? Explain.
Answer:

Question 3.
Use Structure How can you find the approximate number of liters in 1 pint?
Remember: 1 quart = 2 pints
Answer:

Question 4.
How is the conversion from inches to centimeters different from other conversions between customary and metric units?
Answer:

Do You Know How?
In 5-8, find the equivalent measure. Round to the nearest tenth.
Question 5.
5 in. = _______ cm

Answer:
5 in = 12.7 cm.

Explanation:
In the above-given question,
given that,
1 ft = 12 in.
5 in = 12.7 cm.

Question 6.
2 mi ≈ _______ km

Answer:
2 mi = 3.219 km.

Explanation:
In the above-given question,
given that,
2 mi is equal to 3.129 km.
1 mi = 1.5645 km.
so 2 miles = 3.219 km.

Question 7.
113 g ≈ _______ oz

Answer:
113 g = 4 oz.

Explanation:
In the above-given question,
given that,
113 g is equal to 3.986 ounces.

Question 8.
14 kg ≈ ______ lb

Answer:
14 kg = 30 lb.

Explanation:
In the above-given question,
given that,
14 kg = 30.865 pounds.
pounds is equal to lb.
so 14 kg = 30 lb.

Question 9.
Convert 30 gallons to liters by finding an equivalent rate.
Answer:

Question 10.
Approximately how many ounces are equivalent to 1 kilogram?
Answer:

Practice & Problem Solving

In 11-18, find the equivalent measure. Round to the nearest tenth.
Question 11.
9qt ≈ _______ L

Answer:
9 qt = 8.52 l.

Explanation:
In the above-given question,
given that,
1 l = 1.06 qt.
9 qt = 8.52 l.

Question 12.
2 gal ≈ _______ L

Answer:
2 gal = 7.571 liters.

Explanation:
In the above-given question,
given that,
1 l = 0.264 gal.
2 gal = 7.571 liters.

Question 13.
2 in. ≈ _______ cm

Answer:
2 in = 5.08 cm.

Explanation:
In the above-given question,
given that,
3 in = 7.62 cm.
2 in = 5.08 cm.

Question 14.
5 km ≈ _______ mi

Answer:
5 km = 3.1 miles.

Explanation:
In the above-given question,
given that,
2 miles = 3.129 km.
5 km = 3.1 miles.

Question 14.
5 km ≈ ______ mi

Answer:
5 km = 3.1 mi.

Explanation:
In the above-given question,
given that,
5 km = 3.1 mi.

Question 15.
10 L ≈ ______ qt

Answer:
10 L = 10.567 q.

Explanation:
In the above-given question,
given that,
10 L = 10.567 q.
10.567 q = 10 L.

Question 16.
5.5 t ≈ ______ T

Answer:
5.5 t = 5.401 T.

Explanation:
In the above-given question,
given that,
5.5 t = 5.401.
1 tone = 0.982.
5.5 t = 5.401 T.

Question 17.
50 lb ≈ ______ kg

Answer:
50 lb = 23 kg.

Explanation:
In the above-given question,
given that,
lb is equal to pounds.
50 pounds = 22.68 kg.
50 lb = 23 kg.

Question 18.
10 oz ≈ ______ g

Answer:
10 oz = 283.495.

Explanation:
In the above-given question,
given that,
10 oz = 283.495 g.
283.495 g = 10 oz.

Question 19.
A chef at a restaurant uses 12 pounds of butter each day. About how many grams of butter does the chef use each day? Use the conversion factors $$\frac{16 \text { ounces }}{1 \text { pound }}$$ and $$\frac{28.35 \text { grams }}{1 \text { ounce }}$$
Answer:

Question 20.
Reasoning Simone wants to know whether a new chest of drawers will fit next to her bed. The chest she would like to buy is 73 centimeters wide. She knows that her room is 86 inches wide. The bed is 76 inches wide. Will the chest fit next to her bed? Explain.

Answer:
Yes, the chest fit next to her bed.

Explanation:
In the above-given question,
given that,
Simone wants to know whether a new chest of drawers will fit next to her bed.
The chest she would like to buy is 73 centimeters wide.
She knows that her room is 86 inches wide.
The bed is 76 inches wide.
so the chest fit next to her bed.

Question 21.
Be Precise Denali is the highest mountain in the United States. What is its height in meters? Round to the nearest whole number.

Answer:

Explanation:
In the above-given question,
given that,
Denali is the highest mountain in the United States.
Denali is approximately 20,320 ft high.

Question 22.
Construct Arguments Francesca wants to convert 1 foot to centimeters. Use what you know about customary units to explain how she can do this.
Answer:

Question 23.
Higher Order Thinking At the state fair, a person must be at least 138 centimeters tall to ride the roller coaster. Billy wants to ride the coaster. He is 4 feet 7 inches tall. Is Billy tall enough to ride the coaster? Explain.
Answer:

Question 24.
Paul’s car holds a maximum of 19 gallons of gas. About how many liters of gas does Paul need to fill his gas tank?

Answer:
The number of liters of gas does Paul need to fill his gas tank = 10 liters.

Explanation:
In the above-given question,
given that,
Paul’s car holds a maximum of 19 gallons of gas.
9 gallons is remaining as shown.
19 – 9 = 10.
so the number of liters of gas does paul need to fill his gas tank = 10 liters.

Assessment Practice

Question 25.
The posted speed limit is 65 miles per hour. Select all the metric measures that are faster than 65 miles per hour.
☐ 65 km per hour
☐ 97.5 km per hour
☐ 104 km per hour
☐ 105.7 km per hour
☐ 120.3 km per hour

Answer:
Option A is correct.

Explanation:
In the above-given question,
given that,
The posted speed limit is 65 miles per hour.
65 km per hour.
so option A is correct.

Question 26.
Boys competing in the long jump event must jump at least 15 feet to qualify for the state track and field meet. Select all the metric measures that are less than 15 feet.
☐ 6.5 m
☐ 5.0 m
☐ 4.5 m
☐ 3.92 m
☐ 3.5 m

Answer:

### Topic 5 Review

Essential Question
What are ratios and rates? How can you use ratios and rates to describe quantities and solve problems?

Vocabulary Review
Complete each definition and then provide an example of each vocabulary word.

Use Vocabulary in Writing
Explain how you can convert 52 ounces to pounds. Use vocabulary words in your explanation.

Concepts and Skills Review

Lesson 5.1 Understand Ratios

Quick Review
A ratio is a relationship in which for every x units of one quantity there are y units of another quantity. A ratio can be written using the word “to,” a colon, or a fraction bar to separate the two terms.

Example
The ratio of men to women at a small wedding is 6:4. If there are 16 women at the wedding, how many men are at the wedding?
Draw a diagram to represent the ratio. Because 4 boxes represent 16 women, each box represents 4 women.

There are 24 men at the wedding.

Practice
A florist uses 5 red roses for every 2 white roses in her bouquets.
Question 1.
Write the ratio of white roses to red roses in three different ways.

Answer:
The ratio of white roses to red roses is 2:5.

Explanation:
In the above-given question,
given that,
A florist uses 5 red roses for every 2 white roses in her bouquets.
the ratio of white roses to red roses is 2:5.
so the ratio is 2:5.

Question 2.
Write the ratio of red roses to the total number of flowers in three different ways.

Answer:
The ratio of red roses to the total number of flowers is 5:7.

Explanation:
In the above-given question,
given that,
A florist uses 5 red roses for every 2 white roses in her bouquets.
the ratio of red roses to the total number of flowers is 5:7.
so the ratio is 5:7.

Question 3.
If the florist uses 10 red roses in a bouquet, how many white roses does she use?

Answer:
The ratio is 10:4.

Explanation:
In the above-given question,
given that,
A florist uses 5 red roses for every 2 white roses in her bouquets.
if the florist uses 10 red roses in a bouquet.
he can use 4 white roses.
10: 4.
5:2.
so the ratio is 10:4.

Question 4.
If the florist uses 10 white roses in an arrangement, how many red roses does she use?

Answer:
The ratio is 25:10

Explanation:
In the above-given question,
given that,
A florist uses 5 red roses for every 2 white roses in her bouquets.
if the florist uses 10 white roses in an arrangement.
he can use 25 red roses.
25:10.
5:2.
so the ratio is 25:10.

Lesson 5.2 Generate Equivalent Ratios

Quick Review
You can multiply or divide both terms of a ratio by the same nonzero number to find equivalent ratios.

Example
Find two ratios that are equivalent to $$\frac{21}{126}$$
One Way
Multiply.
$$\frac{21 \times 2}{126 \times 2}=\frac{42}{252}$$

Another Way
Divide.
$$\frac{21 \div 3}{126 \div 3}=\frac{7}{42}$$

Practice
In 1-4, find two ratios equivalent to the given ratio.
Question 1.
$$\frac{5}{12}$$

Answer:
The ratios are 10/24 and 1/2.4.

Explanation:
In the above-given question,
given that,
the ratio is 5/12.
5 x 2 = 10.
12 x 2 = 24.
the ratio is 10/24.
divide by 5.
5/5 = 1.
12/5 = 2.4.
so the ratio is 1/2.4

Question 2.
14:32

Answer:
The ratios are 7/16 and 5/12.

Explanation:
In the above-given question,
given that,
the ratio is 14/32.
7 x 2 = 14.
16 x 2 = 32.
the ratio is 7/16.
divide by 2.
10/2 = 5.
24/2 = 12.
so the ratio is 5/12.

Question 3.
3 to 4

Answer:
The ratios are 6/8 and 1/1.3.

Explanation:
In the above-given question,
given that,
the ratio is 3/4.
3 x 2 = 6.
4 x 2 = 8.
the ratio is 6/8.
divide by 3.
3/3 = 1.
4/3 = 1.3.
so the ratio is 1/1.3.

Question 4.
$$\frac{7}{8}$$

Answer:
The ratios are 14/16 and 1/1.14.

Explanation:
In the above-given question,
given that,
the ratio is 7/8.
7 x 2 = 14.
8 x 2 = 16.
the ratio is 14/16.
divide by 7.
7/7 = 1.
8/7 = 1.14.
so the ratio is 1/1.14.

Question 5.
For every 4 bagels sold at a bakery, 7 muffins are sold. How many muffins are sold when the bakery sells 24 bagels? Complete the table.

Answer:
The missing values are 14, 21, 28, 35, and 42.

