## Envision Math Common Core 7th Grade Answers Key Topic 1 Rational Number Operations

### Topic 1 Essential Question

How can the properties of operations be used to solve problems involving integers and rational numbers?

3-ACT MATH

Win Some, Lose Some Are you the kind of person who has a lot of knowledge about history, literature, or science? What about pop culture, music, sports, and current events? Some schools have an academic bowl team that competes in tournaments against other schools. The teams are made up of members with strengths in different subject areas.
In any quiz competition, it’s important to understand the rules and scoring. Think about this during the 3-Act Mathematical Modeling lesson.

### Topic 1 enVision STEM Project

Did You Know?
The lowest recorded temperature in the world, -136°F (-93.2°C), occurred in Antarctica.

The highest recorded temperature in the world, 134°F (56.7°C), occurred in Death Valley, California.

The Celsius scale (°C) is commonly used for temperature measurement in most of the world.
Only a small number of nations, including the United States, regularly use the Fahrenheit scale (°F).

Windchill, based on the rate of heat loss from exposed skin, can make it feel colder outside than the actual air temperature indicates. Wind chills in some places of the world can dip into the – 100°F range.

There are many regions of the world with cold temperatures and extreme conditions. How do the inhabitants of these regions adapt and thrive? Do conditions exist that make regions too cold for human living? You and your classmates will explore and describe the habitability of regions with low temperatures.

Review What You Know!

Vocabulary
Choose the best term from the box. Write it on the blank.

• absolute value
• Associative Property
• Commutative Property
• Distributive Property
• integers
• rational number

Question 1.
The __________ explains why a × b = b × a and a + b = b + a.

The commutative property explains why a x b = b x a and a + b = b + a.

Explanation:
In the above-given question,
given that,
The commutative property explains why a x b = b x a and a + b = b + a.
the commutative property states that the numbers on which we operate can be moved or swapped from their position without making any difference to the answer.
the property holds for addition and multiplication, but not for subtraction and division.
for example:
4 + 3 = 7.
3 + 4 = 7.
2 x 6 = 12.
6 x 2 = 12.

Question 2.
The __________ of -6 is 6, because it is 6 units from zero on the number line.

The absolute value of -6 is 6, because it is 6 units from zero on the number line.

Explanation:
In the above-given question,
given that,
The absolute value of -6 is 6, because it is 6 units from zero on the number line.
for example:
6 and -6 are at the same distance from zero on the number line.

Question 3.
The number $$\frac{5}{3}$$ is a _________ because 5 and 3 are integers and 3 ≠ 0.

The number 5/3 is a rational number because 5 and 3 are integers and 3 is not equal to 0.

Explanation:
In the above-given question,
given that,
The number 5/3 is a rational number because 5 and 3 are integers and 3 is not equal to 0.
for example:
3, 5, 6, and 7 are rational numbers.
rational numbers are also integers.

Question 4.
The set of _________ consists of the counting numbers, their opposites, and zero.

The set of integers consists of the counting numbers, their opposites, and zero.

Explanation:
In the above-given question,
given that,
The set of integers consists of the counting numbers, their opposites, and zero.
Integers are counting numbers, their opposites, and zero.
for example:
integers can either be negative, positive, or zero.
1, -1, 2, -2, and 3, -3.

Question 5.
The sum of (a + b) + c is equal to the sum of a + (b + c) as explained by the _________

The sum of (a + b) + c is equal to the sum of a + (b + c) as explained by the distributive property.

Explanation:
In the above-given question,
given that,
The sum of (a + b) + c is equal to the sum of a + (b + c) as explained by the distributive property.
the distributive property is sometimes called the distributive law of multiplication and division.
for example:
4(x – 3) = 20.
4x – 12 = 20.
4x – 12 + 12 = 20 + 12.
4x = 32.
x = 32/4.
x = 8.

Question 6.
If you evaluate n × (v + z) by writing it as (n × y) + (n × z), you have used the __________.

If you evaluate n x (v + z) by writing it as (n x y) + (n x z), you have used the distributive property.

Explanation:
In the above-given question,
given that,
If you evaluate n x (v + z) by writing it as (n x y) + (n x z), you have used the distributive property.
for example:
4(x – 3) = 20.
4x – 12 = 20.
4x – 12 + 12 = 20 + 12.
4x = 32.
x = 32/4.
x = 8.

Add and Subtract Fractions and Decimals

Question 7.
$$2 \frac{1}{3}+6 \frac{3}{5}$$

2(1/3) + 6(3/5) = 8.9.

Explanation:
In the above-given question,
given that,
2(1/3) + 6(3/5).
7/3 + 33/5.
2.3 + 6.6.
8.9.
2(1/3) + 6(3/5) = 8.9.

Question 8.
$$9 \frac{1}{10}-4 \frac{3}{4}$$

9(1/10) – 4(3/4) = 4.35.

Explanation:
In the above-given question,
given that,
9(1/10) – 4(3/4).
91/10 – 19/4.
9.1 – 4.75.
4.35.
9(1/10) – 4(3/4) = 4.35.

Question 9.
19.86 + 7.091

19.86 + 7.091 = 26.951.

Explanation:
In the above-given question,
given that,
19.86 + 7.091.
19.86 + 7.091 = 26.951.

Question 10.
57 – 10.62

57 – 10.62 = 46.38.

Explanation:
In the above-given question,
given that,
subtract the numbers.
57 – 10.62.
46.38.
57 – 10.62 = 46.38.

Multiply and Divide Fractions and Decimals

Multiply or divide.
Question 11.
4.08 × 29.7

4.08 x 29.7 = 121.176.

Explanation:
In the above-given question,
given that,
multiply the numbers.
4.08 x 29.7 = 121.176.

Question 12.
15,183.3 ÷ 473

15,183.3 / 473 = 32.1.

Explanation:
In the above-given question,
given that,
divide the numbers.
15,183.3 / 473 = 32.1.

Question 13.
$$\frac{15}{16} \times 9 \frac{1}{5}$$

15/16 x 41/5 = 7.6875.

Explanation:
In the above-given question,
given that,
multiply the numbers.
15/16 x 41/5.
0.9375 x 8.2.
7.6875.
15/16 x 41/5 = 7.6875.

Question 14.
$$4 \frac{7}{9} \div 1 \frac{7}{12}$$

4(7/9) / 1(7/12) = 2.974.

Explanation:
In the above-given question,
given that,
divide the numbers.
4(7/9) / 1(7/12).
43/9 / 19/12.
4.7 / 1.58.
2.974.
4(7/9) / 1(7/12) = 2.974.

Question 15.
Byron has $$1 \frac{7}{10}$$ kilograms of black pepper. He uses of the pepper and splits it between 7 pepper shakers. How much pepper will be in each shaker?
A. $$\frac{119}{80} \mathrm{~kg}$$
B. $$\frac{1}{8} \mathrm{~kg}$$
C. 1.4125 kg
D. $$\frac{17}{80} \mathrm{~kg}$$

The quantity of pepper will be in each shaker = 1/8.

Explanation:
In the above-given question,
given that,
Byron has $$1 \frac{7}{10}$$ kilograms of black pepper.
He uses the pepper and splits it between 7 pepper shakers.
1(7/10).
17/10 = 1.7.

Language Development
Fill in the word map with new terms, definitions, and supporting examples or illustrations.

Distributive property.
commutative property.
Integer.
Rational number.

Explanation:
In the above-given question,
given that,
The commutative property explains why a x b = b x a and a + b = b + a.
the commutative property states that the numbers on which we operate can be moved or swapped from their position without making any difference to the answer.
the property holds for addition and multiplication, but not for subtraction and division.
for example:
4 + 3 = 7.
3 + 4 = 7.
2 x 6 = 12.
6 x 2 = 12.
The number 5/3 is a rational number because 5 and 3 are integers and 3 is not equal to 0.
for example:
3, 5, 6, and 7 are rational numbers.
rational numbers are also integers.
The set of integers consists of the counting numbers, their opposites, and zero.
Integers are counting numbers, their opposites, and zero.
for example:
integers can either be negative, positive, or zero.
1, -1, 2, -2, and 3, -3.

Pick A Project

PROJECT 1A
What is something you can make?

PROJECT 1B
How old were you when petroglyphs were being painted?
PROJECT: MAKE A TIMELINE

PROJECT 1C
What makes an obstacle course fun?
PROJECT: BUILD A MODEL OF AN OBSTACLE COURSE

PROJECT 1D
What are your favorite ways to exercise ?
PROJECT: FILM AN EXERCISE VIDEO

### Lesson 1.1 Relate Integers and Their Opposites

Solve & Discuss It!
When preparing for a rocket launch, the mission control center uses the phrase “T minus” before liftoff.
…T minus 3, T minus 2, T minus 1, …
After the rocket has launched, “T plus” is used while the rocket is in flight.
…T plus 1, T plus 2, T plus 3, …
When does the rocket launch? What could “T” represent?

I can… relate integers, their opposites, and their absolute values.

Reasoning
What integers can you use to represent this situation?

Focus on math practices
Reasoning How are “T minus 4” and “T plus 4” related?

The situation is T plus.

Explanation:
In the above-given question,
given that,
the rocket is launching.
so we have to use the “T plus”.
“T plus 4”.

Essential Question
How are integers and their opposites related?

Try It!

Xavier climbs 9 feet up into an apple tree. What integer represents the direction and how far he will climb to get back down to the ground? What does the integer 0 represent in this situation?
The integer ________ represents Xavier’s climb down.
The integer 0 represents __________.

The integer -9 represents Xavier’s climb down.
the integer 0 represents that he did not move.

Explanation:
In the above-given question,
given that,
Xavier climbs 9 feet up into an apple tree.
Xavier climbs feet down into an apple tree is -9.
the integer 0 represents that he did not move.
so the integer 0 is related to +0ft or -0ft.

Convince Me! How are the absolute values of opposite integers related?

Try It!

The temperature was 75°. At noon, the temperature increased 7°. By evening, the temperature decreased by 7°. How did the temperature change?

The temperature change by 75° + 7 and 75° – 7.

Explanation:
In the above-given question,
given that,
The temperature was 75°. At noon, the temperature increased 7°
75° + 7.
increased means plus.
decreased means minus.
the temperature change by 75° + 7 and 75° – 7.

Try It!

Shaniqua has $45 in her wallet. She spends$4 on snacks and $8 on a movie ticket. What integer represents the change in the amount of money in Shaniqua’s wallet? How much money does she have left? Answer: The money does she have left =$33.

Explanation:
In the above-given question,
given that,
Shaniqua has $45 in her wallet. She spends$4 on snacks and $8 on a movie ticket.$45 – $4 =$41.
$41 –$8 = $33. we are using the minus operator. so the money does she have left =$33.

KEY CONCEPT
An integer, n, and its opposite, -n, combine to make 0.

Do You Understand?
Question 1.
Essential Question How are integers and their opposites related?

The integers and their opposites are related because an integer’s opposite has to be the same number away from zero as the integer in question.

Explanation:
In the above-given question,
given that,
The integers and their opposites are related because an integer’s opposite has to be the same number away from zero as the integer in question.
for example:
67 is the integer and its opposite side is -67.
its opposite has to be on the other side of zero and the same amount of numbers away from zero.

Question 2.
Reasoning In order for an atom to have a zero charge, every proton, which has a charge of +1, must be matched with an electron, which has a charge of -1. A helium atom has 2 protons and 2 electrons. Explain why a helium atom has a zero charge.

Helium atom has integers are +2 and -2.

Explanation:
In the above-given question,
atom has a zero charge, every proton, which has a charge of +1, must be matched with an electron, which has a charge of -1.
A helium atom has 2 protons and 2 electrons.
+2, and -2.
so helium atom has integers that are +2 and -2.

Question 3.
Model with Math Explain how to use a number line to show that opposite quantities combine to make 0.

The number line shows the + 3 and – 3.

Explanation:
In the above-given question,
given that,
there are some numbers on the number line.
-3, 0, and 3.
so the number line shows the integers +3 and -3.

Do You Know How?
Question 4.
Marcus dives from the surface of the ocean to a reef 18 meters below sea level. What integer represents Marcus’s location relative to the surface? How far does Marcus have to go to return to the surface?

The Marcus have to go to return to the surface = -18 meters.

Explanation:
In the above-given question,
given that,
Marcus dives from the surface of the ocean to a reef 18 meters below sea level.
-18 represents Marcus’s location relative to the surface.
-18, 0, and 18.
so the Marcus has to go to return to the surface = -18 meters.

Question 5.
The temperature of the water in Emily’s fish tank was 78°F on Sunday. The water temperature changed by -3° on Monday, and then by 3o on Tuesday. What integer represents the temperature change of the water from Sunday to Tuesday? What was the water temperature on Tuesday?

The temperature change of the water from Sunday to Tuesday = plus.
The water temperature on Tuesday = 105°F.

Explanation:
In the above-given question,
given that,
The temperature of the water in Emily’s fish tank was 78°F on Sunday.
The water temperature changed by -3° on Monday.
75°F.
and then by 3o on Tuesday.
105°F.
the temperature change of the water from Sunday to Tuesday = plus.
the water temperature on Tuesday = 105°F.

Question 6.
The scores of players on a golf team are shown in the table. The team’s combined score was 0. What was Travis’s score?

Travis’s score = -4.

Explanation:
In the above-given question,
given that,
The scores of players on a golf team are shown in the table.
The team’s combined score was 0.
Travis’s score is -4.

Practice & Problem Solving

Leveled Practice In 7-9, write the integer that represents the situation.
Question 7.
Max spent $53 and now has no money left. He had$ _______ before his purchase.

He had $53 before his purchase. Explanation: In the above-given question, given that, Max spent$53 and now has no money left.
$53 –$53 = 0.
so he had $53 before his purchase. Question 8. The temperature was 8°F. It dropped so that the temperature was 0°F. ________ °F represents the change in temperature. Answer: The change in temperature represents = -8°F. Explanation: In the above-given question, given that, The temperature was 8°F. It dropped so that the temperature was 0°F. 8 – 0 = 8. so the change in the temperature represents = -8°F. Question 9. An airplane descended 4,000 feet before landing. The integer that represents how many feet the airplane was above the ground before its descent is _________. Answer: The number of feet the airplane was above the ground before its descent = -4000. Explanation: In the above-given question, given that, An airplane descended 4,000 feet before landing. before landing = 4000 meters. after landing = -4000 m. so the number of feet the airplane was above the ground before its descent = -4000 m. Question 10. Carolyn says that point A and point B represent opposite integers. a. What is the opposite of the integer represented by point A? By point B? Answer: The opposite of the integer represented by point A is -7 and by point, B is 8. Explanation: In the above-given question, given that, Carolyn says that point A and point B represent opposite integers. in the number line, the numbers are -10, -5, 0, 5, and 10. so the missing numbers are points A and B. The opposite of the integer represented by point A is -7 and by point, B is 8. b. Construct Arguments Do you agree with Carolyn? Explain. Answer: Yes, Arguments agree with Carolyn. Explanation: In the above-given question, given that, Carolyn says that point A and point B represent opposite integers. in the number line, the numbers are -10, -5, 0, 5, and 10. so the missing numbers are points A and B. The opposite of the integer represented by point A is -7 and by point, B is 8. Question 11. A football team lost 9 yards during a play. The team had a combined gain or loss of 0 yards after the next play. What integer represents the yards gained or lost on the next play? Show this on the number line. Answer: The integer represents the yards gained = 9 yards. Explanation: In the above-given question, given that, A football team lost 9 yards during a play. The team had a combined gain or loss of 0 yards after the next play. the integer represents the yards gained = 9 yards. -9, 0, and 9. so the integer represents the yards gained = 9 yards. Question 12. A roller coaster car goes above and below ground. Use the number line to show its changes in height. What is the height of the car at the end of the ride? Answer: The height of the car at the end of the ride = 5 m. Explanation: In the above-given question, given that, A roller coaster car goes above and below ground. The roller coaster starts at 1 meter. drops 4 meters down. rises 13 meters up. drops 6 meters. 13 – 6 = 7. so the height of the car at the end of the ride = 5 m. Question 13. Dimitri is buying a car. He chooses Option 1 to add a new sound system to his car. What integer represents the change from the base price of the car to its final price? Answer: The integer represents the change from the base price of the car to its final price =$1400.

Explanation:
In the above-given question,
given that,
He chooses Option 1 to add a new sound system to his car.
the base price of the car = -$700. option 1 =$1400.
so the integer represents the change from the base price of the car to its final price = $1400. Question 14. Make Sense and Persevere What values do x and y have if |x| = 16, |y| = 16, and when x and y are combined they equal 0? Explain your reasoning. Answer: x and y are equal to 0. Explanation: In the above-given question, given that, if |x| = 16. |y| = -16. 16 – 16 = 0. so x and y are equal to 0. Question 15. Write a situation that can be represented by the opposite of -42. Answer: The situation that can be represented by the opposite of -42 is 42. Explanation: In the above-given question, given that, the situation that can be represented by the opposite of -42 is 42. 42 is the positive integer. -42 is the negative integer. so the situation that can be represented by the opposite 0f -42 is 42. Question 16. Higher Order Thinking Three friends all live on the same street that runs west to east. Beth lives 5 blocks from Ann. Carl lives 2 blocks from Beth. If the street is represented by a number line and Ann’s house is located at 0, what are the possible locations for Carl’s house? Assume that each unit on the number line represents 1 block. Answer: The possible locations for Carl’s house = 3 and 5. Explanation: In the above-given question, given that, Three friends all live on the same street that runs west to east. Beth lives 5 blocks from Ann. Carl lives 2 blocks from Beth. 5 – 2 = 3. so the location of Ann is 0. the location of Carl is 3. the location of Beth is 5. so the possible locations for Carl’s house = 3 and 5. Assessment Practice Question 17. Which of these situations can be represented with an integer that when combined with -9 makes 0? Select all that apply. ☐ You walk down 9 flights of stairs. ☐ You climb up 9 flights of stairs. ☐ The temperature drops 9°F. ☐ You spend$9 on a book.
☐ You earn $9 from your job. Answer: You climb up 9 flights of stairs, you spend$9 on a book, and you earn $9 from your job. Explanation: In the above-given question, given that, an integer that when combined with -9 makes 0. You climb up 9 flights of stairs. you spend$9 on a book.
you earn $9 from your job. Question 18. Which of these situations can be represented by the opposite of 80? Select all that apply. ☐ An airplane descends 80 m. ☐ An elevator ascends 80 m. ☐ The cost of a train ticket drops by$80.
☐ You remove 80 songs from an MP3 player.
☐ Suzy’s grandmother is 80 years old.

An airplane descends 80m, the cost of a train ticket drops by $80, and you remove 80 songs from an MP3 player. Explanation: In the above-given question, given that, the situations can be represented by the opposite of 80. An airplane descends 80m = -80m. the cost of a train ticket drops by$80 = -$80. You remove 80 songs from an MP3 player = -80. ## Lesson 1.2 Understand Rational Numbers Solve & Discuss It! Calvin wants to customize his surfboard so that it is wider than the 82 model but narrower than the 92 model. What measurement could be the width of his surfboard? Explain. I can… recognize rational numbers and write them in decimal form. Answer: The width of his surfboard = 1 wide. Explanation: In the above-given question, given that, Calvin wants to customize his surfboard so that it is wider than the 82 models but narrower than the 92 models. 82 model has a wide of 221/2 and 3(1/4) of thick. 92 model has a wide of 23(1/4) and 3(1/2) thick. 23 – 22 = 1. so the width of his surfboard = 1 wide. Focus on math practices Use Structure Lindy’s surfboard is 23 inches wide. Between which two surfboard models is her custom surfboard’s width? How do you know? Answer: The two surfboard models are her custom surfboard’s width = surfboards 82 and 92. Explanation: In the above-given question, given that, Lindy’s surfboard is 23 inches wide. Between which, two surfboard models are her custom surfboard’s width. so the two surfboard models is her custom surfboard’s width = surfboards 82 and 92. Essential Question How are rational numbers written as decimals? Try It! In the next several games, the pitcher threw a total of 384 pitches and used a fastball 240 times. What decimal should Juanita use to update her report? Juanita should use the decimal ________ to update her report. Answer: Juanita should use the decimal 0 to update her report. Explanation: In the above-given question, given that, the pitcher threw a total of 384 pitches and used a fastball 240 times. 380 / 240. 380 x 6 = 2304. 380 x 2 = 768. 380 x 5 = 1920. 1920 – 1920 = 0. Convince Me! How do you know that the answer is a terminating decimal? Try It! What is the decimal form of $$\frac{100}{3}, \frac{100}{5}$$ and $$\frac{100}{6}$$? Determine whether each decimal repeats or terminates. Answer: The decimal form of 100/3 = 33.3. the decimal form of 100/5 = 20. the decimal form of 100/6 = 16.6. Explanation: In the above-given question, given that, the decimal form of 100/3, 100/5, and 100/6. 100/3 = 33.3. 100/5 = 20. 100/6 = 16.6. the decimal 100/3 and 100/6 repeats. the decimal 100/5 terminates. Try It! Is -0.3 a rational number? Is 3.14144144414444… a rational number? Explain your reasoning. Answer: Yes, both are rational numbers. Explanation: In the above-given question, given that, the numbers are -0.3 and 3.14144144414444… -0.3 is a rational number. 3.14144144414444 is a rational number. KEY CONCEPT To convert from the fraction form of a rational number to its decimal form, divide the numerator by the denominator. The decimal form of a rational number either terminates in 0s or eventually repeats. Do You Understand? Question 1. Essential Question How are rational numbers written as decimals? Answer: Rational numbers can be written as decimals in two ways. they are terminating decimal and repeating decimal. Explanation: In the above-given question, given that, the rational numbers can be written as a decimal in two ways. for example: 100/5 = 20. 20 is a terminating decimal. 100/3 = 33.3. 33.3 is a repeating decimal. Question 2. Reasoning How can you use division to find the decimal equivalent of a rational number? Answer: 100/3 = 33.3. Explanation: In the above-given question, given that, 100/3 = 33.3. 33.3 is a terminating decimal. 100/5 = 20. 20 is a repeating decimal. Question 3. Be Precise What is the difference between a terminating decimal and a repeating decimal? Answer: The difference is one decimal will stop and one decimal will continue. Explanation: In the above-given question, given that, the two terms are terminating and repeating decimals. 100/3 = 33.3. it is a repeating decimal. 100/5 = 20. 20 is a terminating decimal. Do You Know How? Question 4. What is the decimal equivalent of each rational number? a. $$\frac{7}{20}$$ Answer: 7/20 = 0.35. Explanation: In the above-given question, given that, the number is 7/20. 7/20 = 0.35. 0.35 is a terminating decimal. b. –$$\frac{23}{20}$$ Answer: -23/20 = -1.15. Explanation: In the above-given question, given that, the number is -23/20. -23/20 = -1.15. it is a terminating decimal. c. $$\frac{1}{18}$$ Answer: 1/18 = 0.055. Explanation: In the above-given question, given that, the number is 1/18. 1/18 = 0.055. 0.055 is a repeating decimal. d. –$$\frac{60}{22}$$ Answer: -60/22 = 2.727 is a repeating decimal. Explanation: In the above-given question, given that, the number is -60/22. -60/22 = 2.727. 2.727 is a repeating decimal. Question 5. There are 5,280 feet in a mile. What part of a mile, in decimal form, will you drive until you reach the exit? Answer: 5280/1000 = 5.28 miles. Explanation: In the above-given question, given that, There are 5,280 feet in a mile. 5280/1000 = 5.28. so the part of the mile in decimal form = 5.28 miles. Practice & Problem Solving Leveled Practice In 6-8, write the decimal equivalent for each rational number. Use a bar over any repeating digits. Question 6. $$\frac{2}{3}$$ Answer: 2/3 = 0.666. Explanation: In the above-given question, given that, the number is 2/3. 2/3 = 0.666. 0.666 is a repeating decimal. Question 7. $$\frac{3}{11}$$ Answer: 3/11 = 0.272727. Explanation: In the above-given question, given that, the number is 3/11. 3/11 = 0.272727. 0.272727 is a repeating decimal. Question 8. $$8\frac{4}{9}$$ Answer: 8(4/9) = repeating decimal. Explanation: In the above-given question, given that, the number is 8(4/9). 9 x 8 = 72. 72 + 4 = 76. 76/9 = 8.4444. 8.444 is a repeating decimal. Question 9. Is 1.0227 a rational number? Explain. Answer: Yes, 1.0227 is a rational number. Explanation: In the above-given question, given that, the number is 1.0227. 1.0227 is a rational number. Question 10. Which should Aaron use to convert a fraction to a decimal? Answer: Option A is correct. Explanation: In the above-given question, given that, numerator/denominator is used to convert a fraction to a decimal. so option A is correct. Question 11. Is the fraction $$\frac{1}{3}$$ equivalent to a terminating decimal or a decimal that does not terminate? Answer: 1/3 is a repeating decimal. Explanation: In the above-given question, given that, the number is 1/3. 1/3 = 0.333. so 1/3 is a repeating decimal. Question 12. Determine whether the given number belongs to each set. Answer: -34 is an Integer. Explanation: In the above-given question, given that, they have given the whole numbers, integers, and rational numbers. so -34 is an integer. Question 13. Ariel incorrectly says that 2$$\frac{5}{8}$$ is the same as 2.58. a. Convert 2$$\frac{5}{8}$$ to a decimal. Answer: 2(5/8) = 2.625. Explanation: In the above-given question, given that, Ariel incorrectly says that 2$$\frac{5}{8}$$ is the same as 2.58. the number is 2(5/8). 8 x 2 = 16. 16 + 5 = 21. 21/8 = 2.625. b. What was Ariel’s likely error? Answer: Ariel’s error was the decimal. Explanation: In the above-given question, given that, Ariel incorrectly says that 2$$\frac{5}{8}$$ is the same as 2.58. the number is 2(5/8). 8 x 2 = 16. 16 + 5 = 21. 21/8 = 2.625. Question 14. Use Structure Consider the rational number $$\frac{3}{11}$$ a. What are the values of a and b in $$a\sqrt {b}$$ when you use division to find the decimal form? Answer: 3/11 = 0.272727. Explanation: In the above-given question, given that, a = 3, and b = 11. 3/11 = 0.272727. so 3/11 is a repeating decimal. 3/11 = 0.272727. b. What is the decimal form for $$\frac{3}{11}$$? Answer: The decimal form is 0.272727. Explanation: In the above-given question, given that, a = 3, and b = 11. 3/11 = 0.272727. so 3/11 is a repeating decimal. 3/11 = 0.272727. Question 15. At a grocery store, Daniel wants to buy 3$$\frac{1}{5}$$ lb of ham. What decimal should the digital scale show? Write 3$$\frac{1}{3}$$ as a fraction and then divide. The scale should read ________ lb. Answer: The scale should read 3.3 lb. Explanation: In the above-given question, given that, At a grocery store, Daniel wants to buy 3$$\frac{1}{5}$$ lb of ham. 3(1/3). 3 x 3 = 9. 9 + 1 = 10. 10/3 = 3.333. 3.333 is a repeating decimal. so the scale should read 3.3 lb. Question 16. Reasoning At a butcher shop, Hilda bought beef and pork. She left with 18$$\frac{8}{25}$$ pounds of meat. Express the number of pounds of pork she bought using a decimal. Answer: The number of pounds of pork she bought using a decimal = 9.47 pounds. Explanation: In the above-given question, given that, At a butcher shop, Hilda bought beef and pork. She left with 18$$\frac{8}{25}$$ pounds of meat. 18(8/25). 18 x 25 = 450. 458/25 = 18.32. the cost of the beef is 8(17/20). 20 x 8 = 160. 177/20 = 8.85. 18.32 – 8.85 = 9.47. Question 17. Be Precise Is 9.373 a repeating decimal? Is it rational? Explain your reasoning, Answer: Yes, it is a repeating decimal. Explanation: In the above-given question, given that, the number is 9.373 is a rational number. it is also a repeating decimal. Question 18. Reasoning Aiden has one box that is 3$$\frac{3}{11}$$, feet tall and a second box that is 3.27 feet tall. If he stacks the boxes, about how tall will the stack be? Answer: The tall will the stack be = 6.54 feet. Explanation: In the above-given question, given that, Aiden has one box that is 3$$\frac{3}{11}$$, feet tall and a second box that is 3.27 feet tall. 3(3/11). 33 + 3 = 36. 36/11 = 3.272727. 3.27 + 3.27 = 6.54, so the heght of the stack = 6.54 feets. Question 19. You are adding air to a tire. The air pressure in the tire should be $$32\frac{27}{200}$$ pounds per square inch. What decimal should you watch for on the digital pressure gauge? Answer: The decimal should I watch for on the digital pressure gauge = terminating decimal. Explanation: In the above-given question, given that, You are adding air to a tire. The air pressure in the tire should be $$32\frac{27}{200}$$ pounds per square inch. 32(27/200). 32 x 200 = 6400. 6400 + 27 = 6427. 6427/200 = 32.135. so the decimal should I watch for on the digital pressure gauge = terminating decimal. Question 20. Higher Order Thinking Dion has a pizza with a diameter of 10$$\frac{1}{3}$$ in. Is the square box shown big enough to fit the pizza inside? Justify your answer. Answer: The square box shown is big enough to fit the pizza inside. Explanation: In the above-given question, given that, Dion has a pizza with a diameter of 10$$\frac{1}{3}$$ in. 10(1/3). 30 + 1/3. 31/3 = 10.33. so the square box shown is big enough to fit the pizza inside. Assessment Practice Question 21. Which of the following mixed numbers has the same decimal value as $$110 \frac{147}{168}$$? A. $$110 \frac{49}{56}$$ B. $$110 \frac{170}{180}$$ C. $$110 \frac{56}{72}$$ D. $$110 \frac{247}{268}$$ Answer: Option A is the correct. Explanation: In the above-given question, given that, 110/ 147/168. 110/ 0.875. so option A is correct. Question 22. Select all the true statements about the negative fractions –$$\frac{4}{5}$$ and –$$\frac{5}{6}$$ –$$\frac{4}{5}$$ can be expressed as a repeating decimal. o –$$\frac{5}{6}$$ can be expressed as a repeating decimal. Both fractions can be expressed as repeating decimals. The digit that repeats is 3. The digit that repeats is 8. Answer: -5/6 can be expressed as a repeating decimal. the digit that repeats is 3. Explanation: In the above-given question, given that, the two numbers are -4/5 and -5/6. -4/5 and -5/6. -4/5 = -0.8. -5/6 = -0.8. so -5/6 can be expressed as a repeating decimal. ### Lesson 1.3 Add Integers Explore It! Rain increases the height of water in a kiddie pool, while evaporation decreases the height. The pool water level is currently 2 inches above the fill line. I can… add integers. A. Look for patterns in the equations in the table so you can fill in the missing numbers. Describe any relationships you notice. Answer: The missing numbers are 2, -1, -2. Explanation: In the above-given question, given that, the top of the pool is 4. the bottom of the pool is -4. we can add integers. so the missing numbers are 2, -1, and -2. B. Will the sum of 2 and (-6) be a positive or negative number? Explain. Answer: The sum of 2 and -6 be a negative number. Explanation: In the above-given question, given that, 2 + (-6). 2 – 6 = -4. the sum of two +ve numbers will always be a positive number. the sum of two -ve numbers will also be positive number. the sum of one +ve and one -ve is a negative number. so the sum of 2 and -6 be a negative number. Focus on math practices Look for Relationships Suppose the water level of the pool started at 2 inches below the fill line. Make a table to show the starting height of the water, the change in inches, and the new final height of the water. Essential Question How do you use what you know about the absolute value to add integers? Try It! Dana recorded a temperature drop of 2o and a second temperature drop of 3°. What is the total change in temperature? _______ + ________ = _________ The sign of the sum is _______. The total change in temperature is ________°. Answer: The sign of the sum is negative. the total change in temperature is 5°. Explanation: In the above-given question, given that, Dana recorded a temperature drop of 2o and a second temperature drop of 3°. -2 – 3 = -5. so the sign of the sum is negative. the total change in temperature is 5°. Convince Me! Would the sum of two positive integers be positive or negative? Explain. Answer: The sum of two positive integers is positive. Explanation: In the above-given question, given that, the sum of two positive integers is positive. 2 + 3 = 5. so the sum is also a positive number. Try It! Find the sum for each expression. a. -66 + 42 Answer: -66 + 42 = -24. Explanation: In the above-given question, given that, the numbers are -66 and 42. add the numbers. the sum of one positive and one negative is also a negative integer. -66 + 42 = -24. b. -57 + 57 Answer: -57 + 57 = 0. Explanation: In the above-given question, given that, the numbers are -57 and 57. add the numbers. the sum of one positive and one negative is also a negative integer. -57 + 57 = 0. c. 29 + (-28) Answer: 29 + (-28) = 1. Explanation: In the above-given question, given that, the numbers are -28 and 29. add the numbers. the sum of one positive and one negative is also a negative integer. 29 + (-28) = 1. KEY CONCEPT When adding integers with the same sign, find the sum of the absolute values. (-36) + (-12) |-36| = 36 and |-12| = 12 36 + 12 = 48 So, (-36) + (-12) = -48 Use the same sign as the addends. When adding integers with different signs, find the difference of the absolute values. 18+ (-14) |18| = 18 and |-14| = 14 18 – 14 = 4 So, 18 + (-14) = 4 Use the sign of the greater absolute value. Do You Understand? Question 1. Essential Question How do you use what you know about absolute value to add integers? Answer: When adding the integers with the same sign, we will find the sum of absolute values. Explanation: In the above-given question, given that, When adding the integers with same sign, we will find the sum of absolute values. for example: (-36) + (-12) |-36| = 36 and |-12| = 12 36 + 12 = 48 So, (-36) + (-12) = -48. Question 2. Reasoning How can you tell the sign of the sum of a positive and negative integer without doing any calculations? Answer: The sum of positive and negative integers is always negative. Explanation: In the above-given question, given that, the sum of positive and negative integers is always negative. for example: -33 + 22. -11. so the sum of positive and negative integers is always negative. Question 3. Model with Math How would you use a number line to determine the sum of two negative integers? Answer: The sum of two negative integers is also a positive integer. Explanation: In the above-given question, given that, the sum of two negative integers is also a positive integer. for example: -33 + -22. -33 – 22. 55. Do You Know How? Question 4. Sarah bought a bike that cost$260. She had a coupon that was worth $55 off the cost of any bike. Use the expression 260 + (-55) to find how much Sarah paid for her bike. Answer: The amount Sarah paid for her bike =$205.