Explanation:
In the above-given question,
given that,
For every 4 bagels sold at a bakery, 7 muffins are sold.
the ratio is 1:2.
4 x 1 = 4, 4 x 2 = 8, 4 x 3 = 12, 4 x 4 = 16, 4 x 5 = 20, and 4 x 6 = 24.
7 x 1 = 7, 7 x 2 = 14, 7 x 3 = 21, 7 x 4 = 28, 7 x 5 = 35, and 7 x 6 = 42.
so the missing values are 14, 21, 28, 35, and 42.

Lesson 5.3 Compare Ratios

Quick Review
To compare ratios, make a table to show each ratio and then find a value in which one of the terms is the same in both tables.

Example

Erica can complete more facts than Klayton.

Answer:
The missing value is 4.

Explanation:
In the above-given question,
given that,
the number of days is 1, 2, 3, and 4.
the days of the sun are 2, 4, 6, and 8.
2 x 1 = 2.
2 x 2 = 4.
2 x 3 = 6.
2 x 4 = 8.
so the missing value is 4.

Practice
Question 1.
The school soccer team buys 3 soccer balls for every 2 players. The school volleyball team buys 7 volleyballs for every 5 players. Which team buys more balls per player?

Answer:
The team buys more balls per player = 6:35.

Explanation:
In the above-given question,
given that,
The school soccer team buys 3 soccer balls for every 2 players.
The school volleyball team buys 7 volleyballs for every 5 players.
3 x 2 = 6.
7 x 5 = 35.
so the team buys more balls per player = 6:35.

Question 2.
Jenna walks 12 miles in 5 days. Alex walks 7 miles in 3 days. Who walks more miles per day?

Answer:
The ratio is 60/21.

Explanation:
In the above-given question,
given that,
Jenna walks 12 miles in 5 days. Alex walks 7 miles in 3 days.
12 x 5 = 60.
7 x 3 = 21.
so the ratio is 60/21.

Lesson 5.4 Represent and Graph Ratios

Quick Review
You can solve some ratio problems by making a table of equivalent ratios and then graphing the pairs of values on a coordinate plane.

Example

There will be 4 rainy days if there are 8 sunny days.

Practice
Question 1.
In gym class, the sixth graders walk 2 laps for every 3 laps they run. If the students run 12 laps, how many laps will they walk? Complete the table. Then plot the pairs of values on the coordinate plane.

Answer:
The missing values are 4, 6, and 8.

Explanation:
In the above-given question,
given that,
In gym class, the sixth graders walk 2 laps for every 3 laps they run.
the run laps on the y-axis.
the walk laps on the x-axis.
3 x 2 = 6, 3 x 3 = 9, 3 x 4 = 12.
2 x 2 = 4, 2 x 3 = 6, 2 x 4 = 8.

Lesson 5.5 Understand Rates and Unit Rates

Quick Review
A rate is a ratio that relates two quantities with different units. A unit rate relates a quantity to 1 unit of another quantity. You can use what you know about dividing fractions to write a ratio of fractions as a unit rate.

Example
Write 20 meters in 4 minutes as a rate and as a unit rate.

Practice
Write each statement as a unit rate.
Question 1.
78 miles on 3 gallons

Answer:
The unit rate is 26/1.

Explanation:
In the above-given question,
given that,
78 miles on 3 gallons.
unit rate is 78/3.
3 x 1 = 3.
3 x 26 = 78.
so the unit rate is 26/1.

Question 2.
18 laps in 6 minutes

Answer:
The unit rate is 18.

Explanation:
In the above-given question,
given that,
18 laps in 6 minutes
unit rate is 18/6.
6 x 1 = 6.
3 x 6 = 18.
so the unit rate is 18/1.

Question 3.
48 sandwiches for 16 people

Answer:
The unit rate is 3.

Explanation:
In the above-given question,
given that,
48 sandwiches for 16 people.
unit rate is 48/16.
16 x 1 = 16.
3 x 16 = 48.
so the unit rate is 3.

Question 4.
49 houses in 7 blocks

Answer:
The unit rate is 7.

Explanation:
In the above-given question,
given that,
49 houses in 7 blocks.
unit rate is 49/7.
7 x 1 = 7.
7 x 7 = 49.
so the unit rate is 7.

Question 5.
6 desks in 2 rows

Answer:
The unit rate is 3.

Explanation:
In the above-given question,
given that,
6 desks in 2 rows.
unit rate is 6/2.
2 x 1 = 2.
3 x 2 = 6.
so the unit rate is 3.

Lesson 5.6 Compare Unit Rates

Quick Review
A unit rate compares a quantity to 1 unit of another quantity. To compare unit rates, compare the first terms.

Example
On Pet Day, Meg’s turtle crawled 30 feet in 6 minutes, and Pat’s turtle crawled 25 feet in 5 minutes. Whose turtle crawled at a faster rate?
Write each rate.

Both turtles crawled at the same rate.

Practice
Question 1.
Which is the better value? Circle it.
$5.00 for 4 mangoes$6.00 for 5 mangoes

Answer:
Both have the same value.

Explanation:
In the above-given question,
given that,
$5.00 for 4 mangoes$6.00 for 5 mangoes
$5/4 = 1.25.$6/5 = 1.2.
so both of the values are the same.

Question 2.
Who earned more each month? Circle it.
Atif: $84 over 3 months Jafar:$100 over 4 months

Answer:
Atif earned more than Jafar.

Explanation:
In the above-given question,
given that,
Atif: $84 over 3 months Jafar:$100 over 4 months
84/3 = 28.
100/4 = 25.
so Atif earned more.

Question 3.
Which is a faster rate? Circle it.
3 laps in 5 minutes
4 laps in 7 minutes

Answer:
3 laps in 5 minutes is the faster rate.

Explanation:
In the above-given question,
given that,
3 laps in 5 minutes
4 laps in 7 minutes
3/5 = 0.6.
4/7 = 0.5.
option A is correct.

Question 4.
Which is the better value? Circle it.
3 sandwiches for $15.00 4 sandwiches for$21.00

Answer:
3 sandwiches for $15.00. Explanation: In the above-given question, given that, 3 sandwiches for$15.00
4 sandwiches for $21.00 3/15 = 1/5. 1/5 = 0.2. 4/21 = 0.1. Question 5. Which is the greater rate? Circle it. 6 points in 3 attempts 15 points in 5 attempts Answer: 15 points in 5 attempts. Explanation: In the above-given question, given that, 6 points in 3 attempts 15 points in 5 attempts 6/3 = 2. 15/5 = 3. 15 points in 5 attempts. Lesson 5.7 Solve Unit Rate Problems Quick Review You can use a ratio table or a unit rate to solve problems involving ratios or rates. Example A plane travels at a rate of 780 miles in 2 hours. At this rate, how far will it travel in 3.5 hours? Find the unit rate $$\frac{780 \text { miles } \div 2}{2 \text { hours } \div 2}=\frac{390 \text { miles }}{1 \text { hour }}$$ Find an equivalent rate. $$\frac{390 \text { miles } \times 3.5}{1 \text { hour } \times 3.5}=\frac{1,365 \text { miles }}{3.5 \text { hours }}$$ The plane will travel 1,365 miles in 3.5 hours. Practice Question 1. Doug has 5 hours to make an on-time delivery 273 miles away. Doug drives at a constant speed of 55 miles per hour. Will Doug make the delivery by the deadline? Explain. Answer: Yes, Doug makes the delivery by the deadline. Explanation: In the above-given question, given that, Doug has 5 hours to make an on-time delivery 273 miles away. Doug drives at a constant speed of 55 miles per hour. 273/5 = 54.6. 54.6 x 55 = 3003. Yes, Doug makes the delivery by the deadline. Question 2. Marie has 8 hours to write a 45-page chapter for her book. Marie writes at a constant speed of 4 pages per hour. Will Marie complete the chapter in time? Explain. Answer: Marie takes 90 hours to complete. Explanation: In the above-given question, given that, Marie has 8 hours to write a 45-page chapter for her book. Marie writes at a constant speed of 4 pages per hour. 45 x 8 = 360. 360 /4 = 90. Lesson 5.8 Ratio Reasoning: Convert Customary Units Quick Review You can convert customary measures by finding equivalent rates or by using dimensional analysis. Example How many pints are equivalent to 4 quarts? Find an equivalent rate: 2 pints = 1 quart …….. Identify the conversion rate. So, 8 pints are equivalent to 4 quarts. Practice In 1-4, complete each conversion. Question 1. 2 mi = _______ ft Answer: 2 miles = 10560 feet. Explanation: In the above-given question, given that, 1 mile = 63360 in. 1 ft = 12 in. 2 mi = 63360 x 2. 126720/2 = 10560. Question 2. 144 in. = _______ yd Answer: 144 in = 4.0032 yd. Explanation: In the above-given question, given that, 1 inch = 0.0278 yd. 144 x 0.0278. 4.0032. 144 in = 4.0032 yd. Question 3. 4 oz = _______ lb Answer: 4 oz = 0.0625 lb. Explanation: In the above-given question, given that, 1 oz = 0.0625 lb. 4 oz = 4 x 0.0625. 0.25 lb. Question 4. 3 gal = _______ qt Answer: 3 gal = 12 qt. Explanation: In the above-given question, given that, 1 gal = 4 us qt. 3 gal = 3 x 4. 3 gal = 12 qt. Question 5. The hippo at the zoo weighs 1.5 tons. How many pounds does the hippo weigh? Answer: The weight of the hippo is 3000 pounds. Explanation: In the above-given question, given that, The hippo at the zoo weighs 1.5 tons. 1 ton = 2000 pounds. 1.5 tons = 1.5 x 2000. 3000. The weight of the hippo is 3000 pounds. Lesson 5.9 Ratio Reasoning: Convert Metric Units Quick Review To convert metric units, use the same methods used for converting customary units. Either use the conversion rate to find an equivalent rate or use dimensional analysis. Example Tariq rode his bike 15,100 meters. How many kilometers did he ride his bike? Find an equivalent rate: 1,000 meters = 1 kilometer $$\frac{1,000 m \times 15.1}{1 \mathrm{~km} \times 15.1}=\frac{15,100 \mathrm{~m}}{15.1 \mathrm{~km}}$$ Use dimensional analysis: $$15,100 \mathrm{~m} \times \frac{1 \mathrm{~km}}{1,000 \mathrm{~m} }=\frac{15,100}{1,000} \mathrm{~km}=15.1 \mathrm{~km}$$ Tariq rode 15.1 kilometers. Practice In 1-4, complete each conversion. Question 1. 3 m = _______ mm Answer: 3m = 3000 mm. Explanation: In the above-given question, given that, 1 m = 1000 mm. 3 m = 3 x 1000. 3m = 3000 mm. Question 2. 3,520 mm = _______ cm Answer: 3520 mm = 352 cm. Explanation: In the above-given question, given that, 1mm = 0.1 cm. 3520 mm = 352 cm. Question 3. 4.2 kg = _______ g Answer: 4.2 kg = 4200 g. Explanation: In the above-given question, given that, 1 kg = 1000 g. 4.2 kg = 4.2 x 1000. 4.2 x 1000 = 4200 g. 4.2 kg = 4200 g. Question 4. 300 mL = _______ L Answer: 300 ml = 0.3 l. Explanation: In the above-given question, given that, 1ml = 0.001 l. 300 ml = 300 x 0.001. 300 x 0.001 = 0.3 l. 300 ml = 0.3l. Question 5. Li needs to buy 2 kilograms of apples. If she buys 9 apples that each weigh approximately 150 grams, will she have enough? Explain. Answer: Yes, she has enough. Explanation: In the above-given question, given that, Li needs to buy 2 kilograms of apples. If she buys 9 apples that each weigh approximately 150 grams. 150 x 9 = 1350 g. 2 kg = 2000 g. so she has enough space. Lesson 5.10 Relate Customary and Metric Units Quick Review To convert between metric and customary units, use the conversion rate and find an equivalent rate, or use dimensional analysis. Most conversions will be approximate because, except in the case of inches to centimeters, the conversion rates are approximate. Example Gwen has a cooler that holds 3 quarts. About how many liters does the cooler hold? 1 qt ≈ 0.95 L 3 qt × $$\frac{0.95 \mathrm{~L}}{1 \mathrm{~qt}}$$ = (3 × 0.95) L = 2.85 L Gwen’s cooler holds approximately 2.85 liters. Practice In 1-4, find the equivalent measure. Round to the nearest tenth. Question 1. 100 g ≈ ________ oz Answer: 100 g = 15372.2 oz. Explanation: In the above-given question, given that, 1 gal = 153.722 oz. 100 g = 153.722 x 100. 15372.2. Question 2. 6 ft ≈ _______ m Answer: 6 ft = 1.83 m. Explanation: In the above-given question, given that, 1 ft = 0.305 m. 6 ft = 6 x 0.305. 6 x 0.305 = 1.83. 6 ft = 1.83 m. Question 3. 57 gal ≈ _______ L Answer: 57 gal = 256.5 l. Explanation: In the above-given question, given that, 1 gal = 4.5 l. 57 gal = 57 x 4.5. 57 x 4.5 = 256.5. 57 gal = 256.5 l. Question 4. 27 km ≈ _______ mi Answer: 27 km = 16.767 mi. Explanation: In the above-given question, given that, 1 km = 0.621 mi. 27 km = 27 x 0.621. 27 x 0.621 = 16.767. 27 km = 16.767 mi. Question 5. The science class is raising monarch caterpillars. One of the caterpillars weighs 2.3 ounces. About how many grams does the caterpillar weigh? Round to the nearest tenth. Answer: The caterpillar weighs 65.2 grams. Explanation: In the above-given question, given that, The science class is raising monarch caterpillars. One of the caterpillars weighs 2.3 ounces. 1 ounce = 28.35 grams. 2.3 x 28.35 = 65.205. so the caterpillar weighs 65.2 grams. ### Topic 5 Fluency Practice Pathfinder Shade a path from START to FINISH. Follow the sums and differences in which the digit in the ones place is greater than the digit in the tenths place. You can only move up, down, right, or left. I can… add and subtract multidigit decimals. #### enVision Math Common Core Grade 6 Answer Key ## Envision Math Common Core Grade 4 Answer Key Topic 1 Generalize Place Value Understanding ## Envision Math Common Core 4th Grade Answers Key Topic 1 Generalize Place Value Understanding Essential Questions: How are greater numbers written? How can whole numbers be compared? How are place values related? enVision STEM Project: Caves Do Research Use the Internet or other sources to find the depths in feet of the 5 deepest caves in the world. Journal: Write a Report Include what you found. Also in your report: • Make a place-value chart that includes the five depths. • Write each depth in expanded form. • Use “greater than” or “less than” to compare the depths of two of the caves. Review What You Know Vocabulary Choose the best term from the box. Write it on the blank. • expanded form • place value • number line • rounding • number name • whole numbers Question 1. The numbers 0, 1, 2, 3, 4, and so on are called ___________. Answer: The numbers 0, 1, 2, 3, 4, and so on are called Whole Numbers. Question 2. A number written using only words is written using a __________ Answer: A number written using only words is written using a Number Name . Question 3. Replacing a number with a number that tells about how many or how much is called _________. Answer: Replacing a number with a number that tells about how many or how much is called Place Value Question 4. _______ is the value given to the place of a digit in a number. Answer: Place Value is the value given to the place of a digit in a number. Comparing Numbers Compare each set of numbers using >, <, or =. Question 5. 201 21 Answer: 201 < 21 Explanation : 201 is a three digit number 21 is a two digit number three digit is greater than two digit numbers . Question 6. 313 313 Answer: 313 = 313 Explanation : Step 1 Write the numbers, lining up places. Begin at the left and compare. 313 313 The hundreds digit is the same in both numbers. Step 2 Look at the next digit. Compare the tens. 313 313 The tens digit is also the same in both numbers. Step 3 Look at the next digit. Compare the Ones. 313 313 The Ones digit is also the same in both numbers. so, 313 = 313 . Question 7. 289 290 Answer: Explanation : Step 1 Write the numbers, lining up places. Begin at the left and compare. 289 290 The hundreds digit is the same in both numbers. Step 2 Look at the next digit. Compare the tens. 289 290 8 tens < 9 tens 289 < 290 . Question 8. 7 70 Answer: Explanation : 7 is a one digit number 70 is a two digit number one digit is lesser than two digit numbers . Question 9. 725 726 Answer: 725 < 726 Explanation : Step 1 Write the numbers, lining up places. Begin at the left and compare. 725 726 The hundreds digit is the same in both numbers. Step 2 Look at the next digit. Compare the tens. 725 726 The tens digit is also the same in both numbers. Step 3 Look at the next digit. Compare the Ones. 725 726 5 Ones < 6 Ones so, 725 < 726 . Question 10. 82 82 Answer: 82 = 82 Explanation : Step 1 Write the numbers, lining up places. Begin at the left and compare. 82 82 The hundreds digit is the same in both numbers. Step 2 Look at the next digit. Compare the tens. 82 82 The tens digit is also the same in both numbers. Step 3 Look at the next digit. Compare the Ones. 82 82 The Ones digit is also the same in both numbers. so,82 = 82 Question 11. 614 641 Answer: 614 < 641 Explanation : Step 1 Write the numbers, lining up places. Begin at the left and compare. 614 641 The hundreds digit is the same in both numbers. Step 2 Look at the next digit. Compare the tens. 614 641 1 Tens < 4 Tens . 614 < 641 614 is lesser than 641 . Question 12. 618 618 Answer: 618 = 618 Explanation : Step 1 Write the numbers, lining up places. Begin at the left and compare. 618 618 The hundreds digit is the same in both numbers. Step 2 Look at the next digit. Compare the tens. 618 618 The tens digit is also the same in both numbers. Step 3 Look at the next digit. Compare the Ones. 618 618 The Ones digit is also the same in both numbers. so, 618 = 618 . Question 13. 978 987 Answer: 978 < 987 Explanation : Step 1 Write the numbers, lining up places. Begin at the left and compare. 978 987 The hundreds digit is the same in both numbers. Step 2 Look at the next digit. Compare the tens. 978 987 7 Tens < 8 Tens . 978 < 987 978 is lesser than 987 . Place Value Tell if the underlined digit is in the ones, tens, hundreds, or thousands place. Question 14. 9,482 Answer: 9,482 = 9 Thousands place Question 15. 8,000 Answer: 8,000 = 0 Tens . Question 16. 1,506 Answer: 1,506 = 0 Hundreds . Question 17. 8,005 Answer: 8,005 = 8 thousands Question 18. 5,100 Answer: 5,100 = 1 hundreds . Question 19. 2,731 Answer: 2,731 = 2 Thousands . Rounding Question 20. Construct Arguments Use the number line to describe how to round 450 to the nearest hundred. Answer: 500 Explanation : Nearest 100 of 450 will be 500 . Pick a Project PROJECT 1A How many bones are in your body? Project: Make a Bones Poster PROJECT 1B Would you like to be a construction manager? Project: Design a Building PROJECT 1С Which stadium is your favorite? Project: Create a Stadium Model 3-ACT MATH PREVIEW Math Modeling Page Through ### Lesson 1.1 Numbers Through One Million Solve & Share Mrs. Darcy saved ten$100 bills. How much money did Mrs. Darcy save?
I can … read and write numbers through one million in expanded form, with numerals, and using number names.