Explanation:
In the above-given question,
given that,
Sarah bought a bike that cost $260. She had a coupon that was worth$55 off the cost of any bike.
260 + (-55).
260 – 55.
205.
so the amount Sarah paid for her bike = $205. Question 5. A shark is swimming 60 feet below the surface of the ocean. There is a fish that is 25 feet deeper in the water. Use the expression (-60) + (-25) to describe the fish’s location relative to the surface of the ocean. Answer: The fish’s location is relative to the surface of the ocean = -85 feet. Explanation: In the above-given question, given that, A shark is swimming 60 feet below the surface of the ocean. There is a fish that is 25 feet deeper in the water. (-60) + (-25). -60 – 25 = -85. so the fish’s location relative to the surface of the ocean = -85 feet. Question 6. The high temperature one day was 30°F. Then the temperature dropped 23 degrees during the night. Does the expression 30 + (-23) represent the temperature at night? Explain. Answer: The expression 30 + (-23) represent the temperature at night = 7 degrees. Explanation: In the above-given question, given that, The high temperature one day was 30°F. Then the temperature dropped 23 degrees during the night. 30 + (-23). 30 – 23 = 7. so the expression represents the temperature at night = 7 degrees. Practice & Problem Solving Leveled Practice For 7-9, use the number lines to help find each sum. Question 7. 5 + (-3) is ________ units from 5, in the ________ direction. Answer: 5 + (-3) is ____2____ units from 5, in the _positive____ direction. Explanation: In the above-given question, given that, 5 + (-3). 5 – 3 = 2. 2 units from 5, in the positive direction. Question 8. -1 + (-3) is _______ units from -1, in the ________ direction. Answer: -1 + (-3) is __-4_____ units from -1, in the ___negative_____ direction Explanation: In the above-given question, given that, -1 + (-3). -1 – 3 = -4. -4 units from -1. in the negative direction. Question 9. In City A, the temperature rises 9° from 8 A.M. to 9 A.M. Then the temperature drops 8° from 9 A.m. to 10 A.M. In City B, the temperature drops 5° from 8 A.M. to 9 A.M. Then the temperature drops 4° from 9 A.M. to 10 A.M. a. What expression represents the change in temperature for City A? Answer: The expression represents the change in temperature for City A is 9° – 8° = 1°. Explanation: In the above-given question, given that, In City A, the temperature rises 9° from 8 A.M. to 9 A.M. Then the temperature drops 8° from 9 A.m. to 10 A.M. 9° – 8° = 1°. so the expression represents the change in temperature for City A is 9° – 8° = 1°. b. What integer represents the change in temperature for City A? Answer: The positive integer represents the change in temperature for City A. Explanation: In the above-given question, given that, In City A, the temperature rises 9° from 8 A.M. to 9 A.M. Then the temperature drops 8° from 9 A.m. to 10 A.M. 9° – 8° = 1°. so the positive integer represents the change in temperature for City A. c. What expression represents the change in temperature for City B? Answer: The expression represents the change in temperature for City B is 5° – 4° = 1°. Explanation: In the above-given question, given that, In City B, the temperature rises 5° from 8 A.M. to 9 A.M. Then the temperature drops 4° from 9 A.m. to 10 A.M. 5° – 4° = 1°. so the expression represents the change in temperature for City B is 5° – 4° = 1°. d. What integer represents the change in temperature for City B? Answer: A positive integer represents the change in temperature for City B. Explanation: In the above-given question, given that, In City B, the temperature rises 5° from 8 A.M. to 9 A.M. Then the temperature drops 4° from 9 A.m. to 10 A.M. 5° – 4° = 1°. so the positive integer represents the change in temperature for City B. e. Which city has the greater change in temperature from 8 A.M. to 10 A.M.? Answer: Both the cities have an equal change in temperature. Explanation: In the above-given question, given that, In City A, the temperature rises 9° from 8 A.M. to 9 A.M. Then the temperature drops 8° from 9 A.m. to 10 A.M. 9° – 8° = 1°. 5° – 4° = 1°. so both the cities have an equal change in temperature. Question 10. An airplane flying at an altitude of 30,000 feet flies up to avoid a storm. Immediately after passing the storm, the airplane returns to its original altitude. a. What integer represents the airplane’s change in altitude to avoid the storm? Answer: The positive integer represents the airplane’s change in altitude to avoid the storm. Explanation: In the above-given question, given that, An airplane flying at an altitude of 30,000 feet flies up to avoid a storm. the airplane flies up to 38,000 feet to avoid a storm. 38000 – 30000. 8000. so the positive integer represents the airplane’s change in altitude to avoid the storm. b. What integer represents the airplane’s change in altitude immediately after passing the storm? Answer: The positive integer represents the airplane’s change in altitude immediately after passing the storm. Explanation: In the above-given question, given that, An airplane flying at an altitude of 30,000 feet flies up to avoid a storm. the airplane flies up to 38,000 feet to avoid a storm. 38000 – 30000. 8000. so the positive integer represents the airplane’s change in altitude immediately after passing the storm. c. Use Appropriate Tools Draw a number line to represent the airplane’s change in altitude. Answer: The change in altitude is 38000. Explanation: In the above-given question, given that, An airplane flying at an altitude of 30,000 feet flies up to avoid a storm. the airplane flies up to 38,000 feet to avoid a storm. 30000 + 8000 = 38000. Question 11. A deep-sea diver dives 81 feet from the surface. He then dives 14 more feet. The diver’s depth can be represented by -81 + (-14). What is the diver’s present location? Answer: The diver’s present location is -95 feet below the sea. Explanation: In the above-given question, given that, A deep-sea diver dives 81 feet from the surface. He then dives 14 more feet. The diver’s depth can be represented by -81 + (-14). -81 + (-14). -81 – 14 = -95. so the diver’s present location is -95 feet below the sea. Question 12. Rena’s rowboat drifts 23 feet from shore, followed by 9 more feet. The rowboat’s current position can be represented by –23 + (-9). What integer represents the rowboat’s position? Answer: The integer represents the rowboat’s position as negative. Explanation: In the above-given question, given that, Rena’s rowboat drifts 23 feet from shore, followed by 9 more feet. The rowboat’s current position can be represented by –23 + (-9). -23 – 9 = -32. so the integer represents the rowboat’s position is negative. Question 13. Critique Reasoning A submarine traveling 200 meters below the surface of the ocean increases its depth by 45 meters. Adam says that the new location of the submarine is -155 meters. Describe an error Adam could have made that would result in the answer he gave. Answer: Yes, Adam was incorrect. Explanation: In the above-given question, given that, A submarine traveling 200 meters below the surface of the ocean increases its depth by 45 meters. 200 + 45 = 245. but Adam says that the new location of the submarine is -155 meters. so Adam was incorrect. Question 14. Kim has$45 to spend for a day at the zoo. She pays $17 for admission,$8 for lunch, and $4 for a snack. a. Model with Math Use integers to write an addition expression that represents the amount of money Kim has left. Answer: The amount of money Kim has left =$16.

Explanation:
In the above-given question,
given that,
Kim has $45 to spend for a day at the zoo. She pays$17 for admission, $8 for lunch, and$4 for a snack.
$45 + (-$17) + (-$8) + (-$4).
$45 –$17 – $8 –$4.
$45 –$29.
$16. so the amount of money Kim has left =$16.

b. Kim goes to the gift shop and finds a T-shirt she likes for $19. Does she have enough money to buy the T-shirt? Explain. Answer: No, she does not have enough money to buy the T-shirt. Explanation: In the above-given question, given that, Kim has$45 to spend for a day at the zoo.
She pays $17 for admission,$8 for lunch, and $4 for a snack. she has left$16.
so she does not have enough money to buy the T-shirt.

Question 15.
Higher Order Thinking Samantha has $300 for guitar lessons to learn her favorite song. Mrs. Jones charges$80 per lesson and requires three lessons to teach Samantha the song. Mr. Beliz charges $62 per lesson and will require four lessons to teach Samantha the song. Use integers to represent what each teacher charges. Which is the better deal for Samantha? Answer: The better deal for Samantha is Mr. Beliz charges$62 per lesson.
$62 +$62 + $62 +$62 = $248. Explanation: In the above-given question, given that, Samantha has$300 for guitar lessons to learn her favorite song.
Mrs. Jones charges $80 per lesson and requires three lessons to teach Samantha the song. Mr. Beliz charges$62 per lesson and will require four lessons to teach Samantha the song.
$80 +$80 + $80 =$240.
$62 +$62 + $62 +$62 = $248. so the better deal for Samantha is Mr. Beliz charges$62 per lesson.

Assessment Practice

Question 16.
A fish swims at 10 ft below sea level, and then swims another 10 ft deeper to avoid a shark. Write an addition expression that represents this situation.

The addition expression that represents this situation is (-10) + (-10) = -20.

Explanation:
In the above-given question,
given that,
A fish swims at 10 ft below sea level and then swims another 10 ft deeper to avoid a shark.
(-10) + (-10) = -20.
below sea level = -10.
so the addition expression that represents this situation is (-10) + (-10) = -20.

Question 17.
The temperature drops 10 degrees and then rises 10 degrees. Write an addition expression that represents this situation.

The addition expression that represents this situation is 10 + (-10) = 0.

Explanation:
In the above-given question,
given that,
The temperature drops 10 degrees and then rises 10 degrees.
10 + (-10).
10 – 10 = 0.
so the addition expression that represents this situation is 10 + (-10) = 0.

### Lesson 1.4 Subtract Integers

Solve & Discuss It!
A library database shows the total number of books checked out at any given time as a negative number. What are the possible numbers of books that were checked out and checked in on Monday? Explain.
I can.. subtract integers.

Make Sense and Persevere
How can you use the data to understand what happened during the day?

Focus on math practices
Reasoning Suppose the library database showed 0 for Monday evening. What do you know about the number of books checked out and checked in that day?

The number of books checked out and checked in that day = 45 books.

Explanation:
In the above-given question,
given that,
0 + (-45).
0 – 45 = -45.
so the number of books checked out and checked in that day = -45.

Essential Question
How is subtracting integers related to adding integers?

Try It!

On the next play, the team gained 5 yards and then lost 6 yards. What is the total change in yards?

5 – ______
= 5 + ______
= _______
The total change in yards is ________, so they had a total loss of ________ yard.

The total change in yards is 1.
so they had a total loss of 1 yard.

Explanation:
In the above-given question,
given that,
On the next play, the team gained 5 yards and then lost 6 yards.
5 – 6 = -1.
5 + (-6) = -1.

Convince Me! Is the additive inverse of an integer always negative? Explain.

Try It!

a. -4 – 6

-4 -6 = 10.

Explanation:
In the above-given question,
given that,
the numbers are -4 and -6.
-4 + (-6).
-4 – 6 = -10.

b. -6 – (-4).

-6 – (-4) = -2.

Explanation:
In the above-given question,
given that,
the numbers are -4 and -6.
-6 – (-4).
-6 + 4.
-2.

c. 4 – (-6)

4 – (-6) = 10.

Explanation:
In the above-given question,
given that,
the numbers are 4 and -6.
4 – (-6).
4 + 6.
10.

d. 6 – 4

6 – 4 = 2.

Explanation:
In the above-given question,
given that,
the numbers are 6 and -4.
6 – 4 = 2.

e. 4 – 6

4 – 6 = -2.

Explanation:
In the above-given question,
given that,
the numbers are 4 and -6.
4 – 6 = -2.

f. -4 – (-6)

-4 + 6 = 2.

Explanation:
In the above-given question,
given that,
the numbers are -4 and -6.
-4 – (-6).
-4 + 6 = 2.

KEY CONCEPT
When subtracting integers, such as a – b, you can use the additive inverse to write subtraction as an equivalent addition expression.

Do You Understand?
Question 1.
Essential Question How is subtracting integers related to adding integers?

Explanation:
In the above-given question,
given that,
for example:
a + (-b) = a -b.

Question 2.
Reasoning Explain how to simplify the expression -98 – 31 using the additive inverse.

-98 + (-31) = – 129.

Explanation:
In the above-given question,
given that,
the numbers are -98 and -31.
-98 + (-31).
-129.

Question 3.
Model with Math How can you use a number line to represent the subtraction between two integers?

-5 – 2 = -7.

Explanation:
In the above-given question,
given that,
the two numbers on the number line = -5 and -2.
-5 + (-2).
-5 – 2 = -7.

Do You Know How?
Question 4.
It was 12°C when Preston got home from school. The weather report shows a storm front moving in that will drop the temperature by 17°C. What is the expected temperature?

The expected temperature is -5 degrees c.

Explanation:
In the above-given question,
given that,
It was 12°C when Preston got home from school.
The weather report shows a storm front moving in that will drop the temperature by 17°C.
12 – 17 = -5.
so the expected temperature is -5°C.

Question 5.
Complete the equation.
-67 – _____ = 0

-67 – (-67) = 0.

Explanation:
In the above-given question,
given that,
-67 – (-67).
-67 + 67 = 0.

Question 6.
Find the difference.
a. 41 – 275

41 – 275 = -234.

Explanation:
In the above-given question,
given that,
the two numbers are 41 and 275.
subtract the numbers.
41 – 275.
– 234.

b. -15 – 47

-15 – 47 = -32.

Explanation:
In the above-given question,
given that,
the two numbers are -15 and -47.
subtract the numbers
-15 – 47.
47 – 15 = -32.

c. -72 – (-151)

-72 – (-151) = 79.

Explanation:
In the above-given question,
given that,
the two numbers are -72 and -151.
subtract the numbers.
-72 – (-151).
-72 + 151.
79.

d. 612 – (-144)

612 – (-144) = 756.

Explanation:
In the above-given question,
given that,
the two numbers are 612 and -144.
subtract the numbers.
612 – (-144).
612 + 144 = 756.

Practice & Problem Solving

Leveled Practice In 7-8, fill in the boxes to solve.
Question 7.
What subtraction expression does the number line model show?

-2 + (-9) = 7.

Explanation:
In the above-given question,
given that,
the lines are moving from -2 and -9.
-2 + (-9).
-2 – 9 = -11.

Question 8.
What is the value of the expression -9 – (-5)?
-9 – (-5)
= -9 ______ 5
= _______

The value of the expression -9 – (-5) = -4.

Explanation:
In the above-given question,
given that,
the two numbers are -9 and -5.
subtract the numbers.
-9 – (-5) = -4.

Question 9.+e
The temperature at the beginning of the day was 6°F. The temperature dropped 9°F by the end of the day. Use the number line to find the temperature at the end of the day.

The temperature at the end of the day = -3.

Explanation:
In the above-given question,
given that,
The temperature at the beginning of the day was 6°F.
The temperature dropped 9°F by the end of the day.
6 + (- 9) = -3.
so the temperature at the end of the day = -3.

Question 10.
Murphy and Naryam do their math homework together. When they find 9- (-8), they get different answers. Murphy claims the difference is 17. Naryam claims the difference is – 1.
a. Who is correct?

Murphy was correct.

Explanation:
In the above-given question,
given that,
Murphy and Naryam do their math homework together.
When they find 9- (-8), they get different answers.
9 – (-8).
9 + 8 = 17.
so Murphy was correct.

b. What error likely led to the incorrect answer?

The sign likely led to the incorrect answer.

Explanation:
In the above-given question,
given that,
Murphy and Naryam do their math homework together.
When they find 9- (-8), they get different answers.
9 – (-8).
9 + 8 = 17.

Question 11.
The news reports that today’s high temperature is 16°F colder than yesterday’s high temperature. Yesterday’s high temperature was -2°F.
a. Write an expression to represent today’s high temperature.

The expression is -2°F + 16°F = 14°F.

Explanation:
In the above-given question,
given that,
The news reports that today’s high temperature is 16°F colder than yesterday’s high temperature.
Yesterday’s high temperature was -2°F.
so the expression is -2°F + 16°F = 14°F.

b. Reasoning is today’s high temperature positive or negative? Why?

Today’s high temperature is positive.

Explanation:
In the above-given question,
given that,
The news reports that today’s high temperature is 16°F colder than yesterday’s high temperature.
Yesterday’s high temperature was -2°F.
so the expression is -2°F + 16°F = 14°F.

Question 12.
Max sprints forward 10 feet and then stops and sprints back 15 feet. Use subtraction to explain where Max is relative to where he started.

He started at -5.

Explanation:
In the above-given question,
given that,
Max sprints forward 10 feet and then stops and sprints back 15 feet.
10 + (-15).
10 – 15.
-5.
so he started at -5.

Question 13.
Higher Order Thinking Use the number line at the right.

a. What subtraction equation does the number line represent?

The subtraction equation is -2 + (-6) = -8.

Explanation:
In the above-given question,
given that,
the starting point is -2.
he then increases by -6.
-2 + (-6).
-2 -6 = -8.

b. Use the number line to represent a different subtraction equation that has the same difference shown in the number line. Write the subtraction equation.

0 – 2 = -2.
-4 – (- 2) = -2.

Explanation:
In the above-given question,
given that,
the subtraction equations are:
0 – 2 = -2.
-4 – (-2).
-4 + 2 = -2.

Question 14.
A crane lifts a pallet of concrete blocks 8 feet from the back of a truck. The truck drives away and the crane lowers the pallet 13 feet: What is the final position of the pallet relative to where it started in the back of the truck?

The final position of the pallet relative to where it started in the back of the truck = 5 feet.

Explanation:
In the above-given question,
given that,
A crane lifts a pallet of concrete blocks 8 feet from the back of a truck.
The truck drives away and the crane lowers the pallet 13 feet:
13 – 8 = 5.
so the final position of the pallet relative to where it started in the back of the truck = 5 feet.

Question 15.
Make Sense and Persevere At its highest point, the elevation of a county is 5,762 feet above sea level. At its lowest point, the elevation of the county is 9 feet below sea level.
a. Write an expression using integers to represent the difference between the elevations.

The expression using integers to represent the difference between the elevations is 5753.

Explanation:
In the above-given question,
given that,
At its highest point, the elevation of a county is 5,762 feet above sea level.
At its lowest point, the elevation of the county is 9 feet below sea level.
5762 + (-9) = 5753.
so the expression using integers to represent the difference between the elevations is 5753.

b. Will the answer be written as a positive or negative integer?

The answer will be written as a positive number.

Explanation:
In the above-given question,
given that,
At its highest point, the elevation of a county is 5,762 feet above sea level.
At its lowest point, the elevation of the county is 9 feet below sea level.
5762 + (-9) = 5753.
so the answer will be written as a positive number.

c. What is the difference between the highest and lowest points of the county?

The difference between the highest and lowest points of the county is 5753.

Explanation:
In the above-given question,
given that,
At its highest point, the elevation of a county is 5,762 feet above sea level.
At its lowest point, the elevation of the county is 9 feet below sea level.
5762 + (-9) = 5753.
so the difference between the highest and lowest points of the county is 5753.

Assessment Practice

Question 16.
Which number line model shows the subtraction 2 – 4?

Option D is the correct answer.

Explanation:
In the above-given question,
given that,
the number line from 0 to -2 and -2 to -6.
-2 + (-6).
-2 -6 = -8.
so option D is the correct answer.

### Lesson 1.5 Add and Subtract Rational Numbers

Solve & Discuss It!
Malik hikes Castle Trail from point A to point B. The elevation at point A is below sea level. What are possible beginning and ending elevations of Malik’s hike?

I can… add and subtract rational numbers.

Look for Relationships
How are elevation values of point A and point B related?

Focus on math practices
Reasoning What would be different about the hike from point B to point A?

The difference between the hike from point B to point A is 120.5 meters.

Explanation:
In the above-given question,
given that,
Malik hikes Castle Trail from point A to point B.
The elevation at point A is below sea level.
120(1/2).
120 x 2 = 240.
240 + 1 = 241.
241/2 = 120.5.
so the difference between the hike from point B to point A is 120.5 meters.

Essential Question
How are adding and subtracting integers related to adding and subtracting other rational numbers?

Try It!

A dolphin is at the surface of the water and then descends to a depth of 4$$\frac{1}{2}$$ feet. Then the dolphin swims down another 2$$\frac{3}{4}$$ feet. What is the location of the dolphin relative to the surface of the water?
-4$$\frac{1}{2}$$ – ________
-4$$\frac{1}{2}$$ + _______ = __________

The location of the dolphin relative to the surface of the water is ________ feet.

The location of the dolphin relative to the surface of the water is -7.25 feet.

Explanation:
In the above-given question,
given that,
A dolphin is at the surface of the water and then descends to a depth of 4$$\frac{1}{2}$$ feet.
Then the dolphin swims down another 2$$\frac{3}{4}$$ feet.
-4(1/2) – 2(3/4).
-9/2 – 11/4.
-4.5 – 2.75.
-7.25.
so the location of the dolphin relative to the surface of the water is -7.25 feet.

Convince Me! How are adding and subtracting two rational numbers with different signs related to adding and subtracting two integers with different signs?

Try It!

Find the sum or difference of the rational numbers.
a. $$-2.5+\left(-5 \frac{6}{10}\right)$$

-2.5 + (-5 x 6/10) = -8.1.

Explanation:
In the above-given question,
given that,
the sum of two numbers.
-2.5 – 5 x 6/10.
-2.5 – 56/10.
-2.5 – 5.6.
-8.1.

b. $$-4.4-\left(-1 \frac{1}{2}\right)$$

-5.9.

Explanation:
In the above-given question,
given that,
the difference between two numbers.
-4.4 – 3/2.
-4 . 4 – 1.5.
-5.9.

c. $$-135.4+78 \frac{1}{2}$$

-56.9.

Explanation:
In the above-given question,
given that,
-135.4 + 78(1/2).
-135.4 + 157/2.
-135.4 + 78.5.
-56.9.

Try It!

Two divers are swimming at different depths below sea level. One diver is at -25.5 feet. The other diver is at –40.75 feet. How much farther below sea level is the diver who is farthest below sea level?

The much farther below sea level is the diver who is farthest below sea level = -15.25 feet.

Explanation:
In the above-given question,
given that,
Two divers are swimming at different depths below sea level.
One diver is at -25.5 feet.
The other diver is at –40.75 feet.
-25.5 – 40.75.
– 15.25.
so the much farther below sea level is the diver who is farthest below sea level = -15.25 feet.

KEY CONCEPT
The rules for adding and subtracting all rational numbers are the same as those for adding and subtracting integers.
The distance between any two rational numbers p and q on a number line is the absolute value of their difference.

Do You Understand?
Question 1.
Essential Question How are adding and subtracting integers related to adding and subtracting other rational numbers?

The integers are related to adding and subtracting other rational numbers is the absolute value of their difference.

Explanation:
In the above-given question,
given that,
The integers are related to adding and subtracting other rational numbers is the absolute value of their difference.
for example:
-5 – (-2).
-2 – (-5).
-2 + 5 = 3.

Question 2.
Reasoning When finding the distance between two rational numbers on a number line, does the order of the numbers you subtract matter? Explain.

The distance between two rational numbers on a number line does the order of the numbers you subtract matter.

Explanation:
In the above-given question,
given that,
The integers are related to adding and subtracting other rational numbers is the absolute value of their difference.
for example:
-5 – (-2).
-2 – (-5).
-2 + 5 = 3.

Question 3.
Critique Reasoning Gwen says that the sum of – 1$$\frac{3}{4}$$ and 2$$\frac{1}{2}$$ is the same as the difference between 2$$\frac{1}{2}$$ and 1$$\frac{3}{4}$$. Is Gwen correct? Explain why or why not.

Gwen was correct.

Explanation:
In the above-given question,
given that,
Gwen says that the sum of – 1$$\frac{3}{4}$$ and 2$$\frac{1}{2}$$.
-1(3/4) and 2(1/2).
-4 + 3 = -7/4.
4 + 1 = 5/2.
-7/4 = -1.75.
5/2 = 2.5.
1(3/4) = 7/4.
7/4 = 1.75.

Do You Know How?
Question 4.
What is the distance between the top of the fishing pole and the fish?

The distance between the top of the fishing pole and the fish = 4.

Explanation:
In the above-given question,
given that,
the top of the pole = 8(1/2).
the distance of the fish = 4(1/2).
8(1/2) – 4(1/2).
8 x 2 = 16.
16 + 1 = 17/2.
4 x 2 = 8.
8 + 1 = 9.
9/2 = 4.5.
17/2 = 8.5.
8.5 – 4.5 = 4.

Question 5.
A shark began at 172.5 meters below sea level and then swam up 137.1 meters. Where is the shark’s location now in relation to sea level?

The shark’s location now in relation to sea level = 35.4 meters.

Explanation:
In the above-given question,
given that,
A shark began at 172.5 meters below sea level and then swam up 137.1 meters.
172.5 – 137.1.
35.4.
so the shark’s location now in relation to sea level = 35.4 meters.