Look Back! How did you decide how many zeros you needed to write in your answer?
Answer :
Number of bills = 10 .
Bill Amount = $100 . Total Amount =$100 × 10 = $1000 . Explanation : When multiplying whole numbers by 10, simply add a 0 to the end of the number, so , 100 × 10 = 1000 . Essential Question What Are Some Ways to Write Numbers to One Million? Answer : 10000 × 100 = 1,000,000 . 1000 × 1000 = 1,000,000 . 100000 × 10 = 1,000,000 . Visual Learning Bridge The graph shows the attendance at a ballpark over one year. Write the total attendance in expanded form and using number names. The place-value chart shows periods of three places, starting at the ones period from the right and including the thousands and millions period. Each period is separated by a comma and has three place values: ones, tens, and hundreds. Each digit in 356,039 is written in its place on the chart. Expanded form shows the sum of the values of each digit. Expanded form: 300,000 + 50,000 + 6,000 + 30 + 9 Number name: three hundred fifty-six thousand, thirty-nine is written. Convince Me! Look for Relationships What pattern exists in the three places in each period? Another Example! 21, 125 can be expanded and written in different ways. 20,000 + 1,000 + 100 + 20 + 5 21,000 + 100 + 25 20,000 + 1,100 + 20 + 5 Guided Practice Do You Understand? Question 1. What do you notice about the comma in the number on the previous page? Answer: The commas separates the periods when number name is written . Question 2. Write an example of a number that would include 2 commas. Answer: 2,000,000 = Two millions . Do You Know How? Question 3. Write 7,320 in expanded form. Answer: Expanded form of 7,320 : 7,000 + 300 + 20 + 0 . Question 4. Write 55,426 using number names. Answer: Number name of 55,426 = fifty five thousands four hundred and twenty six . Question 5. In a recent year, 284,604 fans attended the hockey playoffs in Chicago. What digit is in the thousands place in 284,604? Answer: 284,604 = 4 thousands . Independent Practice For 6-8, write each number in expanded form. Question 6. 7,622 Answer: Expanded form of 7,622 : 7,000 + 600 + 20 + 2 . Question 7. 294,160 Answer: Expanded form of 294,160 : 200,000 + 90,000 + 4,000 + 100 + 60 + 0 . Question 8. 43,702 Answer: Expanded form of 43,702 = 40,000 + 3,000 + 700 + 00 + 2 . For 9-11, write each number name. Question 9. 1,688 Answer: Number name of 1,688 : One Thousand six hundred and Eighty Eight . Question 10. 331,872 Answer: Number name of 331,872 : Three hundred and thirty one-thousand and Eight hundred and seventy two . Question 11. 44,444 Answer: Number name of 44,444 : Forty four thousand four Hundred and forty four . Problem Solving Question 12. Letitia wrote one thousand, two hundred four in a place-value chart. What mistake did she make? Answer: one thousand, two hundred four : 1,204 . Explanation : The mistake is that the four should be written in the Ones place not in tens place value . Question 13. Reasoning in 2016, the world’s oldest tree was 5,066 years old. Write the number that is one hundred more using number names. Answer: Age of old tree = 5,066 years One year more than 5,066 years = 5,066 + 100 = 5,166 years . 5,166 : Five thousand One hundred and sixty six . Question 14. Jessica wants to buy a new team jacket that costs$35. If Jessica saves $5 a week for 4 weeks and$4 a week for 3 weeks, will she have enough money to buy the team jacket? Explain.
Answer:
$32 <$ 35 .
No, she doesn’t have enough money she haves $3 less than the required amount . Explanation : Cost of Jacket =$35  .
Number of weeks = 4 weeks
Money saved for each week = $5 Amount saved for 4 weeks = 4 ×$ 5 =  $20 . Number of weeks = 3 weeks Money saved for each week =$4
Amount saved for 4 weeks = 3 × $4 =$12 .
Total Amount saved = $20 +$12 = $32 . No, she doesn’t have enough money she haves$3 less than the required amount .

Question 15.
Vocabulary Drew wrote the following sentence: “A period is a group of any 3 three digits in a number.” Do you agree with Drew? If not, how would you correct him?
Answer:
In place value chart, the digits are grouped in the threes in a big number. The number is read from left to right as ………. billion ………. million ……….. thousands ……….. ones.
The place value chart of the International System is given below:

Question 16.
Higher Order Thinking Two numbers have the same digit in the millions period, the same digits in the thousands period, and the same digits in the ones period. Do these two numbers have the same value? Explain.
Answer:
No,
Explanation :
5,111,666,222
50,111,666,222
Both have same numbers in million,thousands and ones period but they have different numbers in billions periods .

Assessment Practice

Question 17.
Wallace writes the number 72,204 in a place-value chart. Select the places that will be filled on the chart.
☐ Ones
☐ Tens
☐ Thousands
☐ Ten thousands
☐ Hundred thousands
Answer:

Question 18.
Select all that are equal to 96,014.
☐ 96,000 + 10 + 4
☐ 90,000 + 60,000 + 10 + 4
☐ 90,000 + 6,000 + 4
☐ 90,000 + 6,000 + 10 + 4
☐ 96,000 + 14
Answer:

### Lesson 1.2 Place Value Relationships

Solve & Share
Place-value blocks are shown below for 1, 10, and 100. What patterns in the shapes and sizes of the blocks do you see?
I can… recognize that a digit in one place has ten times the value of the same digit in the place to its right.

Look Back! Describe two ways 100 and 10 are related.
Answer :

Essential Question
How Are Place Values Related to Each Other?

Visual Learning Bridge
Kiana had bottle caps. She wants to collect ten times as many bottle caps. How many bottle caps will Kiana have in her collection then?

A hundreds flat represents 100 bottle caps.

To find ten times as many bottle caps, group 10 hundreds flats together.

One thousand is ten times 100.
100 × 10 = 1,000
One hundred is one-tenth of 1,000.
1,000 ÷ 10 = 100
Kiana will have 1,000 bottle caps in her collection.

Convince Me! Generalize Use place-value blocks to model 1 and 10, 10 and 100, 100 and 1,000. What pattern do you see?

Another Example!
Joe scored 2,000 points on a progressive video game. It took him 5 weeks to get his total point value to 20,000. It took him 3 months to get his total point value to 200,000 points. How many times greater than his first score were his points after 5 weeks? After 3 months?
After 5 weeks, Joe’s points were 10 times greater.
After 3 months, Joe’s points were 100 times greater.

2,000 × 10 = 20,000
20,000 × 10 = 200,000
10 × 10 = 100

Guided Practice

Do You Understand?
Question 1.
Is the value of the 2 in 23,406 ten times as great as the value of the 3? Explain.
Answer:
No,
Explanation :
Value of 2 in 23,406 = 20,000 .
Value of 3 in 23,406 = 3,000 .
value of the 2 in 23,406 ten times as great as the value of the 3 = 3,000 × 10 = 30,000 .
So, not equal .

Do You Know How?
For 2, use the relationship between the values of the digits to solve.
Question 2.
Write a number in which the value of the 3 is ten times greater than the value of the 3 in 135,864.
Answer:
345456 .
Explanation:
The value of 3 in 135864 is 30000.We want to write a number that is ten times 30000.We could write any 345456.
Just increase the place value to hundred thousands .

Independent Practice

For 3-5, use the relationship between the values of the digits to solve.
Question 3.
Baseten School District bought 5,000 pencils. They are distributing the pencils evenly to 10 schools in the district. How many pencils will each school get?
Answer:
Number of Pencils = 5,000
Number of schools  = 10 .
Number of pencils each school get = 5,000 ÷ 10 = 500 pencils .
5,000 is ten times the value of 500 .

Question 4.
Place Elementary School is raising money. They raise $90 a week. How long will it take them to raise$900?
Answer:
Amount rise for 1 week = $90 Risen Amount =$900 .
$900 is ten times greater than$90 .
So, Number weeks required to rise Amount = 10 weeks .

Question 5.
A donation of 50 rulers was given to Value Elementary School. The school had 10 times as many erasers donated. How many erasers were donated?
Answer:
Number of Rulers = 50 .
Number of erasers donated =10 times as many erasers donated = 10 × 50 = 500 erasers .

Problem Solving

Question 6.
What can you say about the 3s in 43,862 and 75,398?
Answer:
Place value of 3 in 43,862 = 3,000
Place value of 3 in 75,398 = 300 .
The place value of 3 in 43,862 is ten times greater than the place value in 75,398 .

Question 7.
Critique Reasoning
Mia says in 5,555, all the digits have the same value. Is Mia correct? Explain.
Answer:
No, she is not correct .
Explanation :
Place value from left to right is explained .
Place value of first 5 = 5,000 .
Place value of second 5 = 500 .
Place value of third 5 = 50 .
Place of fourth 5 = 5 .

Question 8.
Number Sense
In 1934, there was an extreme drought in the Great Plains. In the number 1,934, is the value of the 9 in the hundreds place ten times as great as the value of the 3 in the tens place? Explain.
Answer:
No, the place value in Hundreds place is not ten times the place value in tens place .
Explanation :
Place value of 9 in 1934 = 900
Place value of 3 in 1934 = 30 .
9 in the hundreds place ten times as great as the value of the 3 in the tens place = 30 × 10 = 300 .
where 900 and 300 are not equal .

Question 9.
Critique Reasoning Vin says in 4,346, one 4 is 10 times as great as the other 4. Is Vin correct? Explain.
Answer:
No, he is wrong
40 × 100 = 4,000 .
Explanation :
Place of 4 in 4,346 in thousands place = 4,000 .
Place of 4 in 4,346 in tens place = 40 .
Place of 4 in 4,346 in thousands place is 100 times greater than Place of 4 in 4,346 in tens place
40 × 10 = 400 .
40 × 100 = 4,000 .

Question 10.
Describe 2 ways to find the area of the shaded rectangle.

Answer:
Length of shaded rectangle = 3 squares = 3 square unit .
Breadth of Shaded Rectangle = 4 squares = 4 square unit .
Area of rectangle = Length × Breadth = 3 × 4 = 12 unit squares .

Number of squares shaded = 12 square units .
Area of shaded Rectangle = 12 unit squares .

Question 11.
Higher Order Thinking In 448,244, how is the relationship between the first pair of 4s the same as the relationship between the second pair of 4s?
Answer:
448,244
The place of first pair of 44 in 448,244 = 440,000.
The place of Second pair of 44 in 448,244 = 44.
The relationship between the first pair of 4s the same as the relationship between the second pair of 4s = First pair of 44 is 10,000 times greater than Second pair of 44 .
= 44 × 10,000 = 440,000 .

Assessment Practice

Question 12.
Which group of numbers shows the values of the 4s in 44,492?
A. 40,000; 4,000; 400
B. 40,000; 400; 40
C. 4,000; 400; 4
D. 400; 40; 4
Answer:
Option A .
Explanation :
The values of the 4s in 44,492 = 40,000 ; 4,000 ; 400 .

Question 13.
In which number is the value of the red digit ten times as great as the value of the blue digit?
A. 335,531
B. 335,531
C. 335,531
D. 335,531
Answer:

### Lesson 1.3 Compare Whole Numbers

Solve & Share
A robotic submarine can dive to a depth of 26,000 feet. Which oceans can the submarine explore all the way to the bottom? Solve this problem any way you choose.
I can … use place value to compare numbers and record my comparisons using <, =, or >.

Look Back! Which of the oceans listed is the shallowest? Explain.
Answer :
Compare Depths :
28,232 ; 35,840 ; 23,376 .
Step 1:
Write the numbers, lining up places. Begin at the left and compare.
The ten thousands digit 2 is the same in both numbers 28,232 and 23,376 But the number is 3 in 35,840 .
that means 35,840 is the greatest number other than two numbers .
Step 2 :
Look at the next digit. Compare the thousands
28,232
23,376
8 thousands > 3 thousands .
so, 28,232 > 23,376 .
So, 35,840 > 28,232 > 23,376 .

Essential Question
How Do You Compare Numbers?

Visual Learning Bridge
Earth is not perfectly round. The North Pole is 6,356 kilometers from Earth’s center. The equator is 6,378 kilometers from the center. Which is closer to Earth’s center, the North Pole or the equator?

Step 1
Write the numbers, lining up places. Begin at the left and compare.
6,356
6,378
The thousands digit is the same in both numbers.

Step 2
Look at the next digit. Compare the hundreds.
6,356
6,378
The hundreds digit is also the same in both numbers.

Step 3
The first place where the digits are different is the tens place. Compare the tens.
6,356 5 tens < 7 tens,
6,378 so 6,356 < 6,378.
The North Pole is closer than the equator to Earth’s center.

Convince Me! Reasoning is a whole number with 4 digits always greater than or less than a whole number with 3 digits? Explain.

Guided Practice

Do You Understand?
Question 1. Which place do you use to compare the numbers 60,618 and 60,647?
Answer:
To compare the numbers 60,618 and 60,647 we see Tens place value as the ten thousand , thousand and hundred place values are the same .
1 tens < 4 tens .
so, 60,618 < 60,647 .

Question 2.
Morocco has a total area of 442,300 square kilometers. Uzbekistan has a total area of 447,400 square kilometers. Use >, <, or = to compare the two areas.
Answer:
Total Area of Morocco = 442,300 square kilometers.
Total Area of Uzbekistan = 447,400 square kilometers.
Step 1 :
As , hundred thousand and ten thousands are same compare from thousands place .
Step 2 :
Compare the thousands place .
442,300
447,400
2 thousands < 7 thousands .
442,300 < 447,400 .

Do You Know How?
For 3-7, complete by writing >, =, or < in each O.
Question 3.
2,643 2,643
Answer:
2,643 = 2,643
Explanation :
All the thousands , hundreds , tens and ones values are the same .
so, 2,643 = 2,643

Question 4.
62,519 64,582
Answer:
62,519 < 64,582
Explanation :
Step 1 :
Write the numbers, lining up places. Begin at the left and compare.
62,519
64,582
The ten thousands digit is the same in both numbers.
Step 2 :
Compare the next digit , the thousands place .
62,519
64,582
2 thousands < 4 thousands .
62,519 < 64,582 .

Question 5.
218,701 118,692
Answer:
218,701 > 118,692
Explanation :
Write the numbers, lining up places. Begin at the left and compare.
218,701
118,692
The Hundred thousands digit
2 hundred thousand > 1 hundred thousand
218,701 > 118,692 .

Question 6.
32,467 32,467
Answer:
32,467 = 32,467
Explanation :
Step 1 :
All the ten thousand , thousand , hundred , tens and ones digits are same
so, 32,467 = 32,467

Question 7.
19,219 19,209
Answer:
19,219 > 19,209
Explanation :
Step 1 :
All the ten thousand , thousand , hundred digits are same so start comparison from tens place .
Step 2  :
Compare ten place
1 ten > 0 ten
19,219 > 19,209 .

Independent Practice

For 8-12, complete by writing >,=, or < in each .
Question 8.
22,873 22,774
Answer:
22,873 > 22,774
Explanation :
Step 1  :
The given numbers contains same ten thousand and thousand places . compare from hundred place .
Step 2  :
Compare the next digit that is hundred places .
22,873
22,774
8 hundreds > 7 hundreds .
22,873 > 22,774

Question 9.
912,706 912,706
Answer:
912,706 = 912,706
Explanation :
Given numbers have same hundred thousands , ten thousands , thousands , hundreds , tens and ones place values same .
so, 912,706 = 912,706

Question 10.
22,240 2,224
Answer:
22,240 > 2,224
Explanation :
The 5 digit number is always greater than the 4 digit number .
So, 22,240 > 2,224

Question 11.
30,000 + 5,000 + 3 300,000 + 5,000
Answer:
30,000 + 5,000 + 3 300,000 + 5,000
35,003 350,000
35,003 < 350,000
Explanation :
The 6 digit number is always greater than the 5 digit number .
so, 35,003 < 350,000

Question 12.
40,000 + 2,000 + 600 + 6 240,000 + 3,000 + 10
Answer:
40,000 + 2,000 + 600 + 6 240,000 + 3,000 + 10
42,606 243,010
42,606 < 243,010
Explanation :
The 6 digit number is always greater than the 5 digit number .
so, 42,606 < 243,010 .

For 13-17, write which place to use when comparing the numbers.

Question 13.
394,284
328,234
Answer:
394,284 > 328,234
Explanation :
Step 1:
Write the numbers, lining up places. Begin at the left and compare.
The hundred thousands digit 3 is the same in both numbers 394,284 and 328,234 .
Step 2 :
Look at the next digit. Compare the ten thousands place value .
394,284
328,234
9 ten thousands > 2 ten thousands .
So, 394,284 > 328,234

Question 14.
6,716
6,714
Answer:
6,716 > 6,714
Explanation :
Step 1:
Write the numbers, lining up places. Begin at the left and compare.
The thousands digit 6 is the same in both numbers 6,716 and 6,714
Step 2 :
Look at the next digit. Compare the hundreds place value .
The hundreds digit 7 is the same in both numbers 6,716 and 6,714
Step 3 :
Look at the next digit. Compare the tens place value .
The tens digit 1 is the same in both numbers 6,716 and 6,714
Step 4 :
Look at the next digit. Compare the Ones place value .
6 Ones > 4 Ones .
So, 6,716 > 6,714

Question 15.
32,916
32,819
Answer:
32,916 > 32,819
Explanation :
Step 1:
Write the numbers, lining up places. Begin at the left and compare.
The ten thousands digit 3 is the same in both numbers 32,916 and 32,819
Step 2 :
Look at the next digit. Compare the hundreds place value .
The thousands digit 2 is the same in both numbers 32,916 and 32,819
Step 3 :
Look at the next digit. Compare the hundreds place value .
9 hundreds > 8 hundreds .
So, 32,916 > 32,819 .

Question 16.
12,217
11,246
Answer:
12,217 > 11,246
Explanation :
Step 1:
Write the numbers, lining up places. Begin at the left and compare.
The ten thousands digit 1 is the same in both numbers 12,217 and 11,246
Step 2 :
Look at the next digit. Compare the thousands place value .
2 thousands > 1 thousands .
So, 12,217 > 11,246 .

Question 17.
812,497
736,881
Answer:
812,497 > 736,881
Explanation :
Step 1:
Write the numbers, lining up places. Begin at the left and compare.
The hundred thousands digits
8 hundred thousands > 7 hundred thousands .
So, 812,497 > 736,881 .

Problem Solving

For 18-19, use the table at the right.

Question 18.
Which genres at Danny’s Books did NOT sell better than Science?
Answer:
Humor Books did not sell better than Science
Explanation :
As 41,843 > 14,843 .

Question 19.
Which genres at Danny’s Books sold better than Biography?
Answer:
Science , Fantasy and Fiction books sold better than Biography
Explanation :
As , 48143 > 42,843 >41,843 > 41,834 .

Question 20.
Celia bought 3 bags of 4 hamburger buns and 3 bags of 8 hot dog buns. How many hamburger and hot dog buns did Celia buy?
Answer:
Number of Hamburgers buns = 3 bags of 4 hamburger buns = 3 × 4 = 12 buns .
Number of hot dogs buns = 3 bags of 8 hot dog buns = 3 × 8 = 24 buns .
Total Number of hamburger and hot dog buns Celia brought = 12 + 24 buns = 36 buns .

Question 21.
Make Sense and Persevere Write three numbers for which you would use the hundreds place to compare to 35,712.
Answer:
35,712.
The Three numbers are 35,246 ; 35,812 and 35,075 .

Question 22.
enVision® STEM
The Illinoian Stage began about 300,000 years ago. The Wolstonian Stage began about 352,000 years ago. Compare 300,000 to 352,000.
Answer:
300,000 < 352,000.
Explanation :
Step 1:
Write the numbers, lining up places. Begin at the left and compare.
The hundred thousands digit 3 is the same in both numbers 300,000 and 352,000 .
Step 2 :
Look at the next digit. Compare the ten thousands place value .
The ten thousands digit
0 ten thousands < 5 ten thousands .
So, 300,000 < 352,000.

Question 23.
An orchard in Maine has 5,287 apple trees. An orchard in Vermont has 5,729 trees. Use <, >, or = to write a comparison between the number of trees in each orchard.
Answer:
Number of apple trees in Maine orchard = 5,287 apple trees
Number of trees in Vermont orchard = 5,729 trees
5,287 < 5,729
The Vermont orchard have more trees than Maine orchard .

Question 24.
In 2010, the population of Alaska was 710,231. Write this number in expanded form, and write the number name.
Answer:
The population of Alaska = 710,231
Expanded Form : 700,000 + 10,000 + 200 + 30 + 1
Number Name  : Seven Hundred ten thousands Two hundred and thirty one .

Question 25.
Higher Order Thinking Explain how you know 437,160 is greater than 43,716.
Answer:
437,160 > 43,716
Explanation :
437,160 is a 6 digit Number
43,716 is a 5 digit Number .
6 digit number is greater than 5 digit number .

Assessment Practice

Question 26.
Is each comparison true or false?

Answer:

Question 27.
Is each comparison true or false?

Answer:

### Lesson 1.4 Round Whole Numbers

Solve & Share
List 7 numbers that round to 300. Use a variety of numbers. Solve this problem any way you choose.
I can … use place value to round numbers.

Look Back! What is the greatest number between 200 and 300 that is closer to 200 than 300? Explain.
Answer :

Essential Question
How Can You Round Numbers?