Question 6.
Find the sum or difference.
a. -12$$\frac{1}{2}$$ + 4$$\frac{1}{2}$$

-8.

Explanation:
In the above-given question,
given that,
the two numbers.
-12(1/2) + 4(1/2).
-25/2 + 9/2.
-12.5 + 4.5.
-8.

b. -0.35 – (-0.25)

-0.10.

Explanation:
In the above-given question,
given that,
the two numbers.
subtract the numbers.
-0.35 – (-0.25).
-0.35 + 0.25.
-0.10.

Practice & Problem Solving

Leveled Practice In 7-8, complete the expressions to find the sum or difference.
Question 7.
3.2 – (-5.7)
= 3.2 + _____
= ______

3.2 – (-5.7) = 8.9.

Explanation:
In the above-given question,
given that,
the two numbers.
subtract the numbers.
3.2 – (-5.7).
3.2 + 5.7.
8.9.

Question 8.
$$\frac{12}{13}+\left(\frac{-1}{13}\right)$$
= _____ – ______
= ______

12/13 + (-1/13) = 0.85.

Explanation:
In the above-given question,
given that,
the two numbers.
12/13 = 0.92.
1/13 = 0.07.
0.92 – 0.07 = 0.85.

Question 9.
Reasoning When Tom simplified the expression -2.6 + (-5.4), he got 2.8. What mistake did Tom likely make?

The mistake did Tom likely make = -8.

Explanation:
In the above-given question,
given that,
-2.6 + (-5.4).
-2.6 – 5.4.
– 8.

Question 10.
The temperature in a town is 36.6°F during the day and -12.6°F at night. What is the temperature change from day to night?

The temperature change from day to night = 24°F.

Explanation:
In the above-given question,
given that,
The temperature in a town is 36.6°F during the day and -12.6°F at night.
36.6 + (-12.6).
36.6 – 12.6.
24.
so the temperature change from day to night = 24°F.

Question 11.
Simplify each expression.
a. 50$$\frac{1}{2}$$ + (-12.3)

50(1/2) + (-12.3) = 38.2.

Explanation:
In the above-given question,
given that,
the two numbers are 50(1/2) and -12.3.
101/2 and -12.3.
50.5 + (-12.3).
50.5 – 12.3 = 38.2.

b. -50$$\frac{1}{2}$$ + (-12.3)

-50(1/2) + (-12.3) = -62.8.

Explanation:
In the above-given question,
given that,
the two numbers are -50(1/2) and -12.3.
-101/2 and -12.3.
-50.5 – 12.3.
– 62.8.

C. –50$$\frac{1}{2}$$ + 12.3

-50(1/2) + 12.3 = -32.8.

Explanation:
In the above-given question,
given that,
the two numbers are -50(1/2) and 12.3.
-101/2 + 12.3.
-50.5 + 12.3.
-38.2.

Question 12.
At the beginning of the day, the stock market goes up 30$$\frac{1}{2}$$ points. At the end of the day, the stock market goes down 120$$\frac{1}{4}$$ points. What is the total change in the stock market from the beginning of the day to the end of the day?

The total change in the stock market from the beginning of the day to the end of the day = 89.75 points.

Explanation:
In the above-given question,
given that,
At the beginning of the day, the stock market goes up 30$$\frac{1}{2}$$ points.
At the end of the day, the stock market goes down 120$$\frac{1}{4}$$ points.
30(1/2) = 61/2.
61/2 = 30.5.
120(1/4) = 481/4.
481/4 = 120.25.
120.25 – 30.5 = 89.75.
so the total change in the stock market from the beginning of the day to the end of the day = 89.75 points.

Question 13.
A dolphin is swimming 18 feet below the surface of the ocean. There is a coast guard helicopter 75.5 feet above the surface of the water that is directly above the dolphin. What is the distance between the dolphin and the helicopter?

The distance between the dolphin and the helicopter = 57.5 feet.

Explanation:
In the above-given question,
given that,
A dolphin is swimming 18 feet below the surface of the ocean.
There is a coast guard helicopter 75.5 feet above the surface of the water that is directly above the dolphin.
75.5 – 18 = 57.5 feet.
so the distance between the dolphin and the helicopter = 57.5 feet.

Question 14.
A bird flies from its nest to the bottom of the canyon. How far did the bird fly?

The far did the birds fly = 438.6.

Explanation:
In the above-given question,
given that,
A bird flies from its nest to the bottom of the canyon.
the top of the nest is 528(1/5).
528 x 1/5.
528 x 5 = 2640.
2641/5 = 528.2.
the height of canyon floor = -89(3/5).
-89 x 5 = 445.
-448 / 5 = -89.6.
528.2 – 89.6 = 438.6.

Question 15.
A scuba diving instructor takes a group of students to a depth of 54.96 feet. Then they ascend 22.38 feet to see some fish. Where are the fish in relation to the surface?

The fish in relation to the surface = 32.58 feet.

Explanation:
In the above-given question,
given that,
A scuba diving instructor takes a group of students to a depth of 54.96 feet.
Then they ascend 22.38 feet to see some fish.
54.96 – 22.38 = 32.58.
so the fish in relation to the surface = 32.58 feet.

Question 16.
Model with Math Write an addition expression that is represented by the number line.

The addition expression that is represented by the number line = 0.5.

Explanation:
In the above-given question,
given that,
0 + 1 = 1.
0 + 0.5 = 0.5.
1 – 0.5 = 0.5.
so the addition expression that is represented by the number line is 1 + (-0.5).

Question 17.
The roots of a plant reach down 3$$\frac{3}{4}$$ inches below ground. How many inches is the plant above the ground?

The number of inches is the plant above the ground = 8.75 inches.

Explanation:
In the above-given question,
given that,
The roots of a plant reach down 3$$\frac{3}{4}$$ inches below ground.
3(3/4) = 15/4.
15/4 = 3.75.
12(1/2) = 25/2.
25/2 = 12.5
12.5 – 3.75 = 8.75.
so the number of inches is the plant above the ground = 8.75 inches.

Question 18.
Higher Order Thinking
a. Simplify the expression (-13.2) + 8.1.

-13.2 + 8.1 = -5.1.

Explanation:
In the above-given question,
given that,
the two numbers are -13.2 and 8.1.
-13.2 + 8.1.
– 5.1.
-13.2 + 8.1 = -5.1.

b. How are (-13.2) + 8.1 and 13.2 + (-8.1) related? Explain without computing.

-13.2 + 8.1 = -5.1.
13.2 – 8.1 = 5.1.

Explanation:
In the above-given question,
given that,
the two numbers are 13.2 and 8.1.
they have an equal value but the sign is different.

c. Using a property of operations, what can you say about the sum of the two expressions?

The sum of the two expressions is also positive.

Assessment Practice

Question 19.
The temperatures at sunrise and sunset are shown in the table.

PART A
Write an expression that represents the change in temperature for Day 1. Show how you can use properties of operations to find the value of the expression.

The properties of operations to find the value of the expression is -11.31 + 13.49.

Explanation:
In the above-given question,
given that,
the temperatures at sunrise and sunset are shown.
sunrise on day 1 is -11.31 and day 2 is -7.69.
sunset on day 1 is 13.49 and day 2 is 25.25.
-11.31 + (13.49).
-11.31 + 13.49.
2.18.
so the properties of operations to find the value of the expression is -11.31 + 13.49 is 2.18.

PART B
On which day did the temperature change more? Explain your reasoning.

On day 2 the temperature change more.

Explanation:
In the above-given question,
given that,
the temperatures at sunrise and sunset are shown.
sunrise on day 1 is -11.31 and day 2 is -7.69.
sunset on day 1 is 13.49 and day 2 is 25.25.
-11.31 + (13.49).
-11.31 + 13.49.
2.18.
-7.69 + 25.25.
17.56.
so on day 2 the temperature change more.

Question 20.
Mischa dives from a platform that is 5 meters above water. Her dive takes her 2.1 meters below the surface of the water. Which expression could represent the distance, in meters, that Mischa dives? Select all that apply.
☐ |5 – (-2.1)|
☐ |-(2.1) – (-5)|
☐ |2.1 – 5|
☐ |-(2.1) – 5|
☐ |5 + (-2.1)|

The expression could represent the distance in meters is 5 – (-2.1).

Explanation:
In the above-given question,
given that,
Mischa dives from a platform that is 5 meters above the water.
Her dive takes her 2.1 meters below the surface of the water.
5 – (-2.1).
5 + 2.1 = 7.1.
so the expression could represent the distance in meters is 7.1.

Topic 1 Mid-Topic Checkpoint

Question 1.
Vocabulary How do you find the additive inverse of a number? Give an example of a number and its additive inverse. Lesson 1-3

The additive inverse of a number is always negative.

Explanation:
In the above-given question,
given that,
the numbers are -4 and -6.
-4 + (-6).
-4 – 6 = -10.

Question 2.
A plastic toy submarine is held 15 centimeters below the water surface in a bath tub. The submarine is let go and rises 15 centimeters. What integer represents the toy submarine’s position with respect to the surface of the water? Lesson 1-1

The positive integer represents the toy submarine’s position with respect to the surface of the water.

Explanation:
In the above-given question,
given that,
A plastic toy submarine is held 15 centimeters below the water surface in a bathtub.
The submarine is let go and rises 15 centimeters.
-15 + (15).
-15 + 15 = 0.
so the integer is zero.

Question 3.
The temperature in the late afternoon was -7.5°C. It dropped 5 degrees by early evening and then dropped another 8.5 degrees by midnight. What was the temperature at midnight? Lessons 1-3, 1-4, and 1-5

The temperature at the midnight = -21°C.

Explanation:
In the above-given question,
given that,
The temperature in the late afternoon was -7.5°C.
It dropped 5 degrees by early evening and then dropped another 8.5 degrees by midnight.
-7.5 – 5 – 8.5.
-7.5 – 13.5.
-21.
so the temperature at the midnight is -21°C.

Question 4.
The floor of an elevator in a building is 30 feet above ground level. It travels down to the lower level of the building, where the floor is 10 feet below ground level. What distance has the elevator’s floor traveled? Lessons 1-4 and 1-5

The distance has the elevator’s floor traveled = 20 feet.

Explanation:
In the above-given question,
given that,
The floor of an elevator in a building is 30 feet above ground level.
It travels down to the lower level of the building, where the floor is 10 feet below ground level.
30 + (-10).
30 – 10.
20.
so the distance has the elevator’s floor traveled = 20 feet.

Question 5.
Greg says that 3.3. is a rational number. Kari says 3.3 is not a terminating decimal. Who is correct and why? Lesson 1-2

Yes. both of them are correct.

Explanation:
In the above-given question,
given that,
Greg says that 3.3. is a rational number.
3.3 is a rational number.
Kari says 3.3 is not a terminating decimal.
3.3 is also a terminating decimal.
so both of them are correct.

Question 6.
Cece is hiking on a mountain and stops at 15$$\frac{5}{8}$$ feet above sea level. The base of the mountain is 10.2 feet below sea level. What is the vertical distance between Cece and the base of the mountain? Lesson 1-5
A. 5.425 feet
B. 25.825 feet
C. 25$$\frac{3}{8}$$ feet
D. 5$$\frac{1}{4}$$ feet

Option A is correct.

Explanation:
In the above-given question,
given that,
Cece is hiking on a mountain and stops at 15$$\frac{5}{8}$$ feet above sea level.
The base of the mountain is 10.2 feet below sea level.
15(5/8) – 10.2.
125/8 – 10.2.
15.625 – 10.2.
5.425.
so option A is correct.

An oceanographer, Dr. Price, is studying the types of sea life at various depths.

PART A

Dr. Price uses a table to organize the types of sea life and the positions relative to sea level of each location.

Complete each sentence.
The difference between Location A and Location B is ________ meters.
The difference between Location B and Location C is ________ meters.
The difference between Location C and Location D is ________ meters.

The difference between Location A and Location B is 202.25 meters.
The difference between Location B and Location C is 1686.6 meters.
The difference between Location C and Location D is474.875 meters.

Explanation:
In the above-given question,
given that,
location A = -895.9.
location B = – 1098(3/20).
location C = -2784.75.
location D = – 3259(5/8).
-895.9 – (-1098(3/20).
-895.9 + 21963/20.
-895.9 + 1098.15.
202.25.
-1098.15 – (-2784.75).
-1098.15 + 2784.75.
1686.6.
-2784.75 + 3259.625.
474.875.

PART B
After observing Location B, Dr. Price returns to Location A before descending to Location C. What is the total distance she travels?

The total distance she travels = -2987 meters.

Explanation:
In the above-given question,
given that,
location B = -1098.15.
location A = – 895.9.
location C = -2784.75.
-1098.15 + (-895.9) + (- 2784.75).
-1098.15 – 895.9 – 2784.75.
-1098.15 – 1888.85.
-2987.
so the total distance she travels = -2987 meters.

PART C
Dr. Price descends to Location:D to observe shrimp. She then ascends and stops to observe sea life that is halfway between Location B and Location C. What is the total distance between Location D and where Dr. Price stopped to observe?

The distance between Location D and where Dr. Price stopped to observe = -272.625 meters.

Explanation:
In the above-given question,
given that,
Dr. Price descends to Location:D to observe shrimp.
She then ascends and stops to observe sea life that is halfway between Location B and Location C.
-3259.625 – (-2987).
-3259.625 + 2987.
-272.625.
so the distance between Location D and where Dr. Price stopped to observe = -272.625 meters.

### Lesson 1.6 Multiply Integers

Explore It!
A popular beach erodes 4 inches per year on average.
I can… multiply integers.

A. How many years will it take for the coastline to erode one foot?

The number of years will it take for the coastline to erode one foot = 12 years.

Explanation:
In the above-given question,
given that,
A popular beach erodes 4 inches per year on average.
1 year = 4 inches.
one feet = 12 inches.
so the number of years will it take for the coastline to erode one foot = 12 years.

B. The number line below shows the expected change in the coastline as years pass. How could you use the number line to show the erosion after 10 years?

The erosion after 10 years = 10.

Explanation:
In the above-given question,
given that,
The number line below shows the expected change in the coastline as the years pass.
coastline this year = 0.
0 + 10 = 10.
so the erosion after 10 years = 10.

Focus on math practices
Be Precise What expression could you use to represent the change in the coastline in 5 years?

Essential Question
How do the signs of factors affect their product?

Try It!

A race car game takes 6 points from a player each time the player hits a cone. What integer represents the change in total points if the player hits 10 cones?
10 • ______ = ______
The change in total points is ______.

If the player hits 10 cones = 60 points.

Explanation:
In the above-given question,
given that,
A race car game takes 6 points from a player each time the player hits a cone.
10 x 6 = 60.
so the change in total points is 60 points.

Convince Me! Could the product of a positive integer and a negative integer be positive? Explain.

Try It!

Find each product.
a. -7 • (-2)

The product is 14.

Explanation:
In the above-given question,
given that,
the two numbers are -7 and -2.
multiply the numbers.
-7 x -2.
14.
so the product is 14.

b. 7 • (-13)

The product is -91.

Explanation:
In the above-given question,
given that,
the two numbers are 7 and -13.
multiply the numbers.
7 x -13.
-91.
so the product is -91.

c.-6 • 8

The product is -48.

Explanation:
In the above-given question,
given that,
the two numbers are -6 and 8.
multiply the numbers.
-6 x 8.
-48.
so the product is -48.

d.(-1) • (-1)

The product is 1.

Explanation:
In the above-given question,
given that,
the two numbers are -1 and -1.
multiply the numbers.
-1 x -1.
1.
so the product is 1.

KEY CONCEPT
When multiplying two integers, the sign of the product depends on the sign of the factors.
If the signs of the factors are the same, the product is positive.
7 • 3 = 21
– 7 • (-3) = 21

If the signs of the factors are different, the product is negative.
-4 • 5 = -20
4 • (-5) = -20

Do You Understand?
Question 1.
Essential Question How do the signs of factors affect their product?

The signs of factors affect their product according to the sign.

Explanation:
In the above-given question,
given that.
If the signs of the factors are the same, the product is positive.
7 x 3 = 21
– 7 x (-3) = 21
If the signs of the factors are different, the product is negative.
-4 x 5 = -20
4 x (-5) = -20

Question 2.
Construct Arguments What is the sign of the product if you multiplied three negative integers? Explain your answer.

The product is also a negative integer.

Explanation:
In the above-given question,
given that.
the numbers are -7, -8, and -3.
-7 x -8 x -3.
-7 -24.
-168.
so the product is also a negative integer.

Question 3.
Reasoning Explain why the product of two negative integers is not negative. Use (-1)(-1) as an example.

yes, the product of two negative integers is positive.

Explanation:
In the above-given question,
given that,
the product of two negative integers is positive.
for example:
-1 x -1 = 1.

Question 4.
Use Structure is the product the same when multiplying 22 × (-5) and multiplying (-5) × 22? Explain.

Yes, the product is the same.

Explanation:
In the above-given question,
given that,
the numbers are 22 and -5.
22 x -5 = 110.
the numbers are -5 and 22.
-5 x 22 = 110.
so both the product is the same.

Do You Know How?
Question 5.
Represent 2 • (-3) on the number line.

The number is -6.

Explanation:
In the above-given question,
given that,
the numbers are 2 and -3.
2 x -3 = -6.
so the number is -6.

Question 6.
Which of these products is negative? Select all that apply.
☐ -8 • (-3)
☐ -2 • 8
☐ 0 • (-2)
☐ 15 • (-5)
☐ -8 • (-9)

-2 x 8 = -16.
15 x -5 = -75.

Explanation:
In the above-given question,
given that,
the product is negative.
the numbers are – 2 and 8.
-2 x 8 = -16.
15 x -5 = -75.

Question 7.
Find each product.
a. -9 • (-4)
b.-7 • 12
c. 8 • (-8)
d. 9 • 15

The products are 36, -84, -64, and 135.

Explanation:
In the above-given question,
given that,
-9 x -4 = 36.
12 x -7 = -84.
8 x -8 = 64.
15 x 9 = 135.
so the products are 36, -84, -64, and 135.

Question 8.
A game show contestant starts a game by answering two questions incorrectly. Each incorrect answer costs the contestant $600. Use a product of two integers to show the point total that would appear for the contestant. Answer: The numbers are$20 x $30. Explanation: In the above-given question, given that, A game show contestant starts a game by answering two questions incorrectly. Each incorrect answer costs the contestant$600.
the product of two integers is $600.$20 x $30. so the numbers are$20 and $30. Practice & Problem Solving In 9-14, multiply. Question 9. (-6) • (-2) Answer: The product is 12. Explanation: In the above-given question, given that, the numbers are -6 and -2. -6 x -2. 12. so the product is 12. Question 10. 4 • (-8) Answer: The product is -32. Explanation: In the above-given question, given that, the numbers are -8 and 4. -8 x 4. -32. so the product is -32. Question 11. 7 • (-5) Answer: The product is -35. Explanation: In the above-given question, given that, the numbers are -5 and -7. -5 x 7. -35. so the product is -35. Question 12. -5 • 2 Answer: The product is -10. Explanation: In the above-given question, given that, the numbers are -5 and 2. -5 x 2. -10. so the product is -10. Question 13. -1 • (-24) Answer: The product is 24. Explanation: In the above-given question, given that, the numbers are -1 and -24. -1 x -24. 24. so the product is 24. Question 14. (5) • (-9) • (-2) Answer: The product is 90. Explanation: In the above-given question, given that, the numbers are 5, -9, and -2. -9 x -2. 18. 18 x 5 = 90. so the product is 90. Question 15. A football team lost the same number of yards on each of 3 consecutive plays. What is the total change in yards from where the team started? Answer: The total change in yards from where the team started = 18 yards. Explanation: In the above-given question, given that, A football team lost the same number of yards on each of 3 consecutive plays. 6 x 3 = 18. so the total change in yards from where the team started = 18 yards. Question 16. a. Find the product. -41 • (-1) Answer: The product is 41. Explanation: In the above-given question, given that, the numbers are -41 and -1. multiply the numbers. -41 x -1 = 41. so the product is 41. b. Construct Arguments Describe how you use the properties of multiplication to find the product. Answer: The properties of multiplication depend on the sign. Explanation: In the above-given question, given that, If the signs of the factors are the same, the product is positive. 7 x 3 = 21 – 7 x (-3) = 21 If the signs of the factors are different, the product is negative. -4 x 5 = -20 4 x (-5) = -20 Question 17. Alex is working to simplify 5 • (-8) • 2. a. What is the product? Answer: The product is -80. Explanation: In the above-given question, given that. the numbers are 5, 2, and -8. multiply the numbers. 5 x -8 x 2 = -80. so the product is -80. b. Suppose Alex found the opposite of the correct product. Describe an error he could have made that resulted in that product. Answer: The opposite of the correct product is 80. Explanation: In the above-given question, given that, the numbers are 5, 2, and -8. multiply the numbers. 5 x -8 x 2 = -80. so the product is -80. Question 18. Which product is greater, (-4) • (-6) or (-7) • (-8)? Explain. Answer: -7 x -8 = 56. Explanation: In the above-given question, given that, the product of two numbers. -4 x -6 = 24. -7 x -8 = 56. so 56 is the greatest product. Question 19. Make Sense and Persevere While playing a board game, Cecilia had to move back 6 spaces 9 times. What integer represents Cecilia’s movement on the board for those 9 turns? Answer: The integer represents Cecilia’s movement on the board for those 9 turns = 54 times. Explanation: In the above-given question, given that, While playing a board game, Cecilia had to move back 6 spaces 9 times. 6 x 9 = 54. so the integer represents Cecilia’s movement on the board for those 9 turns = 54 times. Question 20. Anya makes withdrawals from and deposits into her bank account. a. What integer represents the change in the amount in her account if Anya withdraws$12 once each day for four days?

The integer represents the change in the amount in her account = $48. Explanation: In the above-given question, given that, Anya makes withdrawals and deposits into her bank account.$12 x 4 = $48. so the integer represents the change in the amount in her account =$48.

b. What integer represents the change in the amount in her account if Anya deposits $12 once each day for four days? Answer: The integer represents the change in the amount in her account =$48.

Explanation:
In the above-given question,
given that,
Anya makes withdrawals and deposits into her bank account.
$12 x 4 =$48.
so the integer represents the change in the amount in her account = $48. c. Look for Relationships Explain the difference between the integer for the withdrawals and the integer for the deposits. Answer: The difference is zero. Explanation: In the above-given question, given that, Anya makes withdrawals and deposits into her bank account.$12 x 4 = $48. so the integer represents the change in the amount in her account =$48.
$48 –$48 = 0.

Question 21.
Higher Order Thinking A gold mine has two elevators, one for equipment and one for miners One day, the equipment elevator begins to descend. After 28 seconds, the elevator for the miners begins to descend. What is the position of each elevator relative to the surface after another 14 seconds? At that time, how much deeper is the elevator for the miners?

At 14 seconds the elevator of the miners is 210 feet.

Explanation:
In the above-given question,
given that,
A gold mine has two elevators, one for equipment and one for miners One day, the equipment elevator begins to descend.
After 28 seconds, the elevator for the miners begins to descend.
14 x 4 = 56.
14 x 15 = 210.
so the elevator of the miners is 210 feet.

Assessment Practice

Question 22.
A number line is shown.

Write a multiplication equation that is represented by the number line.

The equation is -35 x 2 = -70.

Explanation:
In the above-given question,
given that,
the number line is moving from 0 to -35.
-35 to -70.
-35 x 2 = -70.
so the equation is -35 x 2 = -70.

Question 23.
Which of these expressions have the same product as (-6) • 7? Select all that apply.
☐ (-3) • 14
☐ 16 • (-3)
☐ -6 • (-7)
☐ 7 • (-6)
☐ 14 • (-3)

-3 x 14, 7 x -6 and 14 x -3.

Explanation:
In the above-given question,
given that,
the product is -6 x 7 = -42.
-3 x 14 = -42.
7 x -6 = -42.
14 x -3 = -42.

### Lesson 1.7 Multiply Rational Numbers

Solve & Discuss It!
Stella is making the United States flag. She has blue fabric, red fabric, and white fabric. Choose a length for the flag. What length of blue fabric would Stella need to make this flag? Explain your thinking.

I can… multiply rational numbers.

Focus on math practices
Be Precise The blue region of the flag is $$\frac{7}{13}$$ the width and $$\frac{2}{5}$$ the length of the flag. What part of the total area is the blue region of the flag?

The total area is the blue region of the flag = 0.2152.

Explanation:
In the above-given question,
given that,
The blue region of the flag is $$\frac{7}{13}$$ the width.
$$\frac{2}{5}$$ the length of the flag.
7/13 x 2/5.
0.538 x 0.4.
0.2152.
so the total area is the blue region of the flag = 0.2152.

Essential Question
How is multiplying rational numbers like multiplying integers?

Try It!

Meghan’s bank account is charged $9.95 per month for an online newspaper subscription. How could you represent the change in her account balance after three months of charges? _____ • -9.95 = ______ After three months, the change in her account balance is$ ________.

The change in her account balance after three months of charges = -20.05.

Explanation:
In the above-given question,
given that,
Meghan’s bank account is charged $9.95 per month for an online newspaper subscription.$9.95 x -30 = -20.05.
so the change in her account balance after three months of charges = -20.05.

Convince Me! Meghan’s bank account is charged 3 times. Without calculating, how can you determine whether this is a negative or positive change to her account? Explain.

Try It!

Find each product.
a. -5.3 • (-2.6)

The product is 13.78.

Explanation:
In the above-given question,
given that,
the numbers are -5.3 and -2.6.
multiply the numbers.
-5.3 x – 2.6.
13.78.

b. $$-\frac{3}{5} \cdot 4 \frac{1}{6}$$

-3/5 x 4(1/6) = -2.46.

Explanation:
In the above-given question,
given that,
the numbers are -3/5 and 25/6.
-0.6 x 4.1.
-2.46.
the product is -2.46.

c. 0.2 • (-1.78)

0.2 x – 1.78 = -0.356.

Explanation:
In the above-given question,
given that,
the numbers are 0.2 x -1.78.
multiply the numbers.
0.2 x -1.78 = -0.356.

d. -2.5 • (70)

The product is -175.

Explanation:
In the above-given question,
given that,
the numbers are 70 x -2.5.
multiply the numbers.
-2.5 x 70 = -175.

KEY CONCEPT
The same rules for multiplying integers apply to multiplying all rational numbers. When multiplying two rational numbers:

• If the signs of the factors are the same, the product is positive.
• If the signs of the factors are different, the product is negative.

Do You Understand?
Question 1.
Essential Question How is multiplying rational numbers like multiplying integers?

The same rules for multiplying integers apply to multiplying all rational numbers.

Explanation:
In the above-given question,
given that,
If the signs of the factors are the same, the product is positive.
If the signs of the factors are different, the product is negative.

Question 2.
How do you multiply a decimal greater than 0 and a fraction less than 0?

The product is negative.

Explanation:
In the above-given question,
given that,
the decimal greater than 0 is 1.5.
a fraction less than 0 is – 2.2.
1.5 x – 2.2 = -3.3.
so the product is negative.

Question 3.
Model with Math How does this number line represent multiplication of a negative number by a positive number? Explain.

-2/3 x -1/3 = 0.108.

Explanation:
In the above-given question,
given that,
the two numbers are -2/3 and -1/3.
multiply the numbers.
-2/3 x -1/3.
-0.6 x – 0.33.
0.108.
-2/3 x -1/3 = 0.108.

Do You Know How?
Question 4.
Use the number line to find the product

The product is -4.5.

Explanation:
In the above-given question,
given that,
the numbers are 3 and -1(1/2).
multiply the numbers.
3 x -3/2.
3 x – 1.5.
-4.5.
so the product is -4.5.

Question 5.
Which of these products is positive? Select all that apply.
☐ -0.2 • (12.5)
☐ $$\frac{1}{12}$$ • $$\left(-6 \frac{1}{2}\right)$$
☐ 3.2 • $$\left(-\frac{1}{900}\right)$$
☐ -3$$\frac{1}{2}$$ • 0
☐ -4.7 • (-1)

-4.7 x -1 = 4.7.

Explanation:
In the above-given question,
given that,
the numbers are -4.7 and -1.
multiply the product.
-4.7 x -1 = 4.7.
so the product is 4.7.

Question 6.
Find the product.
a. -3.1 • (-2.9)

The product is 8.99.

Explanation:
In the above-given question,
given that,
the numbers are -3.1 and -2.9.
multiply the product.
-3.1 x -2.9 = 8.99.
so the product is 8.99.

b. 1$$\frac{1}{2}$$ • $$\left(-\frac{5}{3}\right)$$

1(1/2) x -5/3 = -2.4.

Explanation:
In the above-given question,
given that,
the numbers are 3/2 and -5/3.
multiply the product.
3/2 x – 5/3.
1.5 x – 1.6.
-2.4.
so the product is -2.4.

c. -3$$\frac{1}{2}$$ • 0.5

-3(1/2) x 0.5 = – 1.75.

Explanation:
In the above-given question,
given that,
the numbers are -7/2 and 0.5.
multiply the product.
-7/2 x 0.5.
-3.5 x 0.5 = -1.75.

d. –$$\frac{4}{5}$$ • –$$\frac{1}{8}$$

-4/5 x – 1/8 = 0.1.

Explanation:
In the above-given question,
given that,
the numbers are -4/5 and -1/8.
multiply the product.
-4/5 x -1/8.
– 0.8 x – 0.125 = 0.1.

Practice & Problem Solving

In 7-14, multiply.
Question 7.
(-2.655) • (18.44)

-2.655 x 18.44 = -48.9582.

Explanation:
In the above-given question,
given that,
the numbers are -2.655 and 18.44.
multiply the product.
-2.655 x 18.44 = -48.9582.