Visual Learning Bridge
James researched 10 facts about Tallahassee, Florida for an assignment. One of the facts he found was the population in Tallahassee for the year 2017. He chose to round the population on his Florida Facts poster. If James rounded the population in 2017 to the nearest thousand, what was the number James displayed?

Round 181,376 to the nearest thousand. 181,376 is between 181,000 and 182,000.

Find the number 181,376 is closer to.
Mark the halfway point on the number line.

181,376 is to the left of the halfway point.
The poster displayed the population as 181,000.

Convince Me!

Critique Reasoning
Ellie says, “When I round these three numbers, I get the same number for two of them.” Anthony says, “Hmmm, when I round these numbers, I get the same number for all three.” Who is correct? Explain.

Answer :
Both are correct .
Explanation :
Ellie statement is right when we round off all the 3 Numbers to Nearest hundred then
1483 is round off to 1500
1250 is round off to 1300
1454 is round off to 1500
That means two numbers have same round off for 1483 and 1454 numbers .
Antony statement is right when we round off the 3 numbers to Nearest thousands then
1483 is round off to 1000
1250 is round off to 1000
1454 is round off to 1000
that means three numbers have same round off .

Another Example!
You can use place value to round. Round 181,376 to the nearest hundred.

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Guided Practice

Do You Understand?
Question 1.
Explain how to round a number when 3 is the digit to the right of the rounding place.
Answer:
Rounding 1435 to nearest Hundred .

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 2.
What number is halfway between 421,000 and 422,000?
Answer:
A number which is halfway between 421,000 and 422,000 is 421,500 .

Do You Know How?
For 3-8, round each number to the place of the underlined digit.
Question 3.
128,955
Answer:
128,955 – Rounding to nearest thousands .

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 4.
85,639
Answer:
85,639 – Rounding to nearest tens .

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 5.
9,924
Answer:
9,924 – Rounding to nearest thousands .

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 6.
194,524
Answer:
194,524 – Rounding to nearest ten thousands

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 7.
160,656
Answer:
160,656 – Rounding to nearest hundred thousands

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 8.
149,590
Answer:
149,590 – Rounding to nearest hundred thousands

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Independent Practice

For 9-24, use a number line or place value to round each number to the place of the underlined digit.
Question 9.
493,295
Answer:
493,295 – Rounding to nearest ten thousands

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 10.
39,230
Answer:
39,230 – Rounding to nearest ten thousands

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 11.
277,292
Answer:
277,292 – Rounding to nearest hundred thousands

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 12.
54,846
Answer:
54,846 – Rounding to nearest Hundred .

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 13.
4,028
Answer:
4,028 – Rounding to nearest tens .

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 14.
638,365
Answer:

638,365 – Rounding to nearest hundred thousands

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 15.
453,280
Answer:
453,280 – Rounding to nearest thousands .

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 16.
17,909
Answer:
17,909 – Rounding to nearest Hundred .

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 17.
956,000
Answer:

956,000 – Rounding to nearest hundred thousands

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 18.
55,460
Answer:
55,460 – Rounding to nearest thousands .

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 19.
321,679
Answer:
321,679 – Rounding to nearest ten thousands

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 20.
417,547
Answer:
417,547- Rounding to nearest tens .

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 21.
117,821
Answer:
117,821 – Rounding to nearest ten thousands

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 22.
75,254
Answer:
75,254 – Rounding to nearest ten thousands

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 23.
949,999
Answer:
949,999 – Rounding to nearest ten thousands

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 24.
666,821
Answer:
666,821 – Rounding to nearest thousands .

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Problem Solving

Question 25.
For each zoo in the table, round the attendance to the nearest hundred thousand.

Answer:

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

Question 26.
Number Sense Write four numbers that round to 700,000 when rounded to the nearest hundred thousand.
Answer:
The four numbers that round to 700,000 when rounded to the nearest hundred thousand are 702,000 ; 745,000; 699,860 ; 692,480 and 734,955 .

Question 27.
Reasoning
A forest ranger correctly rounded the number of visitors to a park to be 120,000 visitors. Write a number that could be the actual number of visitors if he rounded to the nearest ten thousand.
Answer:
The Number the X is rounded to nearest ten thousand is 120,000 visitors .
the Rounding number is  123,500 .

Question 28.
Amy counted the number of boys and girls at a party. She recorded the results in the tally chart below.

How many more boys than girls were at the party?
Answer:
Number of Girls = 3
Number of Boys = 7
Number of more Boys than Girls at the party = 7 – 3 = 4 boys .

Question 29.
Higher Order Thinking
Liz attended class every day since she started school as a kindergartner. She said she has been in school for about 1,000 days. What numbers could be the actual number of school days if Liz rounded to the nearest ten?
Answer:
Number of days attended to school = 1,000 days
Rounded to the nearest ten 1,000 = 1000 days  .

Assessment Practice

Question 30.
Complete the table. Round each number to the given place.

Answer:

Explanation :

• Find the digit in the rounding place.
• Look at the next digit to the right. 3 becomes a 4. If it is 5 or greater, add 1 to the rounding digit. If it is less than 5, leave the rounding digit alone.
• Change all digits to the right of the rounding place to 0.

### Lesson 1.5 Problem Solving

Construct Arguments

Solve & Share

The land areas of three states are shown in the table. Mickey said Alaska’s land area is about 10 times greater than Georgia’s land area. Explain why Mickey is or is not correct. Construct a math argument to support your answer.
I can … construct arguments using what I know about place-value relationships.

Thinking Habits
Be a good thinker! These questions can help you.

• How can I use numbers, objects, drawings, or actions to justify my argument?
• Am I using numbers and symbols correctly?
• Is my explanation clear and complete?

Question :
Look Back! Construct Arguments Mary said Georgia’s land area is about 10 times greater than Hawaii’s land area. Is Mary correct? Construct a math argument to support your answer.
Answer :
Georgia’s land area = 57,513 square miles .
Hawaii’s land area = 6,423 square miles .
10 times area of Hawaii land is = 6,423 square miles × 10 = 60,423 square miles .
57,513 not equal to 60,423
Georgia’s land area not equal to 10 times area of Hawaii land .
So, the statement made is wrong .

Essential Question
How Can You Construct Arguments?

Visual Learning Bridge
The table shows the retail sales per person in three states. Bella says Arizona had more retail sales per person than Massachusetts.

How can you construct a math argument that supports Bella’s conjecture?
I will use what I know about place value to compare numbers.

How can I construct an argument?
I can

• give an explanation that is clear and complete.
• use numbers and symbols correctly in my explanation.
• use numbers, objects, drawings, or actions to justify my argument.
• use a counterexample in my argument.

Here’s my thinking.
Bella’s statement makes sense.
Start with the greatest place value. The digits are the same in the ten thousands place and in the thousands place. The digits are different in the hundreds place, so that place is compared.
$13,637$13,533
600 > 500
S0, $13,637 >$13,533.
Bella is correct. Arizona had more retail sales per person than Massachusetts.

Convince Me!
Construct Arguments Gayle said Arizona had more retail sales than Massachusetts because 7 > 3, so $13,637 >$13,533. Construct an argument to explain whether or not Gayle is correct.
Answer :
The digits are the same in the ten thousands place and in the thousands place. The digits are different in the hundreds place, so that place is compared.
$13,637$13,533
600 > 500 .
So, 13,637 >13,533
Explanation :
Step 1: Always start from the digits at the highest place value.
Step 2: Compare the digits at this place value in both numbers. The number with the bigger digit is greater. Step 3: If the digits are equal, move one place value to the right 👉, and repeat Step 2.

Guided Practice

Construct Arguments Use the table on the previous page. Jorge said Massachusetts has more retail sales per person than lowa.

Question 1.
What numbers would you use to construct an argument supporting Jorge’s conjecture?
Answer:
Sales in Lowa = $13,172 Sales in Massachusetts =$13,533

Question 2.
How could you support Jorge’s conjecture?
Answer:
Start with the greatest place value. The digits are the same in the ten thousands place and in the thousands place. The digits are different in the hundreds place, so that place is compared.
$13,172$13,533
100 < 500
S0, $13,172 <$13,533.
Jorge is correct. Massachusetts had more retail sales per person than Lowa .

Question 3.
Is Jorge’s conjecture true? Justify your answer.
Answer:
Yes, Jorge’s conjecture is true
Explanation :
Start with the greatest place value. The digits are the same in the ten thousands place and in the thousands place. The digits are different in the hundreds place, so that place is compared.
$13,172$13,533
100 < 500
S0, $13,172 <$13,533.
Jorge is correct. Massachusetts had more retail sales per person than Lowa .

Independent Practice

Construct Arguments
The population of Gerald’s city is three hundred thousand, twenty-seven. Gerald wrote the number as 327,000. Emily lives in a city that has a population of three hundred sixteen thousand, forty-two. Gerald concluded that his city’s population is greater than the population of Emily’s city.
Question 4.
Does Gerald’s explanation make sense? Identify any flaws in Gerald’s thinking
Answer:
Yes,
Population of Gerald’s city = three hundred thousand, twenty-seven = 300,027 .
but Gerald wrote as = 327,000 .

Question 5.
Construct a math argument that explains why Gerald did not write the population of his city correctly.
Answer:
Population of Gerald’s city = three hundred thousand, twenty-seven = 300,027 .
but Gerald wrote as = 327,000 .
Gerald wrote the population in wrongly as she taught it as three hundred and twenty-seven  thousands  .

Question 6.
Correct Gerald’s argument. Explain how to compare the populations of Gerald’s and Emily’s cities.
Answer:
Gerald’s Argument is Wrong .
Population of Gerald’s city = three hundred thousand, twenty-seven = 300,027 .
Population in Emily City = three hundred sixteen thousand, forty-two = 360,042 .
Population in Emily city is greater than Population of Gerald’s city .
Explanation :
Start with the greatest place value. The digits are the same in the hundred thousands place . The digits are different in the tens thousands place, so that place is compared.
300,027
360,042
0 < 60,000
So, 300,027 < 360,042 .
Population in Emily city is greater than Population of Gerald’s city

Problem Solving

Performance Task
Planets
The planets in our solar system are different sizes, as shown below. Nora conjectured that Jupiter’s equator is about 10 times as long as Earth’s equator.

Question 7.
Make Sense and Persevere What information do you have?
Answer:
Length of Earth Equator = 40,030 km
Length of Jupiter Equator =439,264 km
Length of Venus Equator = 38,025 km
Length of Mars Equator = 21,297 km

Question 8.
Be Precise What are possible estimates for the lengths of the equators of Jupiter and Earth?