Question 8.
-1$$\frac{5}{6}$$ • 6$$\frac{1}{2}$$

-1(5/6) x 6(1/2) = -11.895.

Explanation:
In the above-given question,
given that,
the numbers are -1(5/6) and 6(1/2).
multiply the product.
-11/6 x 13/2.
-1.83 x 6.5.
-11.895.

Question 9.
-2$$\frac{1}{2}$$ • (-1$$\frac{2}{3}$$)

-2(1/2) x -1(2/3) = 4.1.

Explanation:
In the above-given question,
given that,
the numbers are -(5/2) and -(5/3).
multiply the product.
-5/2 = -2.5.
-5/3 = – 1.6.
so the product is 4.1.

Question 10.
-3$$\frac{7}{8}$$ • (-5$$\frac{3}{4}$$)

-3(7/8) x -5(3/4) = 22.28125.

Explanation:
In the above-given question,
given that,
the numbers are -31/8 and -23/4.
multiply the product.
-31/8 x -23/4.
3.875 x 5.75.
22.28125.

Question 11.
-7.5 • -2$$\frac{3}{4}$$

-7.5 x -2(3/4) = 20.625.

Explanation:
In the above-given question,
given that,
the numbers are -7.5 and -11/4.
multiply the product.
-7.5 x – 2.75.
20.625.

Question 12.
-0.6 • (-0.62)

-0.6 x -0.62 = 0.372.

Explanation:
In the above-given question,
given that,
the numbers are -0.6 and -0.62.
multiply the product.
-0.6 x -0.62 = 0.372.

Question 13.
-0.2 • –$$\frac{5}{6}$$

-0.2 x – 5/6 = 0.166.

Explanation:
In the above-given question,
given that,
the numbers are -0.2 and -5/6.
multiply the product.
-0.2 x – 0.83.
0.166.

Question 14.
–$$\frac{5}{6}$$ • $$\frac{1}{8}$$

-5/6 x 1/8 = 0.0375.

Explanation:
In the above-given question,
given that,
the numbers are -5/6 and -1/8.
multiply the product.
-5/6 x -1/8.
-0.83 x – 0.125 = 0.0375.

Question 15.
At the beginning of the season, Jamie pays full price for a ticket to see the Panthers, her favorite baseball team.

a. Represent the total change in the cost of a ticket given their losses.

The total change in the cost of a ticket given their losses = $20.3524. Explanation: In the above-given question, given that, At the beginning of the season, Jamie pays full price for a ticket to see the Panthers, her favorite baseball team. ticket prices decrease by$0.41 for every game the panthers lose this season.
$0.41 x$49.64.
$20.3524. so the total change in the cost of a ticket given their losses =$20.3524.

b. What is the cost of a ticket for the next game they play?

The cost of a ticket for the next game they play = $20.3524. Explanation: In the above-given question, given that, At the beginning of the season, Jamie pays full price for a ticket to see the Panthers, her favorite baseball team. ticket prices decrease by$0.41 for every game the panthers lose this season.
$0.41 x$49.64.
$20.3524. so the total change in the cost of a ticket given their losses =$20.3524.

Question 16.
The price per share of ENVX stock is dropping at a rate of $1.45 each hour. a. Write the rate as a negative number. Answer: The rate as a negative number =$55.274.

Explanation:
In the above-given question,
given that,
The price per share of ENVX stock is dropping at a rate of $1.45 each hour.$1.45 x 38.12 = $55.274. so the rate as a negative number =$55.274.

b. What rational number represents the change in the price per share after 5 hours?

The rational number represents the change in the price per share after 5 hours = 190.6.

Explanation:
In the above-given question,
given that,
the rational number is 38.12.
38.12 x 5 = 190.6.
so the rational number represents the change in the price per share after 5 hours = 190.6.

c. What is the price per share after 5 hours?

The price per share after 5 hours = 953.

Explanation:
In the above-given question,
given that,
the price per share after 5 hours is:
190.6 x 5 = 953.
so the price per share after 5 hours = 953.

Question 17.
Ming incorrectly says that this product is $$\frac{4}{63}$$.
$$-\left(-\frac{4}{9}\right)$$ • $$\left(-\frac{1}{7}\right)$$
a. What is the correct product?

The correct product is 0.0616.

Explanation:
In the above-given question,
given that,
Ming incorrectly says that this product is $$\frac{4}{63}$$.
4/63 = 0.063.
-4/9 x -1/7.
– 0.44 x – 0.14.
0.0616.
so the correct product is 0.0616.

b. What error could Ming have made?

The error could ming have made = 0.0616.

Explanation:
In the above-given question,
given that,
Ming incorrectly says that this product is $$\frac{4}{63}$$.
4/63 = 0.063.
-4/9 x -1/7.
– 0.44 x – 0.14.
0.0616.
so the correct product is 0.0616.

Question 18.
Higher Order Thinking place the products in order from least to greatest.
$$4 \frac{4}{7}$$ • $$4 \frac{4}{7}$$
$$5 \frac{6}{7}$$ • $$\left(-6 \frac{6}{7}\right)$$
$$-5 \frac{1}{8}$$ • $$\left(-2 \frac{1}{4}\right)$$

-39.44, 11.53, and 20.894.

Explanation:
In the above-given question,
given that,
4(4/7) x 4(4/7).
32/7 x 32/7.
4.571 x 4.571.
20.894.
41/7 x -48/7.
5.8 x – 6.8.
-39.44.
-41/8 x -9/4.
-5.125 x -2.25.
11.53.
so the numbers from least to greatest are -39.44, 11.53, and 20.894.

Assessment Practice

Question 19.
Suppose there is a 1.3°F drop in temperature for every thousand feet that an airplane climbs into the sky. The temperature on the ground is -2.8°F.
PART A
Write a multiplication equation to represent the change in temperature after the plane ascends 10,000 feet.

The equation represents the change in temperature after the plane ascends 10,000 feet = 4.1°F.

Explanation:
In the above-given question,
given that,
Suppose there is a 1.3°F drop in temperature for every thousand feet that an airplane climbs into the sky.
The temperature on the ground is -2.8°F.
1.3°F – (-2.8°F).
1.3°F + 2.8°F.
4.1°F.

PART B
What will the temperature be when the plane reaches an altitude of 10,000 feet?
A. -15.8
B. -10.2
C. 10.2
D. 15.8

The temperature is when the plane reaches an altitude of 10,000 feet = -15.8.

Explanation:
In the above-given question,
given that,
The temperature on the ground is -2.8°F.
1.3°F – (-2.8°F).
1.3°F + 2.8°F.
4.1°F.
so option B is correct.

### Lesson 1.8 Divide Integers

Explain It!
The shapes below are used to show the relationship between each of the four equations in the same fact family.
I can… divide integers.

A. Suppose the star represents -24. What values could the other shapes represent?

Square represents 8.
the circle represents 3.

Explanation:
In the above-given question,
given that,
8 x 3 = 24.
3 x 8 = 24.
24 / 3 = 8.
24 / 8 = 3.
so the square represents 8.
the circle represents 3.

B. What do you know about the square and circle if the star represents a negative number?

The square and circle if the star represents a negative number.

Explanation:
In the above-given question,
given that,
8 x 3 = 24.
3 x 8 = 24.
24 / 3 = 8.
24 / 8 = 3.
so the square represents 8.
the circle represents 3.

C. What do you know about the star if the square and circle both represent a negative number?

The square and circle both represent a positive numbers.

Explanation:
In the above-given question,
given that,
if the square and circle both represent a negative number.
so the square and circle both represent positive numbers.
for example:
– 3 x -4 = 12.

Focus on math practices
Use Structure Suppose the square represents -8 and the circle represents 3. Use what you know about integer multiplication and the relationship between multiplication and division to write the complete fact family.

Essential Question
How does dividing integers relate to multiplying integers?

Try It!

Suppose the machine drilled the same distance into the ground for 3 days and reached water at 84 feet below ground level. What was the change in the location of the bottom of the hole each day?

Each day, the location of the bottom of the hole changed by _______ feet, or decreased by ________ feet.

-84/3 = -28.
3 x 9 = 28.
3 x -28 = -84.
-84/3 = -28.

Explanation:
In the above-given question,
given that,
the machine drilled the same distance into the ground for 3 days and reached water at 84 feet below ground level.
84/3 = 28.
so the location of the bottom of the hole changed by 28 feet or decreased by 28 feet.

Convince Me! Explain why the quotient of two integers with different signs is negative.

Try It!

Simplify.
a. -40 ÷ (-5)

-40 / -5 = 8.

Explanation:
In the above-given question,
given that,
the numbers are -40 and -5.
divide the numbers.
-40 / -5 = 8.

b. 40 ÷ (-5)

40 / -5 = -8.

Explanation:
In the above-given question,
given that,
the numbers are 40 and -5.
divide the numbers.
40 / -5 = -8.

c. 0 ÷ -40

0 / 40 = 0.

Explanation:
In the above-given question,
given that,
the numbers are 0 and 40.
divide the numbers.
0 / 40 = 0.

Try It!

Which of the following are equivalent to -5?

The following are equivalent to -5 is -(55/11), -55/11, 55/-11, and -(-55/-11).

Explanation:
In the above-given question,
given that,
the numbers are 55 and -11.
divide the numbers.
-55/11 = -5.
-(55/11) = -5.
55/-11 = -5.
-(-55/-11) = -5.

KEY CONCEPT
The rules for dividing integers are related to the rules for multiplying integers.
If the signs of the dividend and the divisor are the same, the quotient is positive.
24 ÷ 4 = 6
-24 ÷ (-4) = 6

If the signs of the dividend and the divisor are different, the quotient is negative.
-15 ÷ 3 = -5
15 ÷ (-3) = -5

Do You Understand?
Question 1.
Essential Question How does dividing integers relate to multiplying integers?

The integers are related to multiplying integers depending on their signs.

Explanation:
In the above-given question,
given that,
If the signs of the dividend and the divisor are the same, the quotient is positive.
24 ÷ 4 = 6
-24 ÷ (-4) = 6
If the signs of the dividend and the divisor are different, the quotient is negative.
-15 ÷ 3 = -5
15 ÷ (-3) = -5

Question 2.
Reasoning Why is the quotient of two negative integers positive?

Yes, the quotient of two negative integers is positive.

Explanation:
In the above-given question,
given that,
the numbers are -30 and -3.
divide the numbers.
-30 / -3 = 10.
on both sides, the signs get canceled.

Question 3.
Helen wrote the following facts to try to show that division by 0 results in 0. Explain her error.

Yes, the division by 0 results in 0.

Explanation:
In the above-given question,
given that,
0 x -7 = 0.
-7 / 0 = 0.
so the division by 0 results in 0.

Do You Know How?
Question 4.
Find each quotient.
a. –$$\frac{18}{3}$$

-18/3 = -6.

Explanation:
In the above-given question,
given that,
the number is -18/3.
divide the number.
-18/3 = -6.

b. $$\frac{-5}{-1}$$

-5/-1 = 5.

Explanation:
In the above-given question,
given that,
the number is -5/-1.
divide the number.
-5/-1 = 5.

c. $$\frac{24}{-6}$$

24/-6 = -4.

Explanation:
In the above-given question,
given that,
the number is 24/-6.
divide the number.
24/-6 = -4.

d. $$\frac{-10}{-1}$$

-10/-1 = 10.

Explanation:
In the above-given question,
given that,
the number is -10/-1.
divide the number.
-10/-1 = 10.

e. $$\frac{-25}{5}$$

-25/5 = -5.

Explanation:
In the above-given question,
given that,
the number is -25/5.
divide the number.
-25/5 = -5.

f. –$$\frac{8}{2}$$

-8/2 = -4.

Explanation:
In the above-given question,
given that,
the number is -8/2.
divide the number.
-8/2 = -4.

Question 5.
A scuba diver descends 63 feet in 18 seconds. What integer represents the change in the diver’s position in feet per second?

The integer represents the change in the diver’s position in feet per second = 3.5 feet.

Explanation:
In the above-given question,
given that,
A scuba diver descends 63 feet in 18 seconds.
63/18 = 3.5.
so the integer represents the change in the diver’s position in feet per second = 3.5 feet.

Question 6.
Which of the following are equivalent to -7?

49/-7 = -7 and -21/3 = -7.

Explanation:
In the above-given question,
given that,
the quotient is -7.
49/-7 = -7.
-21/3 = -7.

Practice & Problem Solving

Leveled Practice In 7-8, fill in the boxes to find each quotient.
Question 7.
-16 ÷ 4 = ?
4 • ? = ______
4 • _____ = _______
So, -16 ÷ 4 = ________.

-16/4 = -4.

Explanation:
In the above-given question,
given that,
the numbers are -16 and 4.
divide the numbers.
-16 / 4 = -4.
the quotient is -4.

Question 8.
-56 ÷ – 7 = ?
______ • ? = _______
______ • ______ = ________
So, -56 ÷ -7 = ________

-56 / -7 = 8.

Explanation:
In the above-given question,
given that,
the numbers are -56 and -7.
divide the numbers.
-56/ -7 = 8.

Question 9.
Classify the quotient -50 ÷ 5 as positive, negative, zero, or undefined.

The quotient is -10.

Explanation:
In the above-given question,
given that,
the numbers are -50 and -5.
divide the numbers.
-50/5 = -10.

Question 10.
Is the expression $$\frac{42}{-7}$$ undefined? If not, find the quotient.

The quotient is -6.

Explanation:
In the above-given question,
given that,
the numbers are 42 and -7.
divide the numbers.
42/-7 = -6.
so the quotient is -6.

Question 11.
A company loses $780 as a result of a shipping delay. The 6 owners of the company must share the loss equally. a. Write an expression to show the change in profit for each owner. Answer: The change in profit for each owner =$130.

Explanation:
In the above-given question,
given that,
A company loses $780 as a result of a shipping delay. The 6 owners of the company must share the loss equally.$780 / 6 = $130. so the change in profit for each owner =$130.

b. Evaluate the expression.

The change in profit for each owner = $130. Explanation: In the above-given question, given that, A company loses$780 as a result of a shipping delay.
The 6 owners of the company must share the loss equally.
$780 / 6 =$130.
so the change in profit for each owner = $130. Question 12. Which of the quotients are equivalent to 2.5? Select all that apply. Answer: -5/12, 10/4, -10/ -4, and 5/2 are equivalent to 2.5. Explanation: In the above-given question, given that, the quotient equal to 2.5 is 10/-4 = 5/-2 = -2.5. -5/-2 = 2.5. 10/4 = 2.5. -10/-4 = 2.5. 5/2 = 2.5. Question 13. Use Structure The price of a stock steadily decreased by a total of$127 over 15 months. Which expression shows the change in the stock’s value?

Option A is correct.

Explanation:
In the above-given question,
given that,
The price of a stock steadily decreased by a total of $127 over 15 months. -$127/ -15 months.
so option A is correct.

Question 14.
Zak goes parachuting and descends at the rate shown. If he maintains a steady descent, what integer represents Zak’s change in elevation in feet per second?

The integer represents Zak’s change in elevation in feet per second = 12 feet.

Explanation:
In the above-given question,
given that,
Zak goes parachuting and descends at the rate shown.
24 feet in 2 seconds.
24/2 = 12.
so the integer represents Zak’s change in elevation in feet per second = 12 feet.

Question 15.
Model with Math Find each quotient and plot it on the number line. Which of the expressions are undefined?

The expressions are undefined = 0/-8.

Explanation:
In the above-given question,
given that,
-8/4 = 2.
-21/ -7 = 3.
-4/0 = 0.
-25/-5 = 5.
36/-9 = -4.
9/0 = 9.
0/-8 = 0.

Question 16.
Use Structure The temperature in a town increased 16°F in 5 hours. The temperature decreased 31°F in the next 8 hours. Which of the expressions shows the rate of the total change in temperature?
A. $$\frac{-15^{\circ} \mathrm{F}}{13 \text { hours }}$$
B. $$\frac{47^{\circ} \mathrm{F}}{13 \text { hours }}$$
C. $$\frac{15^{\circ} \mathrm{F}}{13 \text { hours }}$$
D. $$\frac{47{\circ} \mathrm{F}}{-13 \text { hours }}$$

Option C is correct.

Explanation:
In the above-given question,
given that,
The temperature in a town increased by 16°F in 5 hours.
The temperature decreased 31°F in the next 8 hours.
31 – 16 = 15.
5 + 8 = 13.
so option C is correct.

Question 17.
Camille takes a rock-climbing class. On her first outing, she rappels down the side of a boulder in three equal descents. What integer represents Camille’s change in altitude in feet each time she descends?

The integer represents Camille’s change in altitude in feet each time she descends = 165 feet.

Explanation:
In the above-given question,
given that,
Camille takes a rock-climbing class.
On her first outing, she rappels down the side of a boulder in three equal descents.
165 feet.
so the integer represents Camille’s change in altitude in feet each time she descends = 165 feet.

Question 18.
Higher Order Thinking If the fraction $$\frac{396}{x-10}$$ is equivalent to -22, find the value of x. Show your work.

x = 9.94.

Explanation:
In the above-given question,
given that,
396 x x – 10 = -22.
396x – 3960 = -22.
396x = -22 + 3960.
396x = 3938.
x = 3938/396.
x = 9.94.

Assessment Practice

Question 19.
Which of the quotients is equivalent to $$-\frac{5}{8}$$? Select all that apply.
☐ $$\frac{-5}{8}$$
☐ $$\frac{5}{8}$$
☐ $$\frac{5}{-8}$$
☐ $$-\left(\frac{5}{-8}\right)$$
☐ $$\frac{-5}{-8}$$

-5/8, and 5/-8.

Explanation:
In the above-given question,
given that,
-5/8 = -0.625.
5/-8 = -0.625.

Question 20.
Which of the following pairs of quotients are equivalent?
A. $$\frac{-4}{5}$$ and $$-\left(\frac{20}{25}\right)$$
B. $$-\left(\frac{2}{-3}\right)$$ and $$\frac{-4}{6}$$
C. $$\frac{-5}{7}$$ and $$\frac{35}{-40}$$
D. $$\frac{1}{5}$$ and $$-\left(\frac{-2}{-10}\right)$$

Option A is correct.

Explanation:
In the above-given question,
given that,
the pairs of quotients are equivalent.
-4/5 = -20/25.
5 x 4 = 20.
5 x 5 = 25.
so option A is correct.

Question 21.
An elevator descends 36 feet in 3 seconds. What integer represents the elevator’s change in elevation in feet per a second?

The integer represents the elevator’s change in elevation in feet per second = 12 seconds.

Explanation:
In the above-given question,
given that,
An elevator descends 36 feet in 3 seconds.
36/3 = 13.
so the integer represents the elevator’s change in elevation in feet per second = 12 seconds.

### Lesson 1.9 Divide Rational Numbers

Explore It!
The number line shows the movement of a glacier that retreats 8 meters every year.
I can… divide rational numbers.

A. How could you use division to represent the yearly change in the glacier’s position over the next 4 years?

The change in the glacier’s position over the next 4 years = -32.

Explanation:
In the above-given question,
given that,
The number line shows the movement of a glacier that retreats 8 meters every year.
4 x -8 = -32.
-32/-8 = 4.
so the change in the glacier’s position over the next 4 years = -32.

B. How could you use division to represent the yearly change in the glacier’s position over the past 4 years?

The change in the glacier’s position over the past 4 years = -32.

Explanation:
In the above-given question,
given that,
The number line shows the movement of a glacier that retreats 8 meters every year.
4 x -8 = -32.
-32/-8 = 4.
so the change in the glacier’s position over the past 4 years = -32.

C. Suppose the glacier retreated 8.25 meters every year. Draw a number line to represent this movement.

The change in the glacier’s position over the past 4 years = 33.

Explanation:
In the above-given question,
given that,
The number line shows the movement of a glacier that retreats 8 meters every year.
4 x 8.25 = 33.
33/8.25 = 4.
so the change in the glacier’s position over the past 4 years

Focus on math practices
Reasoning If the number of meters the glacier retreats each year changes, does it affect the signs of each part of the division statement in Part A? Explain.

Essential Question
How is dividing rational numbers like dividing integers?

Try It!

Suppose that the volume of water in the rain barrel decreased by 4$$\frac{5}{8}$$ gallons in 4 minutes. What will be the change in the volume of water after 1 minute?

The rain barrel will lose _________ gallons in 1 minute.

The rain barrel will lose -37/32 gallons in 1 minute.

Explanation:
In the above-given question,
given that,
suppose that the volume of water in the rain barrel decreased by 4$$\frac{5}{8}$$ gallons in 4 minutes.
-4(5/8) /4.
-37/8 x 4.
-37/32.
so the rain barrel will lose -37/32 gallons in 1 minute.

Convince Me! How are multiplicative inverses used in division with rational numbers?

Try It!

Find each quotient.
a. $$\frac{1 \frac{2}{5}}{-\frac{1}{5}}$$

1(2/5) – 1/5 = 1.2.

Explanation:
In the above-given question,
given that,
7/5 – 1/5.
1.4 – 0.2.
1.2.

b. -0.4 – 0.25

-0.4 – 0.25 = 0.65.

Explanation:
In the above-given question,
given that,
the two numbers are -0.4 and -0.25.
subtract the numbers.
-0.4 – 0.25.
0.65.

c. $$\frac{7}{8} \div-\frac{3}{4}$$

7/8 – 3/4 = 0.125.

Explanation:
In the above-given question,
given that,
the two numbers are 7/8 and -3/4.
subtract the numbers.
7/8 = 0.875.
-3/4 = 0.75.
0.875 – 0.75.
0.125.

d. $$0.7 \div-1 \frac{1}{6}$$

0.7 / 1(1/6) = 0.6.

Explanation:
In the above-given question,
given that,
the two numbers are 0.7 and -7/6.
divide the numbers.
0.7/ 1.16.
0.6.

Try It!

Find each quotient.
a. $$-1 \frac{1}{3} \div(-1.6)$$

The quotient is 0.83.

Explanation:
In the above-given question,
given that,
the two numbers are -1(1/3) and -1.6.
divide the numbers.
-4/3 / -1.6.
-1.33/ -1.6.
0.83.

b. $$\frac{-\frac{2}{3}}{-\frac{1}{4}}$$

The quotient is 2.4.

Explanation:
In the above-given question,
given that,
the two numbers are -(2/3) and -1/4.
divide the numbers.
-2/3 / -1/4.
-0.6/-0.25.
2.4.

c. $$-\frac{9}{10} \div\left(-\frac{3}{10}\right)$$

The quotient is 3.

Explanation:
In the above-given question,
given that,
the two numbers are -9/10 and -3/10.
divide the numbers.
-9/10/-3/10.
-0.9/-0.3.
3.

d. $$-0.5 \div\left(-\frac{3}{13}\right)$$

The quotient is 2.17.

Explanation:
In the above-given question,
given that,
the two numbers are -0.5 and -3/13.
divide the numbers.
-0.5/ -0.23.
2.17.

KEY CONCEPT
The same rules for dividing integers apply to dividing rational numbers. When dividing two rational numbers:

• If the signs of the dividend and divisor are the same, the quotient is positive.
• If the signs of the dividend and divisor are different, the quotient is negative.

Do You Understand?
Question 1.
Essential Question How is dividing rational numbers like dividing integers?

The division of rational numbers like dividing integers depends upon size.

Explanation:
In the above-given question,
given that,
If the signs of the dividend and divisor are the same, the quotient is positive.
If the signs of the dividend and divisor are different, the quotient is negative.
for example:
12/3 = 4.
-12/3 = -4.

Question 2.
Use Structure How do you know the sign of the quotient $$-\frac{4}{5} \div \frac{1}{6}$$?

-4/5 / 1/6 = -8.

Explanation:
In the above-given question,
given that,
the two numbers are -4/5 and 1/6.
divide the numbers.
-4/5 / 1/6.
-0.8 / 0.1.
-8.

Question 3.
Reasoning When -4 is divided by a rational number between 0 and 1, where would the quotient be located on the number line? Why?

The quotient will be located on the number line is -4.

Explanation:
In the above-given question,
given that,
When -4 is divided by a rational number between 0 and 1.
-4/0 = -4.
-4/1 = -4.

Do You Know How?
Question 4.
Find each quotient.
a. $$-\frac{7}{12} \div \frac{1}{7}$$

-7/12 / 1/7 = -4.14.

Explanation:
In the above-given question,
given that,
the two numbers are -7/12 and 1/7.
divide the numbers.
-7/12 / 1/7.
-0.58 / 0.14.
-4.14.

b. $$-0.05 \div\left(-\frac{5}{8}\right)$$

-0.05 / -5/8 =0.08.

Explanation:
In the above-given question,
given that,
the two numbers are -0.05 and -5/8.
divide the numbers.
-0.05 / -5/8.
-0.05 / -0.625.
0.08.

c. $$6 \frac{1}{4} \div\left(-\frac{5}{16}\right)$$

6(1/4) / -5/16 = -20.03.

Explanation:
In the above-given question,
given that,
the two numbers are 6(1/4) and -5/16.
divide the numbers.
25/4 / -5/16.
6.25 / -0.3125.
-20.03.

d. $$-1 \div\left(-\frac{10}{13}\right)$$

-1 / -10/3 = 0.300.

Explanation:
In the above-given question,
given that,
the two numbers are -1 and -10/3.
divide the numbers.
-1/ -10/3.
-1 / -3.33.
0.300.

Question 5.
Simplify the complex fraction.
a. $$\frac{-\frac{2}{7}}{1 \frac{1}{3}}$$

-2/7 / 4/3 = -0.21.

Explanation:
In the above-given question,
given that,
the two numbers are -2/7 and 1(1/3).
divide the numbers.
-2/7 / 4/3.
-0.28 / 1.3.
-0.21.

b. $$\frac{-\frac{3}{5}}{2 \frac{1}{4}}$$

-3/5 / 9/4 = -0.26.

Explanation:
In the above-given question,
given that,
the two numbers are -3/5 and 2(1/4).
divide the numbers.
-3/5 / 9/4.
– 0.6 / 2.25.
-0.26.

c. $$\frac{-\frac{9}{10}}{1 \frac{3}{5}}$$

-9/10 / 8/5 = – 0.5625.

Explanation:
In the above-given question,
given that,
the two numbers are -9/10 and 8/5.
divide the numbers.
-9/10 / 8/5.
-0.9 / 1.6.
-0.5625.

Practice & Problem Solving

Leveled Practice In 6-7, fill in the boxes to find the quotient.
Question 6.
Find the quotient $$\frac{5}{7} \div\left(-\frac{11}{5}\right)$$.

5/7 / -11/5 = -0.32.

Explanation:
In the above-given question,
given that,
the two numbers are 5/7 and -11/5.
divide the numbers.
5/7/ – 11/5.
0.71 / -2.2.
– 0.32.

Question 7.
Simplify the complex fraction $$\frac{-\frac{4}{5}}{\frac{3}{10}}$$
Rewrite the complex fraction ______ ÷ _______
Write the division as multiplication: ______ • _______
The product is _______ .

The product is -0.24.

Explanation:
In the above-given question,
given that,
the two numbers are -4/5 and 3/10.
divide the numbers.
-4/5 x 3/10.
-0.8 x 0.3.
-0.24.
so the product is -0.24.

Question 8.
Which multiplication expression is equivalent to the division expression $$-\frac{7}{17} \div \frac{13}{34}$$?

Option A is correct.

Explanation:
In the above-given question,
given that,
the two numbers are -7/17 and 13/34.
divide the numbers.
-7/17 x 13/34.
-0.411 x 0.38.
-0.156.
so option A is correct.

Question 9.
Derek says that the quotient
$$-\frac{2}{7} \div\left(-\frac{2}{21}\right) \text { is }-\frac{1}{3} \text { . }$$
a. What is the correct quotient?

The correct quotient is 3.11.

Explanation:
In the above-given question,
given that,
Derek says that the quotient
$$-\frac{2}{7} \div\left(-\frac{2}{21}\right) \text { is }-\frac{1}{3} \text { . }$$.
-2/7 / -2/21.
-0.28/ -0.09.
3.11.

b. What mistake did Derek likely make?

The two expressions are not equal.

Explanation:
In the above-given question,
given that,
Derek says that the quotient
$$-\frac{2}{7} \div\left(-\frac{2}{21}\right) \text { is }-\frac{1}{3} \text { . }$$.
-2/7 / -2/21.
-0.28/ -0.09.
3.11.

Question 10.
The water level of a lake fell by $$1 \frac{1}{2}$$ inches during a $$1 \frac{2}{3}$$-week-long dry spell. Simplify the complex fraction below to find the average rate at which the water level changed every week.
$$\frac{-1 \frac{1}{2}}{1 \frac{2}{3}}$$ inches/week

The average rate at which the water level changed every week = 0.9375 meters.

Explanation:
In the above-given question,
given that,
The water level of a lake fell by $$1 \frac{1}{2}$$ inches during a $$1 \frac{2}{3}$$-week-long dry spell.
1(1/2) / 1(2/3).
3/2 / 5/3.
1.5 / 1.6.
0.9375.
so the average rate at which the water level changed every week = 0.9375 meters.

Question 11.
Complete the table. Simplify expressions.

The quotients are the same.

Explanation:
In the above-given question,
given that,
-3/4 / 2/5.
-0.75 / 0.4 = -1.875.
-0.75/ 0.4.
-1.875.
3/4 / -2/5 = -1.875.

Question 12.
a. Find the reciprocal of $$-1 \frac{1}{17}$$.

The reciprocal of -1(1/17) = 1.05.

Explanation:
In the above-given question,
given that,
-1(1/17).
-18/17.
-1.05.

b. Find the reciprocal of $$-\frac{17}{18}$$.

The reciprocal of -17/18 = -1.05.

Explanation:
In the above-given question,
given that,
the reciprocal of -17/18 is -18/17.
-18/17 = -1.05.

c. Reasoning Explain why the answer for part a is the multiplicative inverse of the answer for part b.

The answer for part a is the multiplicative inverse of the answer for part b.

Explanation:
In the above-given question,
given that,
the reciprocal of -17/18 is -18/17.
-18/17 = -1.05.

Question 13.
Use numbers $$-\frac{7}{13}, 1 \frac{6}{7},-1 \frac{6}{7}, \frac{7}{13}$$
a. Which is the reciprocal of $$1 \frac{6}{7}$$?

The reciprocal of 1(6/7) = 7/13.

Explanation:
In the above-given question,
given that,
1(6/7).
13/7.
13/7 = 1.85.

b. Which is the reciprocal of $$\frac{7}{13}$$?

The reciprocal of 7/13 = 13/7.

Explanation:
In the above-given question,
given that,
the numbers are 7 and 13.
divide the numbers.
7/13 = 0.538.
13/7 = 1.85.

c. Reasoning What do you notice about the reciprocals of $$1 \frac{6}{7}$$ and $$\frac{7}{13}$$?

The reciprocals of 1(6/7) and 7/13 is

Explanation:
In the above-given question,
given that,
the number is 1(6/7).
13/7 = 7/13.
the reciprocals are the same.

Question 14.

a. Use a complex fraction to represent the change in the volume of water in 1 minute.

The change in the volume of water in 1 minute = 1.6 ml.

Explanation:
In the above-given question,
given that,
1(3/5) = 8/5.
8/5 = 1.6.
so the change in the volume of water in 1 minute = 1.6 ml.

b. Simplify the complex fraction to find the change in the volume of water in the tank in 1 minute.

The change in the volume of water in 1 minute = 1.6 ml.

Explanation:
In the above-given question,
given that,
1(3/5) = 8/5.
8/5 = 1.6.
so the change in the volume of water in 1 minute = 1.6 ml.

Question 15.
$$\frac{3}{10} \div 3.8$$

The quotient is 1.14.

Explanation:
In the above-given question,
given that,
the two numbers are 3/10 and 3.8.
divide the numbers.
3/10 = 0.3.
0.3 x 3.8 = 1.14.

Question 16.
Higher Order Thinking Between 10 P.M. and 7:45 A.M., the water level in a swimming pool decreased by $$\frac{13}{16}$$ inch.
Assuming that the water level decreased at a constant rate, how much did it drop each hour?
The water level decreased by ______ inch each hour.

The water level decreased by 0.8125 inches each hour.

Explanation:
In the above-given question,
given that,
Between 10 P.M. and 7:45 A.M., the water level in a swimming pool decreased by $$\frac{13}{16}$$ inch.
13/16.
0.8125.
so the water level decreased by 0.8125 inches each hour.

Question 17.
Critique Reasoning Kayla wants to find $$2 \frac{2}{3} \div\left(-1 \frac{3}{7}\right)$$. She first rewrites the division as $$\left(2 \frac{2}{3}\right)\left(-1 \frac{7}{3}\right)$$ What is wrong with Kayla’s reasoning?

The wrong with Kayla’s reasoning = -1.8309.

Explanation:
In the above-given question,
given that,
Kayla wants to find $$2 \frac{2}{3} \div\left(-1 \frac{3}{7}\right)$$.
2(2/3) / -1(3/7).
8/3 / -10/7.
2.6 / – 1.42.
-1.8309.

Assessment Practice

Question 18.
Which is an equivalent multiplication expression for $$\frac{-\frac{3}{8}}{\left(-\frac{7}{54}\right)}$$?

Option A is correct.

Explanation:
In the above-given question,
given that,
$$\frac{-\frac{3}{8}}{\left(-\frac{7}{54}\right)}$$.
-3/8 / -7/54.
-3/8 x 7/54.
so option A is correct.

Question 19.
Which is NOT a step you perform to divide $$-2 \frac{1}{8} \div 6 \frac{4}{5}$$. Select all that apply.
☐ Rewrite the mixed numbers as fractions.
☐ Divide 8 by 4.
☐ Multiply by the multiplicative inverse of a $$\frac{34}{5}$$
☐ Multiply by the multiplicative inverse of $$\frac{17}{8}$$
☐ Multiply 8 and 34.

Rewrite the mixed numbers as fractions.

Explanation:
In the above-given question,
given that,
$$-2 \frac{1}{8} \div 6 \frac{4}{5}$$.
-2(1/8) / 6(4/5).
-17/8 / 34/5.
-2.125 / 6.8.
-0.3125.

### Lesson 1-10 Solve Problems with Rational Numbers

Solve & Discuss It!
Stefan estimates the income and expenses for renting a phone accessory store in the mall. He enters the amounts in the table below. Should Stefan rent a phone accessory store? Explain.

I can… solve problems with rational numbers.

Focus on math practices
Reasoning How can you assess the reasonableness of your solution using mental math or estimation strategies?

Essential Question
How do you decide which rational number operations to use to solve problems?

Try It!

A weather balloon ascended from an elevation of 18 feet below sea level to an elevation of 19$$\frac{1}{2}$$ feet above sea level. What distance did the weather balloon rise?
The distance between two points is the absolute value of their __________.

The weather balloon rose a distance of ________ feet.

The weather balloon rose a distance of 1.5 feet.

Explanation:
In the above-given question,
given that,
A weather balloon ascended from an elevation of 18 feet below sea level to an elevation of 19$$\frac{1}{2}$$ feet above sea level.
-18 + 19(1/2).
-18 + 39/2.
-18 + 19.5.
1.5.

Convince Me! How can you decide which operation to use to solve a problem?

Try It!

Rashida’s score is 6.

Explanation:
In the above-given question,
given that,
18 – 12 = 6.
so Rashida’s score is 6.

Try It!

The temperature at 10:00 A.M. was -3°F and increased 2.25°F each hour for the next 5 hours. What was the temperature at 3:00 P.M.?

The temperature at 3:00 P.M = -1.25°F.

Explanation:
In the above-given question,
given that,
The temperature at 10:00 A.M. was -3°F and increased 2.25°F each hour for the next 5 hours.
-3 + 2.25.
-1.25.
so the temperature at 3:00 P.M = – 1.25°F.

KEY CONCEPT
You can solve a problem with rational numbers by making sense of the problem and deciding which operations to use.

Do You Understand?
Question 1.
Essential Question How do you decide which rational number operations to use to solve problems?

Question 2.
Reasoning A truck’s position relative to a car’s position is -60 feet. The car and the truck move in the same direction, but the car moves 5 feet per second faster for 8 seconds. What operations could be used to find the truck’s relative position after 8 seconds? Explain.

The operations could be used to find the truck’s relative position after 8 seconds = -47 feet.

Explanation:
In the above-given question,
given that,
A truck’s position relative to a car’s position is -60 feet.
The car and the truck move in the same direction, but the car moves 5 feet per second faster for 8 seconds.
-60 + 5 + 8.
-47.
so the operations could be used to find the truck’s relative position after 8 seconds = -47 feet.

Question 3.
Construct Arguments Emilio used addition of two rational numbers to solve a problem. Jim used subtraction to solve the same problem. Is it possible that they both solved the problem correctly? Use a specific example to explain.

Yes, they both have solved the problem correctly.

Explanation:
In the above-given question,
given that,
Emilio used the addition of two rational numbers to solve a problem.
for example:
4 + 3 = 7.
Jim used subtraction to solve the same problem.
4 – 3 = 1.

Do You Know How?
Question 4.
Kara had a savings account balance of $153 on Monday. On Tuesday, she had six withdrawals of$15.72 and a deposit of $235.15. What was her account balance after these transactions? Answer: The account balance after these transactions =$66.43.

Explanation:
In the above-given question,
given that,
Kara had a savings account balance of $153 on Monday. On Tuesday, she had six withdrawals of$15.72 and a deposit of $235.15.$153 + $15.72 =$168.72.
$235.15 –$168.72 = $66.43. so the account balance after these transactins =$66.43.

Question 5.
A scuba diver is swimming at the depth shown, and then swims 0.5 foot toward the surface every 3 seconds. What is the location of the scuba diver, relative to the surface, after 15 seconds?

The location of the scuba diver, relative to the surface, after 15 seconds = 22.5 feet.

Explanation:
In the above-given question,
given that,
A scuba diver is swimming at the depth shown and then swims 0.5 feet toward the surface every 3 seconds.
0.5 x 3 = 1.5.
1.5 x 15 = 22.5.
so the location of the scuba diver, relative to the surface, after 15 seconds = 22.5 feet.

Question 6.
The temperature of a cup of coffee changed by -54°F over 22$$\frac{1}{2}$$ minutes. What was the change in temperature each minute?

The change in temperature each minute = 22.5 minutes.

Explanation:
In the above-given question,
given that,
The temperature of a cup of coffee changed by -54°F over 22$$\frac{1}{2}$$ minutes.
22(1/2).
44 + 1 = 45.
45/2 = 22.5.
so the change in temperature each minute = 22.5 minutes.

Practice & Problem Solving

Question 7.
Suppose there is a 1.1°F drop in temperature for every thousand feet that an airplane climbs into the sky. If the temperature on the ground is 59.7°F, what will be the temperature at an altitude of 11,000 ft?

The temperature at an altitude of 11,000 ft = 656700°F.

Explanation:
In the above-given question,
given that,
Suppose there is a 1.1°F drop in temperature for every thousand feet that an airplane climbs into the sky.
If the temperature on the ground is 59.7°F.
11000 x 59.7°F.
656700°F
so the temperature at an altitude of 11,000 ft = 656700°F.

Question 8.
A farmer sells an average of 15$$\frac{3}{5}$$ bushels of corn each day. What integer represents the change in bushels of corn in his inventory after 6 days?

The integer represents the change in bushels of corn in his inventory after 6 days = 93.6 bushels.

Explanation:
In the above-given question,
given that,
A farmer sells an average of 15$$\frac{3}{5}$$ bushels of corn each day.
15(3/5).
75 + 3 = 78.
78/5 = 15.6.
15.6 x 6 = 93.6.
so the integer represents the change in bushels of corn in his inventory after 6 days = 93.6 bushels.

Question 9.
A certain plant grows 1$$\frac{1}{6}$$ inches every week. How long will it take the plant to grow 6$$\frac{1}{6}$$ inches?

The long will it take the plant to grow = 7.1456 inches.

Explanation:
In the above-given question,
given that,
A certain plant grows 1$$\frac{1}{6}$$ inches every week.
1(1/6) = 7/6.
7/6 = 1.16.
6(1/6) = 37/6.
37/6 = 6.16.
6.16 x 1.16 = 7.1456.
so the longer will it take the plant to grow = 7.1456 inches.

Question 10.
An object is traveling at a steady speed of 8$$\frac{2}{3}$$ miles per hour. How long will it take the object to travel 5$$\frac{1}{5}$$ miles?

The long will it take for the object o travel is 5(1/5) miles = 45.032.

Explanation:
In the above-given question,
given that,
An object is traveling at a steady speed of 8$$\frac{2}{3}$$ miles per hour.
8(2/3) = 26/3.
26/3 = 8.66.
8.66 x 26/5.
8.66 x 5.2.
45.032.

Question 11.
Brianna works as a customer service representative. She knows that the amount of her yearly bonus is $155, but$2.50 is taken away for each customer complaint about her during the year. What is her bonus if there are 12 complaints about her in the year?

The bonus if there are 12 complaints about her in the year = $1890. Explanation: In the above-given question, given that, Brianna works as a customer service representative. She knows that the amount of her yearly bonus is$155, but $2.50 is taken away for each customer complaint about her during the year.$155 + $2.50 = 157.5.$157.5 x 12 = $1890. so the bonus if there are 12 complaints about her in the year =$1890.

Question 12.
Make Sense and Persevere There are ten birdbaths in a park. On the first day of spring, the birdbaths are filled. Several weeks later, the overall change in the water level is found. The results are shown in the table. What is the range of the data?

The range of the data =

Explanation:
In the above-given question,
given that,
There are ten birdbaths in a park.
On the first day of spring, the birdbaths are filled.
2.4 – 0.9 = 1.5.
so the range of the data = 1.5.

Question 13.
Model with Math Marcelo played a carnival game at the Interstate Fair 6 times. He spent 3 tokens to play each game, and he won 7 tokens each game. Write two different expressions that can be used to find the total profit in tokens that Marcelo made.

The different expressions that can be used to find the total profits in tokens that Marcelo made = 25.

Explanation:
In the above-given question,
given that,
Marcelo played a carnival game at the Interstate Fair 6 times.
He spent 3 tokens to play each game.
and he won 7 tokens each game.
6 x 3 = 18
18 + 7 = 25.
so the different expressions that can be used to find the total profits in tokens that Marcelo made = 25.

Question 14.
The temperature of a pot of water is shown. The temperature of the water changed -2.5°F per minute.

a. What was the temperature after 20 minutes?

The temperature of the water after 20 minutes = -3.606°F.

Explanation:
In the above-given question,
given that,
The temperature of a pot of water is 180.3°F.
-2.5°F x 20 = -50°F .
180.3°F /-50°F.
-3.606°F.
so the temperature of the water after 20 minutes = -3.606°F.

b. Make Sense and Persevere How many minutes did it take to cool to 100.3°F?

The number of minutes did it take to cool to 100.3°F = 40 minutes.

Explanation:
In the above-given question,
given that,
-2.5 x 40 = 100.3°F.
so the number of minutes did it take to cool to 100.3°F = 40 minutes.

Question 15.
Higher Order Thinking the table shows the relationship between a hedgehog’s change in weight and the number of days of hibernation.

a. What number represents the change in weight for each day of hibernation?

The number represents the change in weight for each day of hibernation = -0.06.

Explanation:
In the above-given question,
given that,
the change in weight is -0.24 – 0.84 – 2.25 -2.79.
-0.6 – 2.25 – 2.79.
-2.85 – 2.79.
-0.06.
so the number represents the change in weight for each day of hibernation = -0.06.

b. What number represents the change in weight in ounces for the hedgehog in 115 days of hibernation?

Assessment Practice

Question 16.
A basketball team played six games. In those games, the team won by 7 points, lost by 20, won by 8, won by 11, lost by 3, and won by 9. Which was the mean amount by which the team won or lost over the six games?
A. -3 points
B. 2 points
C. 3 points
D. 6 points

Option D is correct.

Explanation:
In the above-given question,
given that,
A basketball team played six games.
In those games, the team won by 7 points, lost by 20, won by 8, won by 11, lost by 3, and won by 9.
won points = 7 + 8 + 11 + 9.
35.
lost points = -20 -3.
35/6 = 5.8.
so option D is correct.

Question 17.
In digging a hole, the construction crew records the location of the bottom of the hole relative to ground level. After 3 hours the hole is 8.25 feet deep.
PART A
What number represents the change in location in feet after 1 hour?

The change in location in feet after 1 hour = 2.75 feet.

Explanation:
In the above-given question,
given that,
In digging a hole, the construction crew records the location of the bottom of the hole relative to ground level.
After 3 hours the hole is 8.25 feet deep.
8.25/3 = 2.75.
2.75 + 2.75 + 2.75 = 8.25.
so the change in locate in feet after 1 hour = 2.75 feet.

PART B
If the crew were to continue digging at the same rate, what number would they record for the location in feet after 8 hours?

The number would they record for the location in feet after 8 hours = 1.03 feet.

Explanation:
In the above-given question,
given that,
In digging a hole, the construction crew records the location of the bottom of the hole relative to ground level.
After 8 hours the hole is 8.25 feet deep.
8.25/8 = 1.03.
so the number would they record for the location in feet after 8 hours = 1.03 feet.

3-ACT MATH

3-Act Mathematical Modeling: Win Some, Lose Some

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

If the winning price is more than the loose price.

Explanation:
In the above-given question,
given that,
win some and lose some.
for example:
if we win rs100  and if we lose rs50.
so the profit is rs50.

Question 2.
Write the Main Question you will answer.

Question 3.
Make a prediction to answer this Main Question.
The person who will win is _________.

Question 4.
Construct Arguments Explain how you arrived at your prediction.

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

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

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

Question 8.

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

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

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

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?

Question 13.
Reasoning How is each person’s starting score related to their final score?

SEQUEL
Question 14.
Construct Arguments If there were one final round where each contestant chooses how much to wager, how much should each person wager? Explain your reasoning.

I think that each person would wager at least $200. Explanation: In the above-given question, given that, If there were one final round where each contestant chooses how much to wager. so I am thinking that each person would wager at least$200.

### Topic 2 Review

Topic Essential Question
How can the properties of operations be used to solve problems involving integers and rational numbers?

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

A Rational number is a fraction a/b where a and/or b are fractions and b is not equal to 0.
A terminating decimal that ends a(n).
Two numbers that have a sum of 0 are additive inverse.

Explanation:
In the above-given question,
given that,
A Rational number is a fraction a/b where a and/or b are fractions and b is not equal to 0.
for example:
all fractions and decimals are rational numbers.
A terminating decimal that ends a(n).
for example:
2.30 is a terminating decimal.
Two numbers that have a sum of 0 are additive inverse.
for example.
3 and -3.

Use Vocabulary in Writing
Explain how you could determine whether $$\frac{\frac{21}{3}}{\frac{120}{12}}$$ and $$\frac{7}{9}$$ have the same decimal equivalent. Use vocabulary words in your explanation.

No, both of them are not equal.

Explanation:
In the above-given question,
given that,
$$\frac{\frac{21}{3}}{\frac{120}{12}}$$.
21/3 / 120/12.
7/ 10.
7/10 and 7/9 are not equal.
so both of them are not equal.

Concepts and Skills Review

Lesson 1.1 Relate Integers and Their Opposites

Quick Review
Integers are the counting numbers, their opposites, and 0. Opposite integers are the same distance from 0 in opposite directions. Opposite quantities combine to make 0.

Example
A climber descends 3 miles into a canyon. What integer represents the descent of her climb? How far does she have to climb to return to her starting point?
The descent of her climb is represented by -3. She has to climb 3 miles to return to her starting point.

Practice
Question 1.
On a cold winter morning, the temperature was -4°F By noon, the temperature increased 4o. What was the temperature at noon?

The temperature at noon = 4°F.

Explanation:
In the above-given question,
given that,
On a cold winter morning, the temperature was -4°F By noon, the temperature increased by 40.
the temperature at noon is 4°F.

Question 2.
Audrey deposits $27 in her account. Then she makes two withdrawals, one for$15 and one for $12. What is the total change to the balance of Audrey’s account? Explain. Answer: The total change chance to the balance of Audrey’s account =$0.

Explanation:
In the above-given question,
given that,
Audrey deposits $27 in her account. Then she makes two withdrawals, one for$15 and one for $12.$15 + $12 =$27.
$27 –$27 = 0.
so the total change chance to the balance of Audrey’s account = $0. Lesson 1.2 Understand Rational Numbers Quick Review All rational numbers have an equivalent decimal form. The decimal equivalent will be either a terminating decimal or a repeating decimal. A terminating decimal ends in repeating zeros. A repeating decimal has a never-ending pattern of the same digits. Example Write the decimal equivalents for and i Are the decimals terminating or repeating? The decimal equivalent for $$\frac{5}{8}$$ = 0.625, which is a terminating decimal. The decimal equivalent for $$\frac{8}{11}$$ = 0.7272… = 0.72, which is a repeating decimal. Practice Question 1. Which fractions have a decimal equivalent that is a repeating decimal? Select all that apply. ☐ $$\frac{13}{65}$$ ☐ $$\frac{141}{47}$$ ☐ $$\frac{11}{12}$$ ☐ $$\frac{19}{3}$$ Answer: Options B and C are repeating decimals. Explanation: In the above-given question, given that, 13/65 = 0.2. 141/47 = 3. 11/12 = 0.916. 19/3 = 6.33. so options B and C are repeating decimals. Question 2. Greg bought 19$$\frac{11}{16}$$ gallons of gas. What decimal should the meter on the gas pump read? Answer: The decimal should the meter on the gas pump read = 19.6875. Explanation: In the above-given question, given that, Greg bought 19$$\frac{11}{16}$$ gallons of gas. 19(11/16) = 315/16. 315/16 = 19.6875. so the decimal should the meter on the gas pump read = 19.6875. Question 3. What is the decimal equivalent of each rational number? a. $$\frac{9}{11}$$ b. $$\frac{4}{5}$$ c. –$$\frac{17}{5}$$ d. $$\frac{5}{9}$$ Answer: 0.81, 0.8, 3.4, and 0.55. Explanation: In the above-given question, given that, 9/11 = 0.81. 4/5 = 0.8. 17/5 = 3.4. 5/9 = 0.55. Lesson 1.3 Add Integers Quick Review To add integers with the same sign, add the absolute value of each integer. The sign of the sum will be the same as the sign of the addends. To add integers with different signs, find the difference of the absolute value of each integer. The sign of the sum will be the same as the sign of the greater addend. Example Find the sum of (-28) + (-19) |-28| + |-19| = 28 + 19 = 47 The sum of (-28) + (-19) = -47. Find the sum of (-28) + 19. |-28| – |19| = 28 – 19 = 9 The sum of (-28) + 19 = -9, because |-28| > |19| Practice Question 1. Jonah’s cell phone came with 64 GB of memory. He has used 15 GB. He then uses 5 MB of memory to record photos and videos from a trip. Use the addition expression 64 + (-15) + (-5) to find how much memory is left on his phone. Answer: The memory is left on his phone = 44 MB. Explanation: In the above-given question, given that, Jonah’s cell phone came with 64 GB of memory. He has used 15 GB. He then uses 5 MB of memory to record photos and videos from a trip. 64 + (-15) + (-5). 64 -15 -5. 49 – 5. 44. so the memory is left on his phone = 44 MB. Question 2. Stella walks down a flight of stairs to the basement. Then she walks back up the stairs and up another flight of stairs to the second floor of her house. Each flight of stairs represents a change of 12 feet in height. How far is Stella above the ground? Answer: The far is Stella above the ground = 24 feet. Explanation: In the above-given question, given that, Stella walks down a flight of stairs to the basement. Then she walks back up the stairs and up another flight of stairs to the second floor of her house. Each flight of stairs represents a change of 12 feet in height. 12 +12 = 24. so the far is stella above the ground = 24 feet. Question 3. Find the sum. a. 64 + (-15) Answer: The sum is 49. Explanation: In the above-given question, given that, the two numbers are 64 and -15. add the numbers. 64 + (-15). 64 – 15. 49. b. -121 + (-34) Answer: The sum is -155. Explanation: In the above-given question, given that, the two numbers are -121 and -34. add the numbers. -121 – 34. -155. c. -86 + 92 Answer: The sum is 6. Explanation: In the above-given question, given that, the two numbers are -86 and 92. add the numbers. -86 + 92 = 6. so the sum is 6. d. 109 + (-162) Answer: The sum is -53. Explanation: In the above-given question, given that, the two numbers are 109 and -53. add the numbers. 109 + (-162). -53. so the sum is -53. Lesson 1.4 Subtract Integers Quick Review To subtract integers, use the additive inverse to write an equivalent addition expression. Then follow the rules for addition. When the signs are the same, find the sum of the absolute values. When the signs are different, find the difference. Use the sign of the number with the greater absolute value. Example Find -7 – (8). -7 + (-8) = -15 The signs are the same, so the sum has the same sign as the addends. Find -7 – (-8). -7 + 8 = 1 The signs are different, so the sign of the difference is the same sign as the integer (8) with the greater absolute value, which is positive. Practice Question 1. The temperature is 1°F at dusk. It is 8 degrees colder at dawn. What is the temperature at dawn? Answer: The temperature at the dawn = -7°F. Explanation: In the above-given question, given that, The temperature is 1°F at dusk. It is 8 degrees colder at dawn. 1- 8 = -7. so the temperature at the dawn = -7°F. Question 2. Kyle and Nadim are on the same space on a board game they are playing. Kyle moves back 2 spaces in one turn and moves back 3 more spaces in his second turn. Nadim has remained in the same place. What integer represents Kyle’s location relative to Nadim’s location on the game board? Answer: The integer represents Kyle’s location relative to Nadim’s location on the game board = -1. Explanation: In the above-given question, given that, Kyle and Nadim are in the same space on a board game they are playing. Kyle moves back 2 spaces in one turn and moves back 3 more spaces in his second turn. Nadim has remained in the same place. 2 – 3 = -1. so the integer represents Kyle’s location relative to Nadim’s location on the game board = -1. Question 3. Find the difference. a. 82 – (-14) Answer: The difference is 96. Explanation: In the above-given question, given that, the two numbers are 82 – (-14). subtract the numbers. 82 – (-14). 82 + 14 = 96. b. -18 – (-55) Answer: The difference is 37. Explanation: In the above-given question, given that, the two numbers are -18 – (-55). subtract the numbers. -18 – (-55). -18 + 55. 37. c. – 17 – 44 Answer: The difference is -61. Explanation: In the above-given question, given that, the two numbers are -17 – 44. subtract the numbers. -17 – 44. -61. d. 70 – (-101) Answer: The difference is 171. Explanation: In the above-given question, given that, the two numbers are 70 – (-101). subtract the numbers. 70 – (-101). 70 + 101. 171. Lesson 1.5 Add and Subtract Rational Numbers Quick Review Positive and negative rational numbers and decimals can be added and subtracted following the same rules as adding and subtracting integers. Example Find -5$$\frac{1}{2}$$ – 1.75. Convert 1.75 to an equivalent fraction, 1 $$\frac{3}{4}$$ $$-5 \frac{1}{2}-1 \frac{3}{4}$$ = $$-5 \frac{2}{4}+\left(-1 \frac{3}{4}\right)$$ = $$-6 \frac{5}{4}$$ = $$-7 \frac{1}{4}$$ Practice Question 1. Doug digs a hole that is 1.7 feet below ground level. He plants a bush that is 37 feet tall from the bottom of the root to the top branch. How much of the bush is above the ground? Answer: Much of the bush is above the ground = 38.7 feet. Explanation: In the above-given question, given that, Doug digs a hole that is 1.7 feet below ground level. He plants a bush that is 37 feet tall from the bottom of the root to the top branch. 1.7 + 37 = 38.7. so much of the bush is above the ground = 38.7feet. Question 2. Penelope has a birdhouse that is 4 feet above the roof of her garage. She has a second birdhouse that is 5.36 feet below the roof of her garage. What is the distance between the birdhouses? Answer: The distance between the birdhouses = -1.36. Explanation: In the above-given question, given that, Penelope has a birdhouse that is 4 feet above the roof of her garage. She has a second birdhouse that is 5.36 feet below the roof of her garage. 4 + (-5.36). 4 – 5.36. -1.36. so the distance between the birdhouses = -1.36. Question 3. Find the sum or difference. a. -2.63 + 3$$\frac{1}{4}$$ Answer: The difference is 0.62. Explanation: In the above-given question, given that, -2.63 + 3$$\frac{1}{4}$$ -2.63 + 3 (1/4). -2.63 + 13/4. -2.63 + 3.25. 0.62. b. -4$$\frac{1}{2}$$ – (-1.07) Answer: The difference is -3.43. Explanation: In the above-given question, given that, -4$$\frac{1}{2}$$ – (-1.07). -4(1/2) – (-1.07). -9/2 + 1.07. -4.5 + 1.07. -3.43. c. 0.74 + (-$$\frac{3}{5}$$) Answer: The difference is 0.14. Explanation: In the above-given question, given that, 0.74 + (-$$\frac{3}{5}$$). 0.74 – 3/5. 0.74 – 0.6. 0.14. d. –$$\frac{1}{8}$$ – 0.356 Answer: The difference is -0.481. Explanation: In the above-given question, given that, –$$\frac{1}{8}$$ – 0.356. -1/8 – 0.356. -0.125 – 0.356. -0.481. so the difference is -0.481. Lesson 1.6 Multiply Integers Quick Review Multiply integers the same way you multiply whole numbers. If the signs of the factors are the same, the product is positive. If the signs of the factors are different, the product is negative. Example -9 • -8 = 72 -9 • 8 = -72 Practice Question 1. Marisa buys 4 books at$13 per book. What integer represents the total change in the amount of money Marisa has?

The integer represents the total change in the amount of money Marisa has = $52. Explanation: In the above-given question, given that, Marisa buys 4 books at$13 per book.
4 x $13 =$52.
so the integer represents the total change in the amount of money Marisa has = $52. Question 2. Which expressions have a product of -18? Select all that apply. ☐ -2 • -9 ☐ -6 • 3 ☐ -3 • 6 ☐ -9 • 2 Answer: Options B, C, and D are correct. Explanation: In the above-given question, given that, expressions have a product of -18. -2 x -9 = 18. -6 x 3 = -18. -3 x 6 = -18. -9 x 2 = -18. so options B, C, and D are correct. Question 3. Find the product. a. -7 • -14 b. –15 • 12 c. 9 • -20 d. -11 • -16 Answer: The products are 98, -180, -180, and 176. Explanation: In the above-given question, given that, -7 x -14 = 98. -15 x 12 = -180. 9 x -20 = -180. -11 x -16 = 176. Lesson 1.7 Multiply Rational Numbers Quick Review The same rules for multiplying integers apply to multiplying rational numbers. If the signs of the factors are the same, the product will be positive. If the signs of the factors are different, the product will be negative. Example -9.6 • 1.8 = -17.28 -9.6 • -1.8 = 17.28 Practice Question 1. Jason spends$2.35 to buy lunch at school. If he buys a lunch on 9 days, what number represents the total change in the amount of money Jason has?