Answer:
Length of Earth Equator = 40,030 km
Length of Jupiter Equator =439,264 km
length of Jupiter Equator is greater than length of Earth Equator .

Question 9.
Reasoning What is the relationship between the estimates you found for the lengths of the two equators?
Answer:
Length of Earth Equator = 40,030 km
Length of Jupiter Equator =439,264 km
length of Jupiter Equator is greater than length of Earth Equator .

Question 10.
Construct Arguments Construct an argument justifying Nora’s conjecture.
Answer:
No , Nora statement is wrong
Explanation :
Length of Earth Equator = 40,030 km
Length of Jupiter Equator = 439,264 km
Nora conjectured that Jupiter’s equator is about 10 times as long as Earth’s equator.
10 times the Length of Earth Equator = 40,030 km × 10 = 400,300 km .
10 times the Length of Earth Equator is not equal to Length of Jupiter Equator
400,300 # 439,264 kms .
So, Nora statement is wrong

### Topic 1 Fluency Review Activity

Point & Tally
Find a partner. Get paper and pencil. Each partner chooses a different color: Light blue or dark blue.

Partner 1 and Partner 2 each point to a black number at the same time. Both partners multiply those numbers.

If the answer is on your color, you get a tally mark. Work until one partner has twelve tally marks.
I can … multiply within 100.

### Topic 1 Vocabulary Review

Understand Vocabulary
Choose the best term from the box. Write it on the blank.
Word List

• conjecture
• expanded form
• greater than symbol (>)
• Less than symbol (<)
• millions
• period
• place value
• rounding

Question 1.
A group of three digits, separated by commas, starting from the right is called a ___________.
Answer:
A group of three digits, separated by commas, starting from the right is called a period

Question 2.
A process that determines which multiple of 10, 100, 1,000 (and so on) a number is closest to is called __________.
Answer:
A process that determines which multiple of 10, 100, 1,000 (and so on) a number is closest to is calle rounding

Question 3.
A statement that is believed to be true but has not yet been proven is called a ___________.
Answer:
A statement that is believed to be true but has not yet been proven is called a conjecture

Question 4.
The value given to a place a digit has in a number is called its ___________.
Answer:
The value given to a place a digit has in a number is called its place value

Question 5.
In a number, a period of three places to the left of the thousands period is called the __________ period.
Answer:
In a number, a period of three places to the left of the thousands period is called the Millions period.

For each of these terms, give an example and a non-example.
Question 6 – 8.

Answer:

Use Vocabulary in Writing
Question 9.
Describe the value of the 9 in 926,415. Use at least 2 terms from the Word List in your explanation.
Answer:
926,415 – 900,000 – 9 hundred thousands

### Topic 1 Reteaching

Set A pages 5-8

Use a place-value chart to write 301,400.
Expanded form: 300,000 + 1,000 + 400
Number name: three hundred one thousand, four hundred

Remember that each period has hundreds, tens, ones, and the period name.

Write each number in expanded form and using number names.
Question 1.
7,549
Answer:

Expanded form: 7,000 + 500 + 40 + 9
Number name:  Seven thousand, five hundred and forty Nine .

Question 2.
92,065
Answer:

Expanded form: 90,000 + 2,000 + 60 + 5
Number name:  Ninety Two thousand and Sixty Five .

Set B pages 9-12

1,111 = 1,000 + 100 + 10 + 1
As you move left, each 1 × 10 = 10
numeral is 10 times greater 10 × 10 = 100
than the digit on its right. 100 × 10 = 1,000

Remember to use the relationship between the values of the digits.

For 1-2, solve.
Question 1.
How many times greater is the value of the 7 in 70,048 than the value of 7 in 17,992?
Answer:
the value of the 7 in 70,048 = 70,000
the value of 7 in 17,992 = 7,000 .
The value is 7 in 70,048 is 10 times greater than the value of 7 in 17,992  .
7,000 × 10 = 70,000 .
Explanation :
As you move left, each 1 × 10 = 10
numeral is 10 times greater 10 × 10 = 100
than the digit on its right. 100 × 10 = 1,000

Question 2.
Violet has 30 glass tiles. She would like to mosaic tile a tabletop with 10 times that number of tiles. How many tiles does Violet want to use?
Answer:
Number of Violet glass tile = 30
Number of tiles required =10 times that number of tiles = 30 × 10 =  300 titles .

Set C pages 13-16

Use place value to compare 45,423 and 44,897. Start comparing from the left. Look for the first digit that is different.
45,423
44,897
5 > 4
5,000 > 4,000
So, 45,423 > 44,897.

Remember that you can use place value to compare numbers.

Write < or > in the O.
Question 1.
291,846 291,864
Answer:
291,846 <  291,864
Explanation :
Start with the greatest place value. The digits are the same in the hundred thousands ,ten thousands place, thousands place and in the hundred’s place . The digits are different in the Tens place, so that place is compared.
291,846
291,864
40 < 60
So, 291,846 <  291,864 .

Question 2.
66,298 66,298
Answer:
66,298 = 66,298
Explanation :
Both the numbers have same digits in ten thousand, thousand , hundred , tens and ones place.
So, both the numbers are equal .

Question 3.
88,645 87,645
Answer:
88,645 > 87,645
Explanation :
Start with the greatest place value. The digits are the same in ten thousands place  . The digits are different in the thousands place, so that place is compared.
88,645
87,645
8,000 > 7,000
8 thousands > 7 thousands .
So, 88,645 > 87,645

Set D pages 17-20

Round 764,802 to the nearest hundred thousand.

764,802 is to the right of the halfway point.
So, 764,802 rounds to 800,000.

Remember to find the halfway point to help you round.

For 1-4, use number lines or place value to round each number to the place of the underlined digit.
Question 1.
166,742
Answer:
Round 166,742 to the nearest thousand.

166,742 is to the right of the halfway point.
So, 166,742 rounds to 167,000 .

Question 2.
76,532
Answer:
Round 76,532to the nearest thousand.

76,532 is to the right of the halfway point.
So, 76,532 rounds to 77,000 .

Question 3.
5,861
Answer:
Round 5,861 to the nearest thousand.

5,861 is to the right of the halfway point.
So, 5,861 rounds to 6000 .

Question 4.
432,041
Answer:
Round 432,041 to the nearest thousand.

432,041 is to the Left of the halfway point.
So, 432,041 rounds to 432,000 .

Set E pages 21-24

Think about these questions to help you construct arguments.
Thinking Habits

• How can I use numbers, objects, drawings, or actions to justify my argument?
• Am I using numbers and symbols correctly?
• Is my explanation clear and complete?

Remember that you can use math to show why your argument is correct.

According to the 2000 census, the population of a city was 935,426. According to the 2010 census, the population of the same city was 934,578. Taylor says the 2000 population was greater than the 2010 population.
Question 1.
Construct an argument that supports Taylor’s conjecture.
Answer:
Population of the city in 2000 census = 935,426
Population of the city in 2010 census = 934,578
935,426 > 934,578
Taylor statement made is correct .
Explanation :
The digits are the same in the hundred thousands place and in the ten thousands place. The digits are different in the thousands place, so that place is compared.
935,426
934,578
5000 > 4000 .
So, 935,426 > 934,578
Step 1: Always start from the digits at the highest place value.
Step 2: Compare the digits at this place value in both numbers. The number with the bigger digit is greater. Step 3: If the digits are equal, move one place value to the right 👉, and repeat Step 2.

Question 2.
In 1870, the population was seventy-two thousand, five hundred six. Lupita wrote 72,560. Construct a math argument that explains whether Lupita wrote the number correctly.
Answer:
Population in 1870 = seventy-two thousand, five hundred six = 76,506 .
Population in 1870 written by Lupita = 72,560
Lupita wrote the number wrongly as she represented 6 in the tens place like seventy-two thousand, five hundred and sixty .

### Topic 1 Assessment Practice

Question 1.
Choose all the numbers that round to 100,000 when rounded to the nearest hundred thousand.
☐ 9,999
☐ 89,006
☐ 109,999
☐ 119,999
☐ 999,999
Answer:

Question 2.
Which symbol makes the comparison true? Write >,=, or < in the .
111,011 110,111

Answer:
111,011 >110,111
The digits are the same in the hundred thousands place and in the ten thousands place. The digits are different in the thousands place, so that place is compared.
111,011
110,111
1 thousand > 0 thousand
So, 111,011 >110,111
Explanation :
Step 1: Always start from the digits at the highest place value.
Step 2: Compare the digits at this place value in both numbers. The number with the bigger digit is greater. Step 3: If the digits are equal, move one place value to the right 👉, and repeat Step 2.

Question 3.
Write three numbers that round to 40,000 when rounded to the nearest ten thousand.
Answer:
Three numbers that round to 40,000 when rounded to the nearest ten thousand are 38,990 ; 42,640 and 39,999 .

Question 4.
John wrote the numbers 678,901 and 67,890. How many times greater is the value of 7 in 678,901 than the value of 7 in 67,890?
A. 10,000
B. 1,000
C. 100
D. 10
Answer:
Option D
Explanation :
The place value of 7 in 678,901  = 70,000 .
The place value of 7 in 67,890  = 7,000 .
7,000 × 10 = 70,000 .
The place value of 7 in 678,901 is 10 times greater than The place value of 7 in 67,890 .

Question 5.
Look at the numbers in the table.

Which number has one digit that represents ten times the value of the digit to its right? Explain.
Answer:
375,595  has the one digit that represents ten times the value of the digit to its right .
Explanation :
The place value of 375,595 is 5,000
The place value of 375,595 is 500
The place value of 375,595 is 5,000 which is ten times the value of the digit to its right that is The place value of 375,595 is 500 .

Question 6
Write the number for 160,060 in expanded form and using number names.
Answer:
160,060
Expanded form – 100,000 + 60,000 + 60

Question 7.
A. For each number, give the whole number that represents the value of the underlined digit. Write your answers in the boxes.

Answer:

Question :
B. Look at your answers in Part A. In which number is the value of the underlined digit 10 times the value of the digit to the right of it?
A. 155,349
B. 651,907
C. 947,502
D. 317,055
Answer:
Option D .