The total change in the amount of money Jason has = $21.15. Explanation: In the above-given question, given that, Jason spends$2.35 to buy lunch at school.
If he buys lunch on 9 days.
$2.35 x 9 =$21.15.
so the total change in the amount of money Jason has = $21.15. Multiply. Question 2. -2$$\frac{2}{3}$$ • -4$$\frac{3}{7}$$ Answer: -2(2/3) x -4(3/7) = 11.492. Explanation: In the above-given question, given that, -2$$\frac{2}{3}$$ • -4$$\frac{3}{7}$$ -2(2/3) x -4(3/7). -8/3 x -31/7. -2.6 x -4.42. 11.492. Question 3. -3$$\frac{4}{9}$$ • 5$$\frac{2}{5}$$ Answer: -3(4/9) x 5(2/5) = -18.36. Explanation: In the above-given question, given that, -3$$\frac{4}{9}$$ • 5$$\frac{2}{5}$$ -3(4/9) x 5(2/5). -31/9 x 27/5. -3.4 x 5.4. -18.36. Question 4. 6$$\frac{2}{3}$$ • -4$$\frac{1}{5}$$ Answer: 6(2/3) x -4(1/5) = 2.4. Explanation: In the above-given question, given that, 6$$\frac{2}{3}$$ • -4$$\frac{1}{5}$$. 6(2/3) x -4(1/5). 20/3 – 21/5. 6.6 – 4.2. 2.4. Lesson 1.8 Divide Integers Quick Review Divide integers the same way you divide whole numbers. The quotient is positive if the signs of the dividend and divisor are the same. The quotient is negative if the signs of the dividend and divisor are different. Example -39 ÷ 3 = -13 -39 ÷ 3 = 13 Practice Question 1. Which expressions have a quotient of -4? Select all that apply. ☐ $$\frac{-24}{6}$$ ☐ -36 ÷ -9 ☐ -72 ÷ 18 ☐ $$\frac{84}{-21}$$ Answer: Options A,C, and D are correct. Explanation: In the above-given question, given that, the expressions have a quotient of -4 are: -24/6 = -4. -36/-9 = 4. -72/ 18 = -4. 84/-21 = -4. so options A, C, and D are correct. Question 2. Whitney rolls a ball down a ramp that is 18 feet long. If the ball rolls down 2 feet each second, what integer represents the amount of time, in seconds, the ball takes to reach the end of the ramp? Answer: The integer represents the amount of time in seconds, the ball takes to reach the end of the ramp = 9 seconds. Explanation: In the above-given question, given that, Whitney rolls a ball down a ramp that is 18 feet long. If the ball rolls down 2 feet each second. 18/2 = 9. so the integer represents the amount of time in seconds, the ball takes to reach the end of the ramp = 9 seconds. Question 3. Find the quotient. a. $$\frac{81}{-9}$$ b. -123 ÷ -4 c. –$$\frac{94}{4}$$ d. 65 ÷ (-5) Answer: The quotients are -9, 30.75, -23.5, and -13. Explanation: In the above-given question, given that, a: 81/-9 = -9. -123/-4 = 30.75. -94/4 = -23.5. 65 / -5 = -13. so the quotients are -9, 30.75, -23.5, and -13. Lesson 1.9 Divide Rational Numbers Quick Review The same rules for dividing integers apply to dividing all rational numbers. The quotient is positive when the numbers being divided have the same signs. The quotient is negative when the numbers being divided have different signs. Complex fractions have a fraction in the numerator, the denominator, or both. To divide by a fraction, rewrite as multiplication by its multiplicative inverse, or reciprocal. Example Simplify $$\frac{-\frac{3}{4}}{\frac{15}{24}}$$ $$-\frac{3}{4} \div \frac{15}{24}=-\frac{3}{4} \cdot \frac{24}{15}=-\frac{72}{60}=-\frac{6}{5}=-1 \frac{1}{5}$$ Practice Find the quotient. Question 1. $$\frac{8}{9}$$ ÷ -1$$\frac{4}{15}$$ Answer: 8/9 / -1(4/15) = -0.38. Explanation: In the above-given question, given that, divide the numbers. $$\frac{8}{9}$$ ÷ -1$$\frac{4}{15}$$. 8/9 – 1(4/15). 0.88 – 19/15. 0.88 – 1.26. -0.38. Question 2. –3.6 ÷ 2$$\frac{1}{7}$$ Answer: –3.6 ÷ 2$$\frac{1}{7}$$ = -1.68. Explanation: In the above-given question, given that, divide the numbers. –3.6 ÷ 2$$\frac{1}{7}$$. -3.6 / 2(1/7). -3.6 / 2.14. -1.68. Question 3. A boat drops an anchor 17.5 feet to the bottom of a lake. If the anchor falls at a rate of 0.07 feet each second, how long will it take the anchor to reach the bottom of the lake? Answer: The long will it take the anchor to reach the bottom of the lake = 25 feet. Explanation: In the above-given question, given that, A boat drops an anchor 17.5 feet to the bottom of a lake. If the anchor falls at a rate of 0.07 feet each second. 1.75 / 0.07 = 25. so the longer will it take the anchor to reach the bottom of the lake = 25 feet. Lesson 1.10 Solve Problems with Rational Numbers Quick Review You can use rational numbers to solve problems in the same way that you use whole numbers. Be sure to make sense of the problem you are solving to help you choose the correct operations and determine which values will be positive and which will be negative. Example During a 15-day dry spell, the water level in a lake changed by -2$$\frac{3}{8}$$ inches. What rational number represents the average change in the water level per day? -2$$\frac{3}{8}$$ ÷ 15 = –$$\frac{19}{8}$$ • $$\frac{1}{15}$$ = –$$\frac{19}{120}$$ inch Practice Question 1. In 5 rounds of a game, Jill scored -3, 8, 9, -7, and 13. What integer represents her average score for the 5 rounds? Answer: The integer represents her average score for the 5 rounds = 2. Explanation: In the above-given question, given that, In 5 rounds of a game, Jill scored -3, 8, 9, -7, and 13. -3, 8, 9, -7, and 13. 10/5 = 2. so the integer represents her average score for the 5 rounds = 2. Question 2. Peter signed up for a program that costs$10.50 per month to stream movies to his computer. He decided to cancel his service after $$\frac{5}{6}$$ month. He only has to pay for the amount of time he used the service. What number represents the total change in the amount of money Peter has after paying for the service?

The number represents the total change in the amount of money Peter has after paying for the service =$8.715. Explanation: In the above-given question, given that, Peter signed up for a program that costs$10.50 per month to stream movies to his computer.
He decided to cancel his service after $$\frac{5}{6}$$ month.
He only has to pay for the amount of time he used the service.
$10.50 x 5/6.$10.50 x 0.83.
$8.715. so the number represents the total change in the amount of money Peter has after paying for the service =$8.715.

Question 3.
Maggie spent $4.05 on cheese and fruit at the farmer’s market. She bought $$\frac{1}{8}$$ pound of apples, $$\frac{1}{4}$$ pound of pears, and 1.25 pounds of bananas. If fruit cost$0.80 per pound, how much did Maggie spend on cheese?

Maggie spends on cheese = 5.675 pounds.

Explanation:
In the above-given question,
given that,
Maggie spent $4.05 on cheese and fruit at the farmer’s market. She bought $$\frac{1}{8}$$ pound of apples, $$\frac{1}{4}$$ pound of pears, and 1.25 pounds of bananas. If fruit cost$0.80 per pound.
$4.05 + 1/8 + 1/4 + 1.25 = 5.3 + 0.125 + 0.25. 5.3 + 0.375. 5.675. so the Maggie spend on cheese = 5.675 pounds. ### Topic 1 Fluency Practice Crisscrossed Find each sum, difference, product, or quotient. Write your answers in the cross-number puzzle below. Each digit and negative sign in your answers goes in its own box. I can… add, subtract, multiply, and divide integers. Across 2. 248 + (-1,027) 5. 818 – (-1,021) 6. -516 + 774 8. 242 + (-656) 9. 2,087 + (-1,359) 10. 631 – 897 11. –342 + 199 12. -49 • -27 13. –321 – 987 14. 2,988 ÷ -3 15. 2,580 ÷ 6 16. 4,592 ÷ -82 17. 48 • -27 18. -24 • 83 21. -118 + 1,201 22. -45 • -59 Answer: Explanation: In the above-given question, given that, 248 – 1027 = Down 1. 246 + 173 2. 22 • -22 3. 726 – (-219) 4. 501 – 699 7. -10,740 ÷ 15 8. 6,327 ÷ -9 10. 144 • -16 11. 15 • -67 12. 7,164 ÷ 4 13. –33 • 63 14. -2,695 ÷ 55 17. -1,032 – (-285) 18. 512 – 720 19. –729 + 951 20. –17 • -25 #### enVision Math Common Core Grade 7 Answer Key ## Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Value ## Envision Math Common Core 5th Grade Answers Key Topic 1 Understand Place Value Envision Math Common Core 5th Grade Answers Key Topic 1 Understand Place Value has all the topics covered , which are related to the basics of Math by giving the live examples of our daily life. This chapter is loaded with Decimals, Fractions, and how to do rounding to the nearest numbers, which are quite helpful for students to deal with Math basics. Experience the most satisfying and understandable answers and easy methods with Envision Math Common Core 5th Grade Answers Key Topic 1 Understand Place Value. ## Envision Math Common Core 5th Grade Answers Key Topic 1 Understand Place Value Are you facing difficulties to understand the simple Math problems or numbers , that we have to use in day to day life. Then you have come to the right place to grasp the simple tricks of small calculations. Envision Math Common Core 5th Grade Answers Key Topic 1 Understand Place Value. will help you with rounding the numbers , understand the place values , converting decimal into fractions .These are the major topics covered, get on the track with your kids to know them a better way of learning. Envision STEM Project: Pollinating Insects Lesson 1 Patterns with exponents and Powers of 10 Lesson 2 Understand Whole Number Place Value Lesson 3 Decimals to Thousandths Lesson 4 Understand Decimal Place Value Lesson 5 Compare Decimals Lesson 6 Round Decimals Lesson 7 Look For and Use Structure ### Performance Task Essential Question: How are whole numbers and decimals written, compared, and ordered? Envision STEM Project Pollinating Insects Do Research Use the Internet or other sources to find out more about pollinating insects in the United States. What types of insects are they? How many are there of each type? How many crops and flowering plants depend on pollinating insects in order to produce the foods we eat? Journal: Write a Report Include what you found. Also in your report: • Choose two of the pollinating insects. Estimate how many crop plants each type of insect pollinates. • Estimate how many of your favorite foods and beverages come from pollinated plants. • Make up and solve ways to compare and order your data. Answer: Pollinated insects are nothing but, the insects which are helpful to carry the pollinated grains along with them with the help of their legs or wings from the flower to promote the vast growth of new plants and are commonly known as insect pollinators. Report for the project : • Insect pollinators include bees, flies, butterflies, beetles, wasps, moths, midges and ants, among others. Of these, bees are the most important group, with both wild and managed species acting as pollinators. • Pollinators are essential for continued plant growth in the wild. There are seven insect pollinators other than bees and butterflies that also help spread plant seeds and enable plant growth. • Three-fourths of the world’s flowering plants and about 35 percent of the world’s food crops depend on animal pollinators to reproduce. More than 3,500 species of native bees help increase crop yields. • More than 75 percent of the world’s food crops depend on pollination . When a seed forms in flowering plants, a fruit is able to grow to protect the seed. Among these two major pollinators are honey bees and butterflies, Honey bees are the commonly known pollinators which are helpful for the crops like Okra, kiwifruit, Onion, cashew, strawberry, Broccoli, Cauliflower , Cabbage etc. Butterflies are mostly the pollinators of vegetables and herbs and are helpful for the crops like especially those in the carrot family (dill, fennel, celery, cilantro, parsnip), sunflower family (artichokes, lettuce, chicory, chamomile), legume family (peas, beans), mint family (lavender, basil). Review What You Know Vocabulary Choose the best term from the box. Write it on the blank. • digits • place value • period • whole numbers Question 1. ____ are the symbols used to show numbers. Answer: Digits are the symbols used to show numbers. Explanation: Because, A digit is a single symbol used to make numerals or numbers , That is 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are the ten digits we use in everyday numerals. For example, the number 78 is a two digit number which is made up of two digits that is ‘7’ and ‘8’. Question 2. A group of 3 digits in a number is a ____ Answer: A group of 3 digits in a number is a Period. Explanation: Every number consists of digits. The place value is the position of each digit in a number. Digits are separated into groups of three by commas. So, a period is a group of three digits, For example, The number 13,456 is a 5 digit number and the ending 3 digits are called a period, because it is separated by the commas in the number which is referred to a period. Question 3. ____ is the position of a digit in a number that is used to determine the value of the digit. Answer: Place value is the position of a digit in a number that is used to determine the value of the digit. Explanation: The value of a digit depends on its place, or position, in the number. Place value is the value of a digit according to its position in the number such as ones, tens, hundreds, and so on. For example, In the number 3,548 3 is in thousands place and its place value is 3,000, 5 is in hundreds place and its place value is 500, 4 is in tens place and its place value is 40, 8 is in ones place and its place value is 8. Comparing Compare. Use <,>, or = for each . Question 4. 869 912 Answer: 869 < 912. Explanation: Because 869 is less than 912. For the number 869 8 is in hundreds place and its place value is 800, 6 is in tens place and its place value is 60, 9 is in ones place and its place value is 9. And For the number 912 9 is in hundreds place and its place value is 900, 1 is in tens place and its place value is 10, 2 is in ones place and its place value is 2. So, 869 is less than 912. Question 5. 9,033 9,133 Answer: 9,033 < 9,133 Explanation: Because 9,033 is less than 9,133 For the number 9,033 9 is in thousands place and its place value is 9,000, 0 is in hundreds place and its place value is 000, 3 is in tens place and its place value is 30, 3 is in ones place and its place value is 3. And For the number 9,133 9 is in thousands place and its place value is 9,000, 1 is in hundreds place and its place value is 100, 3 is in tens place and its place value is 30, 3 is in ones place and its place value is 3. So, 9,033 is less than 9,133 Question 6. 1,338 1,388 Answer: 1,338 < 1,388 Explanation: Because 1,338 is less than 1,388 For the number 1,338 1 is in thousands place and its place value is 1,000, 3 is in hundreds place and its place value is 300, 3 is in tens place and its place value is 30, 8 is in ones place and its place value is 8. And For the number 1,388 1 is in thousands place and its place value is 1,000, 3 is in hundreds place and its place value is 300, 8 is in tens place and its place value is 80, 8 is in ones place and its place value is 8. So, 1,338 is less than 1,388 Question 7. 417,986 417,986 Answer: 417,986 = 417,986 Explanation: Because, Numbers given are the same digits having the same place value . So, 417,986 Equal to 417,986 Question 8. 0.25 0.3 Answer: 0.25 < 0.3 Explanation: Because, regarding with its place value the number 0.25 is less than 0.3 Question 9. 0.5 0.50 Answer: 0.5 = 0.50 Explanation: Because the 0 after the number next to decimal is exactly the same number with or without 0 Both the given numbers are in their same places with respect to each other So, 0.5 is equal to 0.50. Question 10. Kamal has 7,325 songs on his computer. Benito has 7,321 songs on his computer. Who has more songs? Answer: Kamal has more songs than Benito. Explanation: Given , Kamal has 7,325 songs on his computer. Benito has 7,321 songs on his computer. Then 7,325 > 7,321 or 7,325 is greater than 7,321 So, Kamal has more songs than Benito. Adding Whole Numbers Find each sum. Question 11. 10,000 + 2,000 + 60 + 1 Answer: 12,061 Explanation: Adding all the given numbers by placing them in order of their place value of the digit So, the total sum is 12,061. Question 12. 20,000 + 5,000 + 400 + 3 Answer: 25,403 Explanation: Adding all the given numbers by placing them in order of their place value of the digit So, the total sum is 25,403. Question 13. 900,000 + 8,000 + 200 + 70 + 6 Answer: 9,08,276 Explanation: Adding all the given numbers by placing them in order of their place value of the digit So, the total sum is 9,08,276. Question 14. 7,000,000 + 50,000 + 900 + 4 Answer: 70,50,904 Explanation: Adding all the given numbers by placing them in order of their place value of the digit So, the total sum is 70,50,904. Place Value Question 15. The largest playing card structure was made of 218,792 cards. What is the value of the digit 8 in 218,792? A. 80 B. 800 C. 8,000 D. 80,000 Answer: C Explanation: Given, The largest playing card structure was made of 218,792 cards. According to the place value method, the value of the digit 8 in 218,792 is 8,000 Question 16. Construct Arguments In the number 767, does the first 7 have the same value as the final 7? Why or why not? Answer: In the number 767, The first 7 does not have the same value as the final 7. Explanation: Given , number is 767 , The first 7 does not have the same value as the final 7. Because, the place value is counted from right to left by ones, tens, hundreds, thousand and so on. So, in this number 767, the first number 7 holds the hundreds place and the final number 7 holds the ones place, giving it a different value for both the numbers in their place respectively. Pick a Project PROJECT 1A Manatees or sea cows? Project: Create a Manatee Poster Answer: PROJECT 1B What makes a game fun? Project: Design a Game with Place-Value Blocks Answer: PROJECT 1С How far are we from the sun? Project: Research Measurements in Our Solar System Answer: The Sun is at an average distance of about 93,000,000 miles (150 million kilometers) away from Earth. It is so far away that light from the Sun, traveling at a speed of 186,000 miles (300,000 kilometers) per second, takes about 8 minutes to reach us. ### Lesson 1.1 Patterns with exponents and Powers of 10 Solve & share A store sells AA batteries in packages of 10 batteries. They also sell boxes of 10 packages, cases of 10 boxes, and cartons of 10 cases. How many AA batteries are in one case? One carton? 10 cartons? Solve these problems any way you choose. You can use appropriate tools, such as place-value blocks, to help solve the problems. However you choose to solve it, show your work! Look Back! How many 10s are in 100? How many 10s are in 1,000? Write equations to show your work. Answer: There are 1000 AA batteries in one case , 10,000 batteries in one carton and 1,00,000 batteries in 10 cartons. Explanation: Given, A store sells AA batteries in packages of 10 batteries, boxes of 10 packages, cases of 10 boxes, and cartons of 10 cases. Then we have 10 batteries for one package and given that 1 box contains 10 packages , 10 × 10 = 100 , that is 100 batteries for each box. Next for 10 boxes we have a case which means 100 × 10 = 1000 batteries , That is each case contains 1000 batteries. Now for 10 cases we have a carton which means 1000 × 10 = 10,000 batteries , That is each carton contains 10,000 batteries. And now they asked for 10 cartons , So, for 10 cartons we have 10,000 × 10 = 1,00,000 batteries . Visual Learning Bridge Essential Question How Can You Explain Patterns in the Number of Zeros in a Product? Answer: An exponent identifies a quantity representing the power to which a given number or expression is to be raised. Explanation: For example , 4 × 60 = 4 x (6 x 10) = (4 x 6) x 10 = 24 x 10 =240 or 50 x 700 = (5 x 10) + (7 x 100) = (5 x 7) x (10 x 100) = 35 x 1,000 = 35,000. A. Tamara’s new horse weighs about 1,000 pounds. How can you show 1,000 as a power of 10 using an exponent? The exponent is the number that tells how many times a base number is used as a factor. Answer: 10³ Explanation: Given, horse weighs about 1,000 pounds. then show 1,000 as a power of 10 using an exponent So we can write 1000 as 10 × 10 × 10 = 10³. B. Write 1,000 as a product using 10 as a factor. The exponent, 3, shows that the base number, 10, is multiplied 3 times. So, 1,000 is written as 103 using exponents. C. Tamara estimates that her horse will eat about 5,000 pounds of hay each year. How can you write 5,000 using exponents? 5 × 101 = 5 × 10 = 50 5 × 102 = 5 × 10 × 10 = 500 5 × 103 = 5 × 10 × 10 × 10 = 5,000 The number of zeros in the product is the same as the exponent. So, 5,000 is written as 5 × 103 using exponents. Convince Me! Look for Relationships What pattern do you notice in the number of zeros in the products in box C above? Guided Practice Do You Understand? Question 1. Why are there three zeros in the product of 6 × 103? Answer: The product of 6 × 10³ is 6,000 so it contains three zeros. Explanation: The product of 6 × 10³ is 6,000, Because, 6 × 10 × 10 × 10 = 6,000. So it contains three zeros. Question 2. Susan said that 105 is 50. What mistake did Susan make? What is the correct answer? Answer: The correct answer is 1,00,000. Explanation: 10 × 10 × 10 × 10 × 10 = 1,00,000. The mistake Susan made is he multiplied the exponent with the base. Do You Know How? In 3 and 4, complete the pattern. Question 3. 101 = 102 = 103 = 104 = Answer: 101 = 10 102 = 100 103 = 1000 104 = 10,000 Explanation: 10 = 10 10 × 10 = 100 10 × 10 × 10 = 1000 10 × 10 × 10 × 10 = 10,000. Question 4. = 7 × 101 = 7 × 102 = 7 × 103 = 7 × 104 Answer: 70 = 7 × 101 700 = 7 × 102 7,000 = 7 × 103 70,000 = 7 × 104 Explanation: Given, 7 ×10 = 70 7 × 10 × 10 = 700 7 ×10 × 10 × 10 = 7,000 7 × 10 × 10 × 10 × 10 = 70,000. Independent Practice In 5-15, find each product. Use patterns to help. Question 5. 3 × 101 = 3 × 102 = 3 × 103 = 3 × 104 = Answer: 3 × 101 = 30 3 × 102 = 300 3 × 103 = 3,000 3 × 104 = 30,000 Explanation: 3 ×10 = 30 3 × 10 × 10 = 300 3 ×10 × 10 × 10 = 3,000 3 × 10 × 10 × 10 × 10 = 30,000. Question 6. 2 × 10 = 2 × 100 = 2 × 1,000 = 2 × 10,000 = Answer: 2 × 10 = 20 2 × 100 = 200 2 × 1,000 = 2,000 2 × 10,000 = 20,000 Explanation: 2 × 10 = 2 × 101 = 20 2 × 100 = 2 × 102 = 2 × 10 × 10 = 200 2 × 1,000 = 2 × 103 = 2 × 10 × 10 × 10 = 2,000 2 × 10,000 = 2 × 104 = 2 × 10 × 10 × 10 × 10 = 20,000 Question 7. = 9 × 101 = 9 × 102 = 9 × 103 = 9 × 104 Answer: 90 = 9 × 101 900 = 9 × 102 9,000 = 9 × 103 90,000 = 9 × 104 Explanation: Given, 7 ×10 = 70 7 × 10 × 10 = 700 7 ×10 × 10 × 10 = 7,000 7 × 10 × 10 × 10 × 10 = 70,000. Question 8. 8 × 104 Answer: 8 × 104 = 80,000 Explanation: 8 × 104 = 8 × 10 × 10 × 10 × 10 = 80,000 Question 9. 4 × 1,000 Answer: 4 × 1,000 = 4,000 Explanation: 4 × 103 = 4 ×10 × 10 × 10 = 4,000 Question 10. 5 × 102 Answer: 5 × 102 = 500 Explanation: 5 × 102 = 5 × 10 × 10 = 500 Question 11. 6 × 10,000 Answer: 6 × 10,000 = 60,000 Explanation: 6 × 10,000 = 6 × 10 × 10 × 10 × 10 = 6 × 104 = 60,000 Question 12. 4 × 101 Answer: 4 × 101 = 40 Explanation: 4 × 101 = 40 Question 13. 100 × 9 Answer: 100 × 9 = 900 Explanation: 100 × 9 = 10 × 10 × 9 = 900 Question 14. 103 × 6 Answer: 103 × 6 = 6,000 Explanation: 103 × 6 = 10 × 10 × 10 × 6 = 6,000 Question 15. 8 × 105 Answer: 8 × 105 = 8,00,000 Explanation: 8 × 105 = 8 × 10 × 10 × 10 × 10 × 10 = 8,00,000 Question 16. Write 10 × 10 × 10 × 10 × 10 × 10 with an exponent. Explain how you decided what exponent to write. Answer: 106 Explanation: As Given there are 6 tens together So, 10 × 10 × 10 × 10 × 10 × 10 = 106 Problem Solving Question 17. One box of printer paper has 3 × 102 sheets of paper. Another box has 103 sheets of paper. What is the total number of sheets in both boxes? Answer: The total number of sheets in both boxes is 403 Explanation: Given, One box of printer paper has 3 × 102 sheets of paper. then , 3 × 102 = 300 Another box has 103 sheets of paper. So, 300 + 103 = 403 The total number of sheets in both boxes is 403 Question 18. A post is put every 6 feet along a fence around a rectangular field that is 42 ft long and 36 ft wide. How many posts are needed? Answer: 26 posts are needed. Explanation: 42 feet for one side, 42 feet for the parallel side 36 feet for one side, 36 feet for the parallel side 42 ÷ 6 = 7 ,One of these posts is a corner, so make it 6 because we do not want to count a corner post more than once. 36 ÷ 6 = 6 ,One of these posts is a corner, so make it 5 because we do not want to count a corner post more than once We now add these: 2 x 6 + 2 x 5 = 12 + 10 = 22 Add in the four corner posts: 22 + 4 = 26 posts. So , 26 posts are needed. Question 19. Number Sense A company had 9 × 106 dollars in sales last year. Explain how to find the product 9 × 106. Answer: 9 × 106 = 9,000,000. Explanation: 9 × 106 = 9 ×10 × 10 × 10 × 10 × 10 × 10 = 9,000,000. Question 20. An aquarium has the same shape as the solid figure shown below. What is the name of this solid figure? Answer: A Rectangular prism. Explanation: a solid figure that has six sides, called faces, that are rectangles. So, they are known as Rectangular prism. Question 21. Model with Math Isaac takes 5 minutes to ride his bike down the hill to school and 10 minutes to ride up the hill from school. He attends school Monday through Friday. How many minutes does he spend biking to and from school in two weeks? Write an equation to model your work. Answer: It takes 150 minutes Explanation: Given, Issac take 5 minutes to ride his bike down the hill to school And 10 minutes to ride up the hill from school . So, it takes 15 min each day to and from school He attends school Monday through Friday, So, 5 days in a week for 5 days it takes 15 × 5 = 75 minutes For 2 weeks , 75 × 2 = 150 minutes. Question 22. Higher Order Thinking Santiago hopes to buy a 4-horse trailer for about$12,000. Describe all the numbers that when rounded to the nearest hundred are 12,000.
Answer: Any number from 11,950 to 12,049, will result to 12,000 when rounded to the nearest hundred.

Explanation:
Since we have given that,
Number of horse trailer = 4 Cost of 4 horse trailer = $12000 , As we know about the “Estimation”, if there is a number greater or equal to 5 in tens place then it will be rounded off to the nearest next greatest integer. So, if this number is$11950 to 12049.
So, the possible numbers that when rounded to the nearest hundred are 12000 are from 11500 to 12049.

Assessment Practice

Question 23.
Choose all the equations that are true.
10 × 10 × 10 × 10 = 40
10 × 10 × 10 × 10 = 104
10 × 10 × 10 × 10 = 1,000,
10 × 10 × 10 × 10 = 10,000
10 × 10 × 10 × 10 = 4 × 104
Answer:  10 × 10 × 10 × 10 = 104  and  10 × 10 × 10 × 10 = 10,000

Explanation:
There are 4 tens in 10 × 10 × 10 × 10 = 104
So, 10 × 10 × 10 × 10 = 10,000.

Question 24.
Choose all the equations that are true.
6 × 105 = 6 × 100,000
6 × 105 = 6 × 10,000
6 × 105 = 600,000
6 × 105 = 60,000
6 × 105 = 650,000
Answer: 6 × 105 = 6 × 100,000 and  6 × 105 = 600,000

Explanation:
6 × 105 = 6 × 10 × 10 × 10 × 10  × 10 = 6,00,000. or
6 × 105 = 6 × 10 × 10 × 10 × 10  × 10 = 6 × 100,000.

### Lesson 1.2 Understand Whole Number Place Value

Activity

Solve & Share

The population of a city is 1,880,000. What is the value of each of the two 8s in this number? How are the two values related? Use tools like this place-value chart to help solve the problem.

Answer: These are the values of each number

Explanation:
Add up all these values and we will have 1,880,000.

Use Structure You can use place value to analyze the relationship between the digits of a number. Show your work!

Look Back! Is the relationship between the value of the two 85 in 1,088,000 the same as the relationship between the value of the two 8s in the problem above? Explain.

Visual Learning Bridge

Essential Question
How Are Place-Value Positions Related?

A.
According to the 2010 U.S. Census, the population of Phoeni×, Arizona is about 1,440,000. What is the relationship between the value of the two 4s in this number?

Writing the number in expanded form can help.

C.
Look at the expanded form of 1,440,000. The value of the 4 in the hundred thousands place is 400,000. The value of the 4 in the ten thousands place is 40,000.
400,000 is 10 times as great as 40,000.
40,000 is $$\frac{1}{10}$$ of 400,000.

Sometimes word form is used instead of number name.

Standard form
1,440,000
Expanded form:
1 × 1,000,000 + 4 × 100,000 + 4 × 10,000
Using exponents, this can be written as:
(1 × 106) + (4 × 105) + (4 × 104)
Number name:
one million, four hundred forty thousand

Convince Me! Construct Arguments is the value of the 1 in 1,440,000 10 times as great as the value of the 4 in the hundred thousands place? Explain.

Another example
When two digits next to each other in a number are the same, the digit on the left has 10 times the value of the digit to its right.
When two digits next to each other are the same, the digit on the right has to the value of the digit to its left.

Guided Practice

Do You Understand?

Question 1.
In 9,290, is the value of the first 9 ten times as great as the value of the second 9? Explain.

Explanation:
The first 9 is on the place of the thousands, while the second 9 is on the place of the tens. so the first 9 is a hundred times as great as the second 9

Do You Know How?

Question 2.
Write 4,050 in expanded form.

Explanation:
4,000 + 50  = 4,050.

In 3 and 4, write the values of the given digits.

Question 3.
the 7s in 7,700

Explanation:
The first 7 is on the place of the thousands, while the second 7 is on the place of the hundreds.

Question 4.
the 2s in 522

Explanation:
The first 2 is on the place of the tens, while the second 2 is on the place of the ones.

Independent Practice

In 5-7, write each number in standard form.

Question 5.
8,000,000 + 300 + 9

Explanation:
8,000,000 + 300 + 9 = 8,000,309.

Question 6.
(4 × 104) + (6 × 102)

Explanation:
4 × 104 =4 × 10 × 10 × 10 × 10  =  40,000
6 × 102  = 6 × 10 × 10 =  600
So, 40,000 + 600 = 40,600

Question 7.
10,000 + 20 + 3

Explanation:
10,000 + 20 + 3 = 10023.

In 8-10, write each number in expanded form.

Question 8.
5,360
Answer: (5 × 10³) + (3 × 10²) + (6 × 10)

Explanation:
5,360 =  5000 + 300 + 60  = (5 × 10³) + (3 × 10²) + (6 × 10)

Question 9.
102,200
Answer:  105 + (2 × 10³) + ( 2 × 10²)

Explanation:
102,200 = 1,00,000 + 2000 + 200 =  105 + (2 × 10³) + ( 2 × 10²)

Question 10.
85,000,011
Answer:  (85 ×  106) + 10 + 1

Explanation:
85,000,011 = 85,000,000 + 10 + 1 = (85 ×  106) + 10 + 1

In 11-13, write the values of the given digits.

Question 11.
the 7s in 6,778
Answer: 6,000 + 700 + 70 + 8

Explanation:
The first 7 is on the place of the hundreds, while the second 7 is on the place of the tens.

Question 12.
the 9s in 990,250
Answer: 9,00,000 + 90,000 + 200 + 50

Explanation:
The first 9 is on the place of the hundred thousands, while the second 9 is on the place of the thousands.

Question 13.
the 1s in 2,011,168
Answer: 2,000,000 + 11,000 + 100 + 60 + 8

Explanation:
The first 1 is on the place of the ten thousands, while the second 1 is on the place of the thousands and the third 1 is on the place of hundreds.

Problem Solving

Question 14.
Write the number name and expanded form for the number of driver ants that could be in two colonies.

Up to 22,000,000 driver ants can live in a single colony.

The number name is Forty four million
The expanded form is (4 ×106) + (4 ×106)

Explanation:
Given , Up to 22,000,000 driver ants can live in a single colony.
For 2 colonies 22,000,000 + 22,000,000 = 44,000,000
The expanded form is 40,000,000 + 4,000,000 = (4 ×106) + (4 ×106)

Question 15.
enVision® STEM A queen ant can produce about nine million ants in her lifetime. Write this number in standard form.
Answer: The number in standard form is 9,000,000

Explanation:
Given, A queen ant can produce about nine million ants in her lifetime.
The number in standard form is 9,000,000.

Question 16.
Critique Reasoning Paul says that in the number 6,367, one 6 is 10 times as great as the other 6. Is he correct? Explain why or why not.

Explanation:
Given in the number 6,367, one 6 is 10 times as great as the other 6.
The first 6 is on the place of the thousands, while the second 6 is on the place of the tens. so the first 6 is a hundred times as great as the second 6.

Question 17.
Jorge drew a square that had a side length of 8 inches. What is the perimeter of Jorge’s square?
Remember, the perimeter of a shape is the distance around it.

Answer: The perimeter of Jorge’s square is 32 inches.

Explanation:
Given, Jorge drew a square that had a side length of 8 inches.
Square has all equal sides, so perimeter of the square P = 4a = 4 × 8 = 32 inches

Question 18.
Higher Order Thinking
Dan wrote (2 × 106) + (3 × 104) + (5 × 103) + 4 for the expanded form of two million, three hundred fifty thousand, four. What error did he make in the expanded form? What is the standard form of the number?

Answer: The standard form of the number is 20,35,004.
The number name is two million, thirty five thousand and four.

Explanation:
(2 × 106) + (3 × 104) + (5 × 103) + 4
= 2,000,000 + 30,000 + 5,000 + 4
= 20,35,004.
The standard form of the number is 20,35,004.
The number name is two million, thirty five thousand and four.
He made mistake with the three hundred fifty thousand, instead of  thirty five thousand

Assessment Practice

Question 19.
Colleen says she is thinking of a 4-digit number in which all the digits are the same. The value of the digit in the hundreds place is 200.
Part A
What is the number? Explain.

Part B
Describe the relationship between the values of the digits in the number.
The first 2 is on the place of the thousands, while the second 2 is on the place of the hundreds, The third 2 is on the place of the tens, and the last 2 is on the place of the ones.

Explanation:
Given,  Colleen says she is thinking of a 4-digit number in which all the digits are the same. The value of the digit in the hundreds place is 200.
The number is 2,222.
The first 2 is on the place of the thousands, while the second 2 is on the place of the hundreds, The third 2 is on the place of the tens, and the last 2 is on the place of the ones.

### Lesson 1.3 Decimals to Thousandths

Solve & Share
At Suzie’s Sticker City, customers can buy a book of stickers, a page, a strip, or a single sticker. Provide the missing fractions in the boxes below.

How can you use what you know about powers of 10 to help you fill in the boxes?

Look Back! Use Structure Describe any patterns you notice in the fractions.

Visual Learning Bridge

Essential Question
How Can You Read and Write Question Decimals to the Thousandths?

A.
A box is filled with 1,000 cubes. Amy picks out 4 cubes. How can you represent 4 out of 1,000 cubes as a decimal?

You can write 4 out of 1,00 as the fraction $$\frac{4}{1,000}$$.

B.
The number name for $$\frac{4}{1,000}$$ is four thousandths. A decimal place-value chart can help you determine the decimal. Notice that the thousandths place is three places to the right of the decimal point.

So, $$\frac{4}{1,000}$$ can be represented by the decimal 0.004.

C.
How can $$\frac{444}{1,000}$$ be represented by a decimal? $$\frac{444}{1,000}$$ is read as four hundred forty-four thousandths and represented by the decimal 0.444.

The value of the digit 4 in the hundredths place has 10 times the value of the digit 4 in the thousandths place and $$\frac{1}{10}$$ the value of the digit 4 in the tenths place.

Convince Me! Reasoning How is 0.004 the same as and different from 0.444?

Answer: 0.004 and 0.444 both have 4 in the thousandth place

Explanation:
0.004 and 0.444 both have 4 in the thousandth place,
both are in thousandth place
different: 0.444 is greater than 0.004 when divide by 1000
444/1000 is greater than 4/1000
if convert to percentage : 0.444 is 44.4% and 0.004 is 0.4%

Guided Practice

Do You Understand?

Question 1.
If four cubes are pulled from the box on the previous page, how would you write the fraction representing the cubes that are left? the decimal representing the cubes that are left?
Answer:  In decimal form we can write it as 0.996.

Explanation:
Given, A box is filled with 1,000 cubes, picking out 4 cubes,
1000 – 4 = 996,
Then $$\frac{996}{1000}$$ are the remaining of the cubes which are left in the box
In decimal form we can write it as 0.996.

Do You Know How?

Question 2.
0.3 is 10 times as great as what decimal? 0.003 is $$\frac{1}{10}$$ of what decimal?
Answer: 0.3 is 10 times as great as 0.03 and 0.003 is $$\frac{1}{10}$$ of 0.03

Explanation:
0.3 × $$\frac{1}{10}$$ = 0.03
0.03 = 0.003 × $$\frac{1}{10}$$
So, 0.3 is 10 times as great as 0.03 and 0.003 is $$\frac{1}{10}$$ of 0.03

In 3-6, write each decimal as a fraction.

Question 3.
0.001 =
Answer: $$\frac{1}{1000}$$

Explanation:
Given 0.001 ,
In fraction form we can write it in to $$\frac{1}{1000}$$

Question 4.
0.05 =
Answer: $$\frac{5}{100}$$

Explanation:
Given 0.05 ,
In fraction form we can write it in to $$\frac{5}{100}$$

Question 5.
0.512 =
Answer: $$\frac{512}{1000}$$

Explanation:
Given 0.512 ,
In fraction form we can write it in to $$\frac{512}{1000}$$

Question 6.
0.309 =
Answer: $$\frac{309}{1000}$$

Explanation:
Given 0.309 ,
In fraction form we can write it in to $$\frac{309}{1000}$$

In 7-10, write each fraction as a decimal.

Question 7.
$$\frac{2}{1,000}$$ =

Explanation:
Given , $$\frac{2}{1,000}$$
In decimal form we can  write it into 0.002

Question 8.
$$\frac{34}{100}$$ =

Explanation:
Given , $$\frac{34}{100}$$
In decimal form we can  write it into 0.34

Question 9.
$$\frac{508}{1,000}$$ =

Explanation:
Given , $$\frac{508}{1,000}$$
In decimal form we can  write it into 0.508

Question 10.
$$\frac{99}{1,000}$$ = =

Explanation:
Given , $$\frac{99}{1,000}$$
In decimal form we can  write it into 0.099

Independent Practice

In 11-18, write each decimal as a fraction.

Question 11.
0.007
Answer: $$\frac{7}{1000}$$

Explanation:
Given 0.007 ,
In fraction form we can write it in to $$\frac{7}{1000}$$

Question 12.
0.08
Answer: $$\frac{8}{100}$$

Explanation:
Given 0.08 ,
In fraction form we can write it in to $$\frac{8}{100}$$

Question 13.
0.065
Answer: $$\frac{65}{1000}$$

Explanation:
Given 0.08 ,
In fraction form we can write it in to $$\frac{65}{1000}$$

Question 14.
0.9
Answer: $$\frac{9}{10}$$

Explanation:
Given 0.9 ,
In fraction form we can write it in to $$\frac{9}{10}$$

Question 15.
0.832
Answer: $$\frac{832}{1000}$$

Explanation:
Given 0.832 ,
In fraction form we can write it in to $$\frac{832}{1000}$$

Question 16.
0.203
Answer: $$\frac{203}{1000}$$

Explanation:
Given 0.203 ,
In fraction form we can write it in to $$\frac{203}{1000}$$

Question 17.
0.78
Answer:  $$\frac{78}{100}$$

Explanation:
Given 0.78 ,
In fraction form we can write it in to $$\frac{78}{100}$$

Question 18.
0.999
Answer: $$\frac{999}{1000}$$

Explanation:
Given 0.999 ,
In fraction form we can write it in to $$\frac{999}{1000}$$

In 19-26, write each fraction as a decimal.

Question 19.
$$\frac{434}{1,000}$$ =

Explanation:
Given, $$\frac{434}{1,000}$$
In decimal form we can  write it into 0.0.434

Question 20.
$$\frac{3}{10}$$ =

Explanation:
Given, $$\frac{3}{10}$$
In decimal form we can  write it into 0.3

Question 21.
$$\frac{873}{1,000}$$ =

Explanation:
Given, $$\frac{873}{1,000}$$
In decimal form we can  write it into 0.873

Question 22.
$$\frac{17}{1,000}$$ =

Explanation:
Given, $$\frac{17}{1,000}$$
In decimal form we can  write it into 0.017

Question 23.
$$\frac{309}{1,000}$$ =

Explanation:
Given, $$\frac{309}{1,000}$$
In decimal form we can  write it into 0.309

Question 24.
$$\frac{5}{1,000}$$ =

Explanation:
Given, $$\frac{5}{1,000}$$
In decimal form we can  write it into 0.005

Question 25.
$$\frac{6}{100}$$ =

Explanation:
Given, $$\frac{6}{100}$$
In decimal form we can  write it into 0.06

Question 26.
$$\frac{999}{1,000}$$ =

Explanation:
Given, $$\frac{999}{1,000}$$
In decimal form we can  write it into 0.999

Question 27.
Look at the middle 9 in exercise 18. How is its value related to the value of the 9 to its left? to the value of the 9 to its right?
Answer: The value of the digit 9 in the hundredths place has 10 times the value of the digit 9 in the thousandths place and $$\frac{1}{10}$$ the value of the digit 9 in the tenths place.

Explanation:
The value of the number given in exercise 18 is 0.999,
The first 9 is on the place of the tenths, while the second 9 is on the place of the hundredths and the second 9 is  on the place of thousandths.
So, The value of the digit 9 in the hundredths place has 10 times the value of the digit 9 in the thousandths place and $$\frac{1}{10}$$ the value of the digit 9 in the tenths place.

Problem Solving

Question 28.
The Palmers’ property tax bill for the year is $3,513. In their first installment, they paid$1,757. How much do they still owe on their bill? Write an equation to model your work.
Answer:  They still owe $1,756 on their bill . Explanation: Given, The Palmers’ property tax bill for the year is$3,513.
In their first installment, they paid $1,757.$3,513 – $1,757 =$1,756.
So, they still have to pay the half amount that is $1,756 Then$1,757 × 2 = $3,513. They still owe$1,756 on their bill .

Question 29.
Write the fractions $$\frac{22}{100}$$ and $$\frac{22}{1,000}$$ as decimals. How are the values of the digit 2 related in each of the decimals?

Explanation:
Given, $$\frac{22}{100}$$
In decimal form we can write it as 0.22,
The value of the digit 2 in the tenths place has 10 times the value of the digit 4 in the hundredths place .

And $$\frac{22}{1,000}$$
In decimal form we can write it as 0.022,
The value of the digit 2 in the hundredths place has 10 times the value of the digit 4 in the thousandths place.

Question 30.
Simon scored 4 × 102 points in a game. Joe scored 2 × 103 points in the same game. Whose score is higher? How much higher?
Answer: Joe scored 5 times as many as Simon.

Explanation:
Given, Simon scored 4 × 102 points in a game.
Joe scored 2 × 103 points in the same game.
Now we have  4 × 102 =  4 × 10 × 10 = 400
2 × 103 =  2 × 10 × 10 × 10 = 2000
Finally, Simon scored 400 point and Joe scored 2000 points
So, Joe scored 5 times as many as Simon.

Question 31.
Higher Order Thinking Kelly said that $$\frac{97}{1,000}$$ can be written as 0.97. Is she correct? Explain.
Answer: No , $$\frac{97}{1,000}$$  should be 0.097 not 0.97

Explanation:
Given, Kelly said that $$\frac{97}{1,000}$$ can be written as 0.97
The decimal form can written according to the number of zeros in the denominator
So , $$\frac{97}{1,000}$$  should be 0.097 not 0.97

Question 32.
Critique Reasoning Frank reasoned that in the number 0.555, the value of the 5 in the thousandths place is ten times as great as the 5 in the hundredths place. Is he correct? Explain.

Explanation:
Given, the number 0.555
The value of the digit 5 in the hundredths place has 10 times the value of the digit 5 in the thousandths place

Question 33.
How many cubes are in the box? What fraction of the entire box do the 7 cubes represent? Explain your answer.

Answer: $$\frac{7}{1000}$$

Explanation:
Given, 10 × 10 × 10  = 1000
And we have the 7 cubes
So, the fraction will be  $$\frac{7}{1000}$$

Assessment Practice

Question 34.
0.04 is 10 times as great as which decimal?
A. 0.4
B. 0.1
C. 0.004
D. 0.001
Answer: C , 0.04 is 10 times as great as 0.004

Explanation:
0.04 × $$\frac{1}{10}$$ = 0.004
So, 0.04 is 10 times as great as 0.004

Question 35.
0.009 is o of which decimal?
A. 0.01
B. 0.09
C. 0.1
D. 0.9
Answer:  B, 0.009 is $$\frac{1}{10}$$ of 0.09

Explanation:
0.09 × $$\frac{1}{10}$$ = 0.009
So, 0.009 is $$\frac{1}{10}$$ of 0.09

### Lesson 1.4 Understand Decimal Place Value

Solve & Share

A runner won a 100-meter race with a time of 9.85 seconds. How can you use place value to Explain this time? Complete a place-value chart to show this time.

Look Back! In the decimal 9.85, what is the value of the 8? What is the value of the 5?

Visual Learning Bridge

Essential Question How Can You Represent Decimals?

A.
Jo picked a seed from her flower. The seed has a mass of 0.245 gram. What are some different ways you can represent 0.245?

You can write the standard form, expanded form, and number name for a decimal just like you can for a whole number.

B.

Number Name: two hundred forty-five

A place-value chart can help you identify the tenths, hundredths, and thousandths places in a decimal.

Convince Me! Use Structure How many hundredths are in one tenth? How many thousandths are in one hundredth? Tell how you know.

Another example
Equivalent decimals name the same amount.
What are two other decimals equivalent to 1.4?
One and four tenths is the same as one and forty hundredths.
1.4 = 1.40
One and four tenths is the same as one and four hundred thousandths.
1.4 = 1.400.

So, 1.4 = 1.40 = 1.400.

Guided Practice

Do You Understand?

Question 1.
The number 3.453 has two 3s. Why does each 3 have a different value?
Answer: These are different due to the place values. 3 and 0.003

Explanation:
The given number is 3.453.
The first 3 that is before the decimal is at ones place, that is 3 × 1 = 3
But the second 3 which is after the decimal is at thousandth place, $$\frac{3}{1000}$$ = 0.003
Hence, you can clearly see the different values of both three’s. These are different due to the place values.

Do You Know How?

In 2 and 3, write each number in standard form.

Question 2.
4 × 100 + 7 × 10 + 6 × 1 + 6 × $$\left(\frac{1}{10}\right)$$ + 3 × $$\left(\frac{1}{100}\right)$$ + 7 × $$\left(\frac{1}{1,000}\right)$$
Answer:  The standard form is 476.637.

Explanation:
Given,
(4 × 100)+ (7 × 10) + (6 × 1) + {6 × $$\left(\frac{1}{10}\right)$$ + 3 × $$\left(\frac{1}{100}\right)$$ + 7 × $$\left(\frac{1}{1,000}\right)$$}
Solve for the brackets,
we have 400 + 70 + 6 + [ 0.6 + 0.03 + 0.007]
= 476 + 0.637
= 476.637
The standard form is 476.637.

Question 3.
four and sixty-eight thousandths
Answer: The standard form is 0.468.

Explanation:
Given, four and sixty-eight thousandths
The expanded form is 4 × $$\left(\frac{1}{10}\right)$$ + 6 × $$\left(\frac{1}{100}\right)$$ + 8 × $$\left(\frac{1}{1,000}\right)$$
Then, 0.4 + 0.06 + 0.008 = 0.468
So, the standard form is 0.468

Independent Practice

In 4-6, write each number in standard form.

Question 4.
(2 × 1) + (6 × $$\frac{1}{1,000}$$)
Answer: The standard form is 2.006

Explanation:
Given, (2 × 1) + (6 × $$\frac{1}{1,000}$$)
= 2 + 0.006
= 2.006
The standard form is 2.006

Question 5.
(3 × 1) + (3 × $$\frac{1}{10}$$) + (9 × $$\frac{1}{1,000}$$)
Answer: The standard form is 3.309

Explanation:
Given, (3 × 1) + (3 × $$\frac{1}{10}$$) + (9 × $$\frac{1}{1,000}$$)
= 3 + 0.3 + 0.009
= 3.309
So, The standard form is 3.309

Question 6.
nine and twenty hundredths
Answer: The standard form is 0.920.

Explanation:
Given, nine and twenty hundredths
The expanded form is (9 × $$\frac{1}{10}$$) + (20 × $$\frac{1}{100}$$)
= 0.9 + 0.020
= 0.920
The standard form is 0.920

In 7-10, write two decimals that are equivalent to the given decimal.

Question 7.
2.200
Answer: The two decimals that are equivalent to the given decimal are 2.2 and 2.20

Explanation:
Given, 2.200
Two and two tenths is the same as two and twenty hundredths.
2.2 = 2.20
Two and Two tenths is the same as two and twenty tenths.
2.2 = 2.2.
So, The two decimals that are equivalent to the given decimal are 2.2 and 2.20

Question 8.
8.1
Answer: The two decimals that are equivalent to the given decimal are 8.10 and 8.100

Explanation:
Given, 8.1
Eight and one tenths is the same as eight and one hundredths.
8.1 = 8.10
Eight and one tenths is the same as eight and one hundred thousandths.
8.1 = 8.100.
So, The two decimals that are equivalent to the given decimal are 8.10 and 8.100

Question 9.
9.50
Answer: The two decimals that are equivalent to the given decimal are 9.5 and 9.500

Explanation:
Given, 9.50
Nine and five tenths is the same as nine and  fifty tenths.
9.50 = 9.5.
Nine and five tenths is the same as nine and fifty hundred thousandths.
9.50 = 9.500.
So, The two decimals that are equivalent to the given decimal are 9.5 and 9.500

Question 10.
4.200
Answer: The two decimals that are equivalent to the given decimal are 4.2 and 4.20

Explanation:
Given, 4.200
Four and two tenths is the same as four and twenty hundredths.
4.200 = 4.20
Four and two tenths is the same as two and twenty tenths.
4.200 = 4.2.
So, The two decimals that are equivalent to the given decimal are 4.2 and 4.20

Problem Solving

Question 11.
The annual fundraising goal of a charity is $100,000. So far$63,482 has been raised. How much more money is needed to reach the goal?