Question 8.
Rhode Island has about three hundred fifty-six thousand acres of forested land. What is this number in standard form rounded to the nearest ten thousand?
A. 350,000
B. 400,000
C. 360,000
D. 356,000
Answer:
Rhode Island Area = three hundred fifty-six thousand = 356,000
356,000 rounded to the nearest ten thousand = 360,000 .

Question 9.
Which one of the following comparisons is correct?
A. 65,215 > 65,512
B. 292,200 < 229,200
C. 890,242 < 890,224
D. 101,111 < 111,111
Answer:
Comparisons which are correct is Option D .

Question 10.
Write <, =, or > to complete a true comparison for each pair of numbers.
72,013 _______ 72,103
87,210 _______ 87,210
126,999 _______ 152,999
400,602 _______ 400,062
147,634 _______ 146,734
Answer:
72,013 < 72,103
87,210 = 87,210
126,999 < 152,999
400,602 > 400,062
147,634 > 146,734

Question 11.
The table shows the areas of four states.

A. Which of the 4 states has the least area? the greatest area? Write the number name for the area of each of these states.
Answer:
Kansas have Least Area= 82,278 square miles .
Montana have Greatest Area = 147,042 square miles .
82,278 = Eighty two thousand two hundred and seventy eight .
147,042 = one hundred and forty seven thousand and forty two .

B. Draw a place-value chart. Record Kansas’s area. Explain how the value of the 2 in the thousands place compares with the value of the 2 in the hundreds place.
Answer:

The value of 2 in one thousands place = 2,000 .
The value of 2 in hundreds place = 200 .
2,000 = 200 × 10
The value of 2 in one thousands place is ten times greater than The value of 2 in hundreds place .

### Topic 1 Performance Task

Video Games
Tanji, Arun, and Juanita are playing a video game with 3 levels. The opportunity to earn points increases as the levels of the game increase. To keep track of their progress, Tanji, Arun, and Juanita record and examine their scores at each level.
Question 1.
Use the Level 1 table to answer the following questions.

Part A
Question :
Tanji noticed he was the only player with 3s in his Level 1 score. What are the values of the 3s in Tanji’s score?
Answer:
Tanji’s Score = 4,337
the values of the 3s in tanji score = 300 and 30 .

Part B
Question :
Arun noticed the 5s in his score were next to each other. Describe the relationship between the 5s in Arun’s score.
Answer:
Arun’s score = 5,519
the values of the 5 s in Arun’s score =5,000 and 500 .
5,000 is ten times the 500 .
the value of one 5 in his score is ten times greater than the value of the other 5

Part C
Question :
Juanita says the value of one 8 in her score is ten times greater than the value of the other 8. Construct an argument and draw a place-value chart to determine if Juanita is correct.
Answer:
Juanita’s Score = 2,868

the value of one 8 in her score is ten times greater than the value of the other 8 is wrong because
the value of one 8 in her score is hundred times greater than the value of the other 8 .
The value of 8 in 2,868 is 800 .
The value of 8 in 2,868 is 8 .
800 = 8 × 100 .

Question 2.
Use the Level 2 table to answer the following questions.

Part A – Question
Juanita had the greatest score at Level 2, followed by Tanji and Arun. Write each player’s score in expanded form to compare each score by place value.
Answer:
Juanita’s Score = 60,114 = 60,000 + 100 + 10 + 4 .
Arun’s Score = 39,207 = 30,000 + 9,000 + 200 + 7
Tanji’s Score = 56,899 = 50,000 + 6,000 + 800 + 90 + 9

Part B – Question
Write each player’s score using number names.
Answer:
Juanita’s Score = 60,114 = Sixty thousand One hundred and fourteen .
Arun’s Score = 39,207 = Thirty nine thousand two hundred and seven .
Tanji’s Score = 56,899 = Fifty Six thousand eight hundred and ninety nine .

Part C – Question
Use >, =, or < to write comparisons between the Level 2 scores.
Answer:
Juanita’s Score = 60,114
Arun’s Score = 39,207
Tanji’s Score = 56,899
60,114 > 56,899  > 39,207 .

Part D – Question
Arun noticed his Level 2 score has a greater value in the thousands place than Tanji’s and Juanita’s Level 2 scores. Round Arun’s score to the nearest thousand.
Answer:
Arun’s Score = 39,207
Rounded to Nearest thousand is
39,207 = 39,000.

## Envision Math Common Core 8th Grade Answers Key Topic 7 Understand And Apply The Pythagorean Theorem

Topic Essential Question
How can you use the Pythagorean Theorem to solve problems?
Answer:
The Pythagorean Theorem is used to calculate the steepness of slopes of hills or mountains. A surveyor looks through a telescope toward a measuring stick a fixed distance away, so that the telescope’s line of sight and the measuring stick form a right angle.

3-ACT MATH OOO

Go with the Flow
You may have noticed that when you double the base and the height of a triangle, the area is more than doubled. The same is true for doubling the sides of a square or the radius of a circle. So what is the relationship? Think about this during the 3-Act Mathematical Modeling lesson.

### Topic 7 enVision STEM Project

Did You Know?
Over two billion people will face water shortages by 2050 according to a 2015 United Nations Environment Program report.

Rainwater can be collected and stored for use in irrigation, industrial uses, flushing toilets, washing clothes and cars, or it can be purified for use as everyday drinking water.
This alternative water source reduces the use of fresh water from reservoirs and wells.

Using water wisely saves money on water and energy bills and extends the life of supply and wastewater facilities.

Roofs of buildings or large tarps are used to collect rainwater.
A rainwater collection system for a building roof that measures 28 feet by 40 feet can provide 700 gallons of water-enough water to support two people for a year—from a rainfall of 1.0 inch.

Even a 5 foot by 7-foot tarp can collect 2 gallons of water from a rainfall total of only 0.1 in.

The rainwater harvesting market is expected to grow 5% from 2016 to 2020.

Your Task: Rainy Days

Rainwater collection is an inexpensive way to save water in areas where it is scarce. One inch of rain falling on a square roof with an area of 100 ft² collects 62 gallons of water that weighs over 500 pounds. You and your classmates will research the necessary components of a rainwater collection system. Then you will use what you know about right triangles to design a slanted roof system that will be used to collect rainwater.
Answer:
It is given that
Rainwater collection is an inexpensive way to save water in areas where it is scarce. One inch of rain falling on a square roof with an area of 100 ft² collects 62 gallons of water that weighs over 500 pounds
Now,
The necessary components of a rainwater collection system are:
A) Catchments B) Coarse mesh C) Gutters D) Conduits E) First-flushing F) Filter G) Storage facility H) Recharge Structures

### Topic 7 GET READY!

Review What You Know!

Vocabulary
Choose the best term from the box to complete each definition.
cube root
diagonal
isosceles triangle
perimeter
right triangle
square root

Question 1.
The __________ of a number is a factor that when multiplied by itself gives the number.
Answer:
We know that,
The “Square root” of a number is a factor that when multiplied by itself gives the number
Hence, from the above,
We can conclude that the best term to complete the given definition is a “Square root”

Question 2.
A _________ is a line segment that connects two vertices of a polygon and is not the side.
Answer:
We know that,
A “Diagonal” is a line segment that connects two vertices of a polygon and is not the side
Hence, from the above,
We can conclude that the best term to complete the given definition is a “Diagonal”

Question 3.
The _________ of a figure is the distance around it.
Answer:
We know that,
The “Perimeter” of a figure is the distance around it
Hence, from the above,
We can conclude that the best term to complete the given definition is the “Perimeter”

Question 4.
A ___________ is a triangle with one right angle.
Answer:
We know that,
A “Right triangle” is a triangle with one right angle
Hence, from the above,
We can conclude that the best term to complete the given definition is a “Right angle”

Simplify Expressions with Exponents

Simplify the expression.
Question 5.
32 + 42
Answer:
The given expression is: 32 + 42
So,
32 + 42
= (3 × 3) + (4 × 4)
= 9 + 16
= 25

Question 6.
22 + 52
Answer:
The given expression is: 22 + 52
So,
2² + 52
= (2 × 2) + (5 × 5)
= 4 + 25
= 29

Question 7.
102 – 82
Answer:
The given expression is: 102 – 82
So,
102 – 82
= (10 × 10) – (8 × 8)
= 100 – 64
= 36

Square Roots

Determine the square root.
Question 8.
$$\sqrt {81}$$
Answer:
The given expression is: $$\sqrt{81}$$
Hence,
$$\sqrt{81}$$ = 9

Question 9.
$$\sqrt {144}$$
Answer:
The given expression is: $$\sqrt{144}$$
Hence,
$$\sqrt{144}$$ = 12

Question 10.
$$\sqrt {225}$$
Answer:
The given expression is: $$\sqrt{225}$$
Hence,
$$\sqrt{225}$$ = 15

Distance on a Coordinate Plane

Determine the distance between the two points.
Question 11.

Answer:
The given graph is:

From the given graph,
The given points are: (2, 5), (7, 5)
Now,
Compare the given points with (x1, y1), (x2, y2)
We know that,
Distance between 2 points = √(x2 – x1)2 + (y2 – y1)2
= √(7 – 2)2 + (5 – 5)2
= $$\sqrt{5²}$$
= 5 units
Hence, from the above,
We can conclude that the distance between the given points is: 5 units

Question 12.

Answer:
The given graph is:

From the given graph,
The given points are: (3, 2), (3, 9)
Now,
Compare the given points with (x1, y1), (x2, y2)
We know that,
Distance between 2 points =√(x2 – x1)2 + (y2 – y1)2
= √(3 – 3)2 + (9 – 2)2
= $$\sqrt{7²}$$
= 7 units
Hence, from the above,
We can conclude that the distance between the given points is: 7 units

Language Development

Complete the word map using key terms, examples, or illustrations related to the Pythagorean Theorem and its Converse.

Answer:

### Topic 7 PICK A PROJECT

PROJECT 7A
Where would you like to bike ride in your neighborhood?
PROJECT: PLAN A METRIC CENTURY RIDE

PROJECT 7B
What designs have you seen on kites?
PROJECT: BUILD A KITE

PROJECT 7C
What buildings in your community have unusual shapes as part of their structure or design?
PROJECT: MAKE A SCRAPBOOK

PROJECT 7D
What geometric designs have you noticed on your clothes?
PROJECT: DESIGN A FABRIC TEMPLATE