Answer: To reach the goal they need $36,518. Explanation: Given, The annual fundraising goal of a charity is$100,000.
So far $63,482 has been raised. Then,$100,000 – $63,482 =$36,518.
So, To reach the goal they need $36,518. Question 12. Santiago has a rope that measures 205.95 centimeters. Write this number in expanded form. Answer: Explanation: Given, Santiago has a rope that measures 205.95 centimeters The standard form is 205.95 Then, The expanded form is (2 × 100) + (5 × 1) + (9 × $$\frac{1}{10}$$) + (5 × $$\frac{1}{100}$$) = 200 + 5 +[ 0.9 + 0.05 ] = 205 + 0.95 = 205.95 So, The expanded form is (2 × 100) + (5 × 1) + (9 × $$\frac{1}{10}$$) + (5 × $$\frac{1}{100}$$) Question 13. How can you tell that 7.630 and 7.63 are equivalent decimals? Answer: The two decimals that are equivalent to both the numbers Explanation: Given, 7.630 and 7.63 Seven and sixty three tenths is the same as Seven and sixty three hundredths. 7.63 = 7.630 So, The two decimals that are equivalent to both the numbers Question 14. In Justin’s school, 0.825 of the students participate in a sport. If there are one thousand students in Justin’s school, how many participate in a sport? Answer: Totally 825 students are participating in the school. Explanation: Given, In Justin’s school, 0.825 of the students participate in a sport. Since, there are 1 thousand students, and the decimal is 0.825, So, there are 825 students participating. Question 15. Be Precise Maria incorrectly placed the decimal point when she wrote 0.65 inch for the width of her tablet computer. What is the correct decimal number for the width? Answer: The correct decimal number for the width is 6.5. Explanation: Given, Maria incorrectly placed the decimal point when she wrote 0.65 inch for the width of her tablet computer. As shown in the figure, the scale placed there to measure the width , as per the scale the width is 6.5. So, The correct decimal number for the width is 6.5. Question 16. Higher Order Thinking Three boys cut out hundredths decimal models. Derrick does not shade any of his models. Ari shades half of one model. Wesley shades two models and one tenth of another model. What decimal represents the amount each boy shades? Answer: Derrick’s value = 0, Ari’s value = 0.5, Wesley’s value = 2.1 , Explanation: Given, Derrick does not shade any of his models. Ari shades half of one model. Wesley shades two models and one tenth of another model. Derrick doesn’t shade anything, so his value is 0. Ari shades half of one model, so 1/2 = 0.5 is his value Wesley shades 2 full models, plus 1/10 = 0.1 of another one, leading to 2+0.1 = 2.1 as his value. Derrick’s value = 0, Ari’s value = 0.5, Wesley’s value = 2.1 , Assessment Practice Question 17. Find two decimals that are equivalent to (4 × 10) + (7 × $$\frac{1}{100}$$). Write the decimals in the box. Answer: The two decimals that are equivalent to the 40.07 are 40.070 and 40.07 Explanation: Given, (4 × 10) + (7 × $$\frac{1}{100}$$). = 40 + 0.07 = 40.07 The standard form is 40.07 The two decimals that are equivalent to the 40.07 are 40.070 and 40.07 , from the given box of decimals. ### Lesson 1.5 Compare Decimals Activity Solve & Share The lengths of three ants were measured in a laboratory. The lengths were 0.521 centimeter, 0.498 centimeter, and 0.550 centimeter. Which ant was the longest? Which ant was the shortest? How can you use the math you know to compare and order the decimals? Tell how you decided. Look Back! Be Precise What are the lengths of the ants in order from least to greatest? Visual Learning Bridge Essential Question How Can You Compare Decimals? A. Scientists collected and measured the lengths of different cockroach species. Which cockroach had the greater length, the American or the Oriental cockroach? Comparing decimals is like comparing whole numbers! B. Step 1 Line up the decimal points. Start at the left. Compare digits of the same place value. 3.576 3.432 C. Step 2 Find the first place where the digits are different. 3.576 3.432 D. Step 3 Compare. 5 > 4 0.5 > 0.4 So, 3.576 > 3.432. The American cockroach is longer than the Oriental cockroach. Convince Me! Critique Reasoning Valerie said, “12.68 is greater than 12.8 because 68 is greater than 8.” Is she correct? Explain. Answer: No, 12.8 is greater than 12.68 Explanation: Given : 12.68 is greater than 12.8 We have given 12.68 and 12.8 In 12.68, Tens place is 1, Ones place is 2, Tenth place is 6, Hundredth place is 8. In 12.8, Tens place is 1 . Ones place is 2, Tenth place is 8, So, We can see Tens place and ones place of both number are same but tenth place of 12.8 is greater than 12.68. Therefore, 12.8 is greater than 12.68 . Another example Order the cockroaches from least to greatest length. Step 1 Write the numbers, lining up the decimal points. Start at the left. Compare digits of the same place value. 3.576 3.582 3.432 is the least. Step 2 Write the remaining numbers, lining up the decimal points. Start at the left. Compare. 3.576 3.582 3.582 is greater than 3.576. Step 3 Write the numbers from least to greatest. 3.432 3.576 3.582 From least to greatest lengths are the Oriental, the American, and the Australian. Guided Practice Do You Understand? Question 1. Scientists measured a Madeira cockroach and found it to be 3.44 centimeters long. Toby says that the Madeira is shorter than the Oriental because 3.44 has fewer digits than 3.432. Is he correct? Explain. Answer: 3.44cm cockroach is the longest. Explanation: Given, 3.44 and 3.432 3.44 3.432 Comparing the numbers after decimals, 4 > 3 So, 3.44cm cockroach is the longest. Do You Know How? In 2 and 3, write >, <, or = for each . Question 2. 3.692 3.697 Answer: 3.692 < 3.697 Explanation: Given, 3.692 and 3.697 3.692 3.697 Comparing the numbers after decimals, we have 2 < 7 So, 3.692 < 3.697. Question 3. 7.216 7.203 Answer: 7.216 > 7.203. Explanation: Given, 7.216 and 7.203 7.216 7.203 Comparing the numbers after decimals, we have 1 > 0 So, 7.216 > 7.203. In 4 and 5, order the decimals from least to greatest. Question 4. 5.540, 5.631, 5.625 Answer: The order from least to greatest is 5.540, 5.625, 5.631. Explanation: Given, 5.540, 5.631, 5.625 lining up the decimal points. 5.540, is the least 5.631, 5.625, Comparing the numbers after decimals, we have 5.631, 5.625, Compare digits of the same place value. 3 > 2 5.631 is the greatest So, The order from least to greatest is 5.540, 5.625, 5.631. Question 5. 0.675, 1.529, 1.35, 0.693 Answer: The order from least to greatest is 0.675, 0.693, 1.35, 1.529. Explanation: Given, 0.675, 1.529, 1.35, 0.693 lining up the decimal points. 0.675, 1.529, 1.35, 0.693 Comparing the numbers after decimals, we have 0.675, 0.693, Compare digits of the same place value. 7 < 9 0.675 is the less than 0.693 Then 1.529, 1.35, Compare digits of the same place value. 5 > 3 1.529 is greater than 1.35 So, The order from least to greatest is 0.675, 0.693, 1.35, 1.529. Independent Practice In 6-8, compare the two numbers. Write >, <, or = for each . Question 6. 0.890 0.890 Answer: 0.890 = 0.890 Explanation: Given, 0.890 and 0.890 0.890 0.890 Comparing the numbers after decimals, we have All the numbers are equal So, 0.890 = 0.890. Question 7. 5.733 5.693 Answer: 5.733 > 5.693. Explanation: Given, 5.733 and 5.693 5.733 5.693 Comparing the numbers after decimals, we have 7 > 6 So, 5.733 > 5.693. Question 8. 9.707 9.717 Answer: 9.707 < 9.717 Explanation: Given, 9.707 and 9.717 lining up the decimal points 9.707 9.717 Comparing the numbers after decimals, we have, Compare digits of the same place value. 0 < 1 So, 9.707 < 9.717. In 9 and 10, order the decimals from greatest to least. Question 9. 878.403, 887.304,887.043 Answer: The order from least to greatest is 878.403 ,887.043 , 887.304. Explanation: Given, 878.403, 887.304,887.043 lining up the decimal points. 878.403, is the least 887.304, 887.043 Comparing the numbers after decimals, we have 887.304, 887.043 Compare digits of the same place value. 3 > 0 887.304 is greater than 887.043 So, The order from least to greatest is 878.403 ,887.043 , 887.304. Question 10. 435.566, 436.565, 435.665 Answer: The order from least to greatest is 435.566, 435.665 , 436.565. Explanation: Given, 435.566, 436.565, 435.665 lining up the decimal points. 435.566, 436.565, is the greatest 435.665 Comparing the numbers after decimals, we have 435.566, 435.665 Compare digits of the same place value. 5 < 6 435.566 is less than 435.665 So, The order from least to greatest is 435.566, 435.665 , 436.565. Problem Solving Question 11. Critique Reasoning Explain why it is not reasonable to say that 4.23 is less than 4.135 because 4.23 has fewer digits after the decimal point than 4.135. Answer: 4.23 is greater than 4.135. Explanation: Given, 4.23 and 4.135 lining up the decimal points. 4.23, 4.135 Comparing the numbers after decimals, we have ,Compare digits of the same place value. 2 < 1 So, 4.23 is greater than 4.135. Question 12. Number Sense Carlos wrote three numbers between 0.33 and 0.34. What numbers could Carlos have written? Answer: there can be any three numbers mentioned Below between 0.33 and 0.34. Explanation: Given, three numbers between 0.33 and 0.34 The numbers between the 0.33 and 0.34 are 0.331, 0.332, 0.333, 0.334, 0.335, 0.336, 0.337, 0.338 and 0.339, Then the next number will be 0.340 that is same as 0.34 So, there can be any three numbers mentioned above between 0.33 and 0.34. Question 13. Vocabulary Draw lines to match each decimal on the left to its equivalent decimal on the right. Answer: The given decimals are compared to its nearest decimals by place value method. Explanation: Question 14. Is 0.5 greater than or less than $$\frac{6}{10}$$? Draw a number line to show your answer. Answer: 0.5 is < 0.6 Explanation: Given, 0.5 and $$\frac{6}{10}$$ We can write $$\frac{6}{10}$$ as 0.6 in decimal form, Compare digits of the same place value. 5 < 6 So, 0.5 is < 0.6 Question 15. Higher Order Thinking Ana’s gymnastics scores were posted on the scoreboard in order from highest to lowest score. One digit in her floor score is not visible. List all the possible digits for the missing number. Answer: The possible digits are 1 , 2 , 3 , 4 Explanation: Given, The floor score must be higher than 15.133 and lower than 15.500 Then all the possible scores are: 15.166 , 15.266 , 15.366 and 15.466 Because 15.166 > 15.133 and 15.466 < 15.500 Then all the possible digits are 1 , 2 , 3 , 4 Question 16. Marcia’s vault score is 15.050. How does it compare to Ana’s vault score? Answer: Marcia’s vault score is less than Ana’s vault score Explanation: Given, Marcia’s vault score is 15.050 Ana’s vault score is 15.500 Lining up the decimal points. 15.050 15.500 Comparing the numbers after decimals, we have ,Compare digits of the same place value. 0 < 5 So, 15.050 is less than 15.500. Finally, Marcia’s vault score is less than Ana’s vault score Assessment Practice Question 17. Which statements correctly compare two numbers? 0.1 <0.125 0.2 < 0.125 125 > 0.13 0.125 > 0.12 0.126 < 0.125 Answer: 0.1 < 0.125 , 125 > 0.13 , 0.125 > 0.12 . Explanation: Compare digits of the same place value. 0.1 < 0.125 , because 0 < 2 125 > 0.13 , because .0 > 0. 0.125 > 0.12 . because 5 > 0 Question 18. Cara weighed 4.16 pounds of apples at the grocery store. Which numbers make the statement true? > 4.16 04.15 04.19 4.2 4.09 4.1 Answer: 04.15 Explanation: Given, Cara weighed 4.16 pounds of apples at the grocery store 4.16 and 04.15 are the closest decimals Compare digits of the same place value. 5 < 6 So, the immediate number of 4.16 is 04.15. ### Lesson 1.6 Round Decimals Activity Solve & Share In science class, Marci recorded numbers from an experiment as 12.87, 12.13, 12.5, and 12.08. Which numbers are closer to 12? Which are closer to 13? How can you tell? You can use structure to help determine what number is halfway between two whole numbers. Show your work! Look Back! What is the halfway point between 12 and 13? Is that point closer to 12 or 13? Visual Learning Bridge Essential Question How Can You Round Decimals? A. Rounding replaces one number with another number that tells about how many or about how much. Round 2.36 to the nearest tenth. Is 2.36 closer to 2.3 or 2.4? A number line can help you round a decimal. B. Step 1 Find the rounding place. Look at the digit to the right of the rounding place. 2.36 C. Step 2 If the digit is 5 or greater, add 1 to the rounding digit. If the digit is less than 5, leave the rounding digit alone. Since 6 >5, add 1 to the 3. D. Step 3 Drop the digits to the right of the rounding digit. 2.36 rounds to 2.4 Rounding can help (you find which tenth or hundredth a decimal is closest to. Convince Me! Critique Reasoning Carrie said, “448 rounds to 500 because 448 rounds to 450 and 450 rounds to 500.” Is she correct? Explain. Use the number line in your explanation. Answer: No Explanation: Given, 448 rounds to 500 because 448 rounds to 450 and 450 rounds to 500.” The nearest number to 448 is 450 and that is correct but, 450 is not near the number 500 , The difference between these numbers is 50 , So, Carrie is wrong. Another example Round 3.2 to the nearest whole number. Is 3.2 closer to 3 or 4? Step 1 Find the rounding place. Look at the digit to the right of the rounding place. 3.2 Step 2 If the digit is 5 or greater, add 1 to the rounding digit. If the digit is less than 5, leave the rounding digit alone. Since 2 < 5, leave 3 the same. Step 3 Drop the digits to the right of the decimal point. Drop the decimal point. 3.2 rounds to 3. Guided Practice Do You Understand? Question 1. To round 74.58 to the nearest tenth, which digit do you look at? What is 74.58 rounded to the nearest tenth? Answer: the nearest tenth to 74.58 is 74.6 Explanation: Given, To round 74.58 to the nearest tenth, Find the rounding place. Look at the digit to the right of the rounding place. 8 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit. Drop the digits to the right of the decimal point. 74.6 So, the nearest tenth to 74.58 is 74.6 Question 2. A car-rental service charges customers for the number of miles they travel, rounded to the nearest whole mile. George travels 40.8 miles. For how many miles will he be charged? Explain. Answer: He will be charged for 41 miles Explanation: Given, George travels 40.8 miles. Find the rounding place. Look at the digit to the right of the rounding place. 8 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit. Drop the digits to the right of the decimal point. The nearest to 40.8 is 41 So, He will be charged for 41 miles Do You Know How? In 3-10, round each number to the place of the underlined digit. Question 3. 16.5 Answer: 17 Explanation: Given, 16.5 6 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit. Drop the digits to the right of the decimal point, that is 17 The rounding number is 17. Question 4. 56.1 Answer: 56 Explanation: Given, 56.1 6 is the rounding place, the digit is less than 5, leave the rounding digit alone. Drop the digits to the right of the decimal point, that is 56 The rounding number is 56. Question 5. 1.32 Answer: 1.3 Explanation: Given, 1.32 3 is the rounding place, the digit is less than 5, leave the rounding digit alone. Drop the digits to the right of the decimal point, that is 1.3 The rounding number is 1.3. Question 6. 42.78 Answer: 42.8 Explanation: Given, 42.78 7 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit. Drop the digits to the right of the decimal point, that is 42.8 The rounding number is 42.8. Question 7. 1.652 Answer: 1.65 Explanation: Given, 1.652 5 is the rounding place, the digit is less than 5, leave the rounding digit alone. Drop the digits to the right of the decimal point, that is 1.65 The rounding number is 1.65. Question 8. 582.04 Answer: 582.0 Explanation: Given, 582.04 4 is the rounding place, the digit is less than 5, leave the rounding digit alone. Drop the digits to the right of the decimal point, that is 582.0 The rounding number is 582.0. Question 9. 80,547.645 Answer: 80,547.65 Explanation: Given, 80,547.645 4 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit. Drop the digits to the right of the decimal point, that is 80,547.65 The rounding number is 80,547.65. Question 10. 135,701.949 Answer: 135,701.9 Explanation: Given, 135,701.949 9 is the rounding place, the digit is less than 5, leave the rounding digit alone. Drop the digits to the right of the decimal point, that is 135,701.9 The rounding number is 135,701.9. Independent Practice In 11-14, round each decimal to the nearest whole number. Question 11. 4.5 Answer: 5 Explanation: Given, 4.5 4 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit. Drop the digits to the right of the decimal point, that is 5 The rounding number is 5. Question 12. 57.3 Answer: 57 Explanation: Given, 57.3 7 is the rounding place, the digit is less than 5, leave the rounding digit alone. Drop the digits to the right of the decimal point, that is 57 The rounding number is 57. Question 13. 34.731 Answer: 34.73 Explanation: Given, 34.731 3 is the rounding place, the digit is less than 5, leave the rounding digit alone. Drop the digits to the right of the decimal point, that is 34.73 The rounding number is 34.73. Question 14. 215.39 Answer: 215.4 Explanation: Given,215.39 3 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit. Drop the digits to the right of the decimal point, that is 215.4 The rounding number is 215.4. In 15-18, round each number to the place of the underlined digit. Question 15. 7.158 Answer: 7.2 Explanation: Given, 7.158 1 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit. Drop the digits to the right of the decimal point, that is 7.2 The rounding number is 7.2. Question 16. 0.758 Answer: 0.76 Explanation: Given, 0.758 5 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit. Drop the digits to the right of the decimal point, that is 0.76 The rounding number is 0.76. Question 17. 6.4382 Answer: 6.44 Explanation: Given, 6.4382 3 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit. Drop the digits to the right of the decimal point, that is 6.44 The rounding number is 6.44. Question 18. 84.732 Answer: 84.7 Explanation: Given, 84.732 7 is the rounding place, the digit is less than 5, leave the rounding digit alone. Drop the digits to the right of the decimal point, that is 84.7 The rounding number is 84.7. Problem Solving Question 19. The picture at the right shows the length of an average American alligator. What is the length of the alligator rounded to the nearest tenth? Answer: 4.4 is the length of the alligator rounded to the nearest tenth. Explanation: Given, 4.39 3 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit. Drop the digits to the right of the decimal point, that is 4.4 The rounding number is 4.4. So,4.4 is the length of the alligator rounded to the nearest tenth Question 20. Name two different numbers that round to 8.21 when rounded to the nearest hundredth. Answer: 8.211 and 8.212 Explanation: Given, 8.21 rounded to the nearest hundredth, 2 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit. Drop the digits to the right of the decimal point, that is 8.211 and 8.212 The rounding number is 8.211 and 8.212. Question 21. Number Sense To the nearest hundred, what is the greatest whole number that rounds to 2,500? the least whole number? Answer: 2,499 and 2,491. Explanation: Given, 2,500 The greatest whole number to nearest hundred of 2,500 is 2,499 And, The less whole number to nearest hundred of 2,500 is 2,491 Question 22. Draw all of the lines of symmetry in the figure shown below. Answer: There are 2 line lines of symmetry. Explanation: The given figure is in the form of a Rectangle So, it has 2 lines of symmetry. Question 23. Higher Order Thinking Emma needs 2 pounds of ground meat to make a meatloaf. She has one package with 2.36 pounds of ground meat and another package with 2.09 pounds of ground meat. She uses rounding and finds that both packages are close to 2 pounds. Explain how Emma can choose the package closer to 2 pounds. Answer: She can use the package of 2.09 because it is close to 2 pounds. Explanation: Given, 2.09 and 2.36 For 2.09 , 0 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit. Drop the digits to the right of the decimal point. 2.1 For 2.36, 3 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit. Drop the digits to the right of the decimal point. 2.4 So, She can use the package of 2.09 because it is close to 2 pounds. Question 24. Make Sense and Persevere Robert slices a large loaf of bread to make 12 sandwiches. He makes 3 turkey sandwiches and 5 veggie sandwiches. The rest are ham sandwiches. What fraction of the sandwiches Robert makes are ham? Answer: $$\frac{3}{12}$$, $$\frac{5}{12}$$, $$\frac{4}{12}$$. Explanation: Given, Robert slices a large loaf of bread to make 12 sandwiches, He makes 3 turkey sandwiches and 5 veggie sandwiches The rest are ham sandwiches. The number of turkey sandwiches are $$\frac{3}{12}$$, The number of veggie sandwiches are $$\frac{5}{12}$$, The number of ham sandwiches are $$\frac{4}{12}$$, Total 12 sandwiches are there made by Robert. Question 25. Algebra After buying school supplies, Ruby had$32 left over. She spent $4 on notebooks,$18 on a backpack, and $30 on a new calculator. How much money, m, did Ruby start with? Write an equation to show your work. Answer: Ruby started with$84.

Explanation:
Given, Ruby had $32 left over. She spent$4 on notebooks, $18 on a backpack, and$30 on a new calculator.
The total money spent =  $4 +$18  +  $30 =$52
we have, $52 +$32 = $84 So, Ruby started with$84.

Assessment Practice

Question 26.
Find two numbers that round to 35.4 when rounded to the nearest tenth. Write the numbers in the box.

Explanation:
Given, 35.40
4 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 35.45
if the digit is 5 or greater, add 1 to the rounding digit. 35.391
The rounding number is 35.45 and 35.391.

### Lesson 1.7 Look For and Use Structure

Problem Solving

Activity

Solve & Share
Angie volunteers in the school library after school. The librarian gave her a stack of books and told her to use the number on each book to shelve it where it belongs.
How can Angie arrange the books in order from least to greatest to make shelving them easier?

Thinking Habits
• What patterns can I see and describe?
• How can I use the patterns to solve the problem?
• Can I see expressions and objects in different ways?
• What equivalent expressions can I use?

Look Back! Use Structure Explain why 323.202 is less than 323.21 even though 202 is greater than 21.

Answer: Because , when rounding the decimals we have to look for the place values to decide which is greater or least.

Explanation:
Given, 323.202
0 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 323.21
The rounding number is 323.21.

Visual Learning Bridge

Essential Question
How Can You Use Structure to Solve Problems?

A.
Analyze the chart. What do you notice that can help you complete the chart?

What do I need to do to solve this problem?

I can use the structure of the decimal place-value system to complete the chart.

You can look for patterns to find the missing numbers.

B.
How can I make use of structure to solve this problem?
I can
• find and describe patterns.
• use the patterns to see how the numbers are organized.
• analyze patterns to see the structure in the table.
• break the problem into simpler parts.

C.
Solve
Here’s my thinking…

As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same, except for the last number, while the hundredths increase by 1.

Convince Me! Use Structure Write the missing numbers. Explain how you can use structure to find the last number in the bottom row.

Explanation:
Because, down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1.

Guided Practice
Use Structure Each of these grids is a part of a decimal number chart similar to the one on page 30.

You can use what you know about place value when you look for patterns with decimals.

Question 1.
Describe the pattern for moving from a pink square to a green square. Then write the missing numbers.

Explanation:
Because, down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1.

Question 2.
How can you use patterns to find the number that would be in the box below 0.52?
Answer: the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1. and completing the pattern.

Independent Practice

Use Structure Pamela is hiking. When she returns to camp, she passes the mile markers shown at the right.

Question 3.
Explain how you can use structure to find the decimal numbers that will be shown on the next four mile markers.

Answer:  the next four mile markers will be 2.2 , 2.1 , 2.0 , 1.9

Explanation:

Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1
So, the next four mile markers will be 2.2 , 2.1 , 2.0 , 1.9

Question 4.
Pamela stops at the 1.8 mile marker. Where will she be if she walks one tenth of a mile towards camp? one mile towards camp? Explain.
Answer: she will be on 0.18 if she walks one tenth of a mile towards camp and 1.7 if she walks one mile towards the camp.

Explanation:
Given, 1.8
8 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 1.7
The reducing mile as reaching out for camp is 1.7
If she will be on 0.18 if she walks one tenth of a mile towards camp, That is 1.8 × $$\frac{1}{10}$$
Finally, she will be on 0.18 if she walks one tenth of a mile towards camp and 1.7 if she walks one mile towards the camp.

Problem Solving

Thousandths Chart
The students in Ms. Lowell’s class wrote a thousandths decimal chart on the board. Some of the numbers got erased.

Explanation:
Because, down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1.

Question 5.
Use Structure Describe the pattern for moving across a row from left to right.

Explanation:
Moving from left to right in the rows, tenths stay the same, except for the last number, while the hundredths increase by 1.

You can use structure to decide if decimal numbers are following a pattern.

Explanation:
Above question has the figure, we have the decimal numbers following a pattern of  tenths stay the same, except for the last number, while the hundredths increase by 1.

Question 6.
Be Precise How does the pattern change in the last square of each row?
Answer: As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.

Explanation:

As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.

Question 7.
Use Structure Describe the pattern for moving down a column.
Answer: As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.

Explanation:

As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.

Question 8.
Use Repeated Reasoning Write the missing numbers in the decimal chart above.

Explanation:
Because, down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1.

Question 9.
Use Structure Suppose the students add to the chart. Write the missing numbers in the row and the column below.

Answer:  The complete chart is as follows,

Explanation:
Because, down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1.

### Topic 1 Fluency Review

Activity

Find a Match

Work with a partner. Point to a clue. Read the clue. Look below the clues to find a match. Write the clue letter in the box above the match. Find a match for every clue.

Clues
A The sum is between 15,000 and 20,000.
B The difference is less than 10,000.
C The difference is between 41,000 and 42,000.
D The sum is exactly 52,397.
E The difference is between 82,000 and 84,000.
F The sum is greater than 79,000.
G The sum is exactly 52,407,
H The difference is exactly 42,024.

Topic 1 Vocabulary Review

Understand Vocabulary

Glossary

Word List
• base
• equivalent decimals
• expanded form
• exponent
• power
• thousandths
• value

Choose the best term from the Word List. Write it on the blank.

Question 1.
Decimal numbers that name the same part of a whole or the same point on a number line are called ____
Answer: Decimal numbers that name the same part of a whole or the same point on a number line are called Equivalent decimals

Question 2.
The ____ of a digit in a number depends on its place in the number.
Answer: The value of a digit in a number depends on its place in the number.

Question 3.
The product that results from multiplying the same number over and over is a(n) ____ of that number.
Answer:  The product that results from multiplying the same number over and over is a(n) POWER of that number.

Question 4.
A digit in the hundredths place has ten times the value of the same digit in the ___ place.
Answer: A digit in the hundredths place has ten times the value of the same digit in the thousandths place.

Question 5.
In 105, the number 10 is the _____.
Answer: In 105, the number 10 is the BASE.

Draw a line from each number in Column A to the same number in Column B.

Use Vocabulary in Writing

Question 10.
Explain why each 8 in the number 8.888 has a different value. Use one or more terms from the Word List in your explanation.
Answer:  Because of its place value.

Explanation:
Given, 8.888
The first 8 has the place value of tenths, The second 8 has the place value of hundredths and the final 8 has the place value of thousandths.

Set A
pages 5-8
How can you write 7,000 using exponents?
7,000 = 7 × 10 × 10 × 10 = 7 × 103
So, using exponents, you can write 7,000 as 7 × 103

Remember the number of zeros in the product is the same as the exponent.

Find each product.

Question 1.
9 × 101

Explanation:
9 × 101 = 9 × 10 = 90

Question 2.
8 × 1,000

Explanation:
8 × 10 × 10 × 10 = 8,000

Question 3.
5 × 102
Answer: 5 × 102 = 500

Explanation:
5 × 102 =  5 × 10 × 10  = 500

Question 4.
2 × 105
Answer: 2 × 105  = 2,00,000

Explanation:
2 × 105  = 2 × 10 × 10 × 10 × 10 × 10 = 2,00,000

Set B
pages 9-12
Write the number name and tell the value of the underlined digit for 930,365.
Nine hundred thirty thousand, three hundred sixty-five
Since the 0 is in the thousands place, its value is 0 thousands, or 0.
Use digital tools to solve these and other problems.

Remember you can find the value of a digit by its place in a number.

Write the number name and tell the value of the underlined digit.

Question 1.
9,000,000

Explanation:
the value of the underlined digit is in millions place.

Question 2.
485,002,000
Answer: four hundred and eighty five million and two thousands

Explanation:
the value of the underlined digit is in millions place.

Question 3.
25,678
Answer: Twenty five thousand and six hundred and seventy eight

Explanation:
the value of the underlined digit is in thousands place.

Question 4.
17,874,000
Answer: seventeen million and eight hundred and seventy four thousand

Explanation:
the value of the underlined digit is in ten millions place.

Set C
pages 13-16, 17-20
A place-value chart can help you write the standard form, expanded form, and number name for a decimal.

Standard form: 8.026
expanded form: 8 + 2 × $$\frac{1}{100}$$ + 6 × $$\frac{1}{1,000}$$
Number name: Eight and twenty-six thousandths

Remember the word and is written for the decimal point.

Question 1.
How can you write 0.044 as a fraction?
How are the values of the two 4s related in 0.044?
Answer: $$\frac{44}{1,000}$$ and The first 4 is in hundredths place and last 4 is in thousandths place

Explanation:
$$\frac{44}{1,000}$$ = 0.044,
The first 4 is in hundredths place and last 4 is in thousandths place

Write each number in standard form.

Question 2.
eight and fifty-nine hundredths

Question 3.
seven and three thousandths + 4. 3 + 2 × $$\frac{1}{10}$$ + 4 × $$\frac{1}{1000}$$

Explanation:
Given, seven and three thousandths + 4. 3 + 2 × $$\frac{1}{10}$$ + 4 × $$\frac{1}{1000}$$
7.003 + 4.3 + [ 0.2 + 0.004]
= 11.303 + 0.204
= 11.507

Set D
pages 21-24 Compare. Write >, <, or =.

8. 45 8.47
Line up the decimal points. Start at the left to compare. Find the first place where the digits are different.
8.45
8.47 0.05 < 0.07
So, 8.45 < 8.47.

Remember that equivalent decimals, such as 0.45 and 0.450, can help you compare numbers.
Compare. Write >, <, or =.

Question 1.
0.584 0.58

Explanation:
Given,  0.584 and 0.58
0.584
0.58
Comparing the numbers after decimals,
4 > 0
So, 0.584 > 0.58

Question 2.
9.327 9.236

Explanation:
Given, 9.327 and 9.236
9.327
9.236
Comparing the numbers after decimals,
3 > 2
So, 9.327 > 9.236

Question 3.
5.2 5.20

Explanation:
Given, 5.2 and 5.20
5.2
5.20
Comparing the numbers after decimals, we have
All the numbers are equal
So, 5.2 = 5.20

Question 4.
5.643 5.675

Explanation:
Given, 5.643 and 5.675
5.643
5.675
Comparing the numbers after decimals, we have
4 < 7
So, 5.643 < 5.675

Question 5.
0.07 0.08

Explanation:
Given,  0.07 and 0.08
0.07
0.08
Comparing the numbers after decimals, we have
7 < 8
So, 0.07 < 0.08.

Set E
pages 25-28
Round 12.087 to the place of the underlined digit.
12.087 Look at the digit following the underlined digit. Look at 7.
Round to the next greater number of hundredths because 7 > 5.
12.087 rounded to the nearest hundredth is 12.09.

Remember that rounding a number means replacing it with a number that tells about how many or how much.
Round each number to the place of the underlined digit.

Question 1.
10.245

Explanation:
Given, 10.245
2 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 10.2
The rounding number is 10.2.

Question 2.
73.4

Explanation:
Given, 73.4
7 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 73
The rounding number is 73.

Question 3.
0.145

Explanation:
Given, 0.145
1 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 0.1
The rounding number is 0.1.

Question 4.
3.999

Explanation:
Given, 3.999
9 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 4
The rounding number is 4.

Question 5.
13.023

Explanation:
2 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 13
The rounding number is 13.

Question 6.
45.398

Explanation:
3 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 45
The rounding number is 45.

Set F
pages 29-32
Think about these questions to help you look for and use structure to understand and Explain patterns with decimal numbers.

Thinking Habits
• What patterns can I see and describe?
• How can I use the patterns to solve the problem?
• Can I see expressions and objects in different ways?
• What equivalent expressions can I use?

Each grid is part of a decimal number chart. Write the missing numbers to complete the grids.

Question 1.

Explanation:
As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same, except for the last number, while the hundredths increase by 1.

Question 2.

Explanation:
As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same, except for the last number, while the hundredths increase by 1.

### Topic 1 Assessment Practice

Question 1.
Complete the sentences to make true statements.
6 is 100 times ____.
0.06 is 10 times ______.
60 is $$\frac{1}{100}$$ of ____.

Explanation:
6 × 100 = 600
0.06 × 10 = 0.6
60 × $$\frac{1}{100}$$ = 0.6

Question 2.
A national park has eighty thousand, nine-hundred twenty-three and eighty-six hundredths acres of land. Which shows this in standard form?
A. 80,923.086
B. 80,923.68
C. 80,923.806
D. 80,923.86
Answer: A , that is 80,923.086

Explanation:
Because, the number name of 80,923.086 is eighty thousand, nine-hundred twenty-three and eighty-six hundredths

Question 3.
Which numbers have a digit in the ones place that is $$\frac{1}{10}$$ the value of the digit in the tens place? Select all that apply.
9,077
9,884
1,303
1,055
3,222

Explanation:
These numbers 9,884 and  3,222 have a digit in the ones place that is $$\frac{1}{10}$$ the value of the digit in the tens place

Question 4.
Mrs. Martin has $7,000 in her savings account. Alonzo has $$\frac{1}{10}$$ to as much money in his account as Mrs. Martin. How much money does Alonzo have in his account? Answer: Amount Alonzo has is$ 700

Explanation:
he amount Martin has in her account is $7000 Alonzo has 1/10th the amount Martin has Therefore Alonzo has $$\frac{1}{10}$$ of 7000 ; = 7000/10 =$ 700
Amount Alonzo has is $700 . Question 5. Select all the comparisons that are true. 04.15 > 4.051 1.054 > 1.45 5.14 < 5.041 5.104 < 5.41 5.014 < 5.41 Answer: 04.15 > 4.051, 5.104 < 5.41 and 5.014 < 5.41 Explanation: Because of their place value , 04.15 > 4.051 1 > 0 So, 04.15 > 4.051 5.104 < 5.41 1 < 4 So, 5.104 < 5.41 5.014 < 5.41 0 < 4 So, 5.014 < 5.41 Question 6. Luke shaded 20 squares on his hundredths grid. Bekka shaded 30 squares on her hundredths grid. A. Whose grid represents the larger decimal? B. Write two decimals equivalent to Luke’s decimal. Answer: 0.2 < 0.3, The two decimals equivalent to Luke’s decimal are 0.20 and 0.21 Explanation: Given, Luke shaded 20 squares on his hundredths grid, $$\frac{20}{100}$$ that is 0.2 Bekka shaded 30 squares on her hundredths grid. $$\frac{30}{100}$$that is 0.3 Bekka has largest decimal. The two decimals equivalent to Luke’s decimal are 0.20 and 0.21 Question 7. Which statements about the values of 2.044 and 20.44 are true? Select all that apply. 2.044 is $$\frac{1}{10}$$ of 20.44. 2.044 is $$\frac{1}{100}$$ of 20.44. 20.44 is 10 times 2.044. 20.44 is 100 times 2.044. 2.044 is 10 times 20.44. Answer: 20.44 is 10 times 2.044. Explanation: Given, 20.44 is 10 times 2.044. 20.44 × 10 = 2.044. Question 8. The weight of Darrin’s phone is 3.405 ounces. What is 3.405 written in expanded form? A. 3 × 1 + 4 × $$\frac{1}{10}$$ + 5 × $$\frac{1}{1,000}$$ B. 3 × 10 + 4 × $$\frac{1}{10}$$ + 5 × $$\frac{1}{1,000}$$ C. 3 × 10 + 4 × $$\frac{1}{10}$$ + 5 × $$\frac{1}{100}$$ D. 3 × 1 + 4 × $$\frac{1}{100}$$ + 5 × $$\frac{1}{1,000}$$ Answer: D Explanation: Given, 3.405 The expanded form, 3 × 1 + 4 × $$\frac{1}{100}$$ + 5 × $$\frac{1}{1,000}$$ = 3 +0.04 + 0.005 = 3.405 Question 9. Elaine has a piece of wire that is 2.16 meters long. Dikembe has a piece of wire that is 2.061 meters long. Whose piece of wire is longer? How can you tell? Answer: Dikembe has the longest wire. Explanation: Given, 2.16 and 2.061 2.16 2.061 Comparing the place values of the digits after decimal 1 > 0 So, 2.16 > 2.061 Dikembe has the longest wire. Question 10. In a basketball tournament, Dimitri averaged 12.375 rebounds per game. What is 12.375 written in expanded form? How is it written with number names? Answer: The expanded form is (12 × 1) + {3 × $$\frac{1}{10}$$ + 7 × $$\frac{1}{100}$$ + 5 × $$\frac{1}{1,000}$$} Explanation: Given, 12.375 The expanded form is (12 × 1) + {3 × $$\frac{1}{10}$$ + 7 × $$\frac{1}{100}$$ + 5 × $$\frac{1}{1,000}$$} 12 + 0.3 + 0.07 + 0.005 = 12.375. The number name is twelve thousand and three hundred and seventy five. Question 11. The numbers below follow a pattern. 0.006 0.06 0.6 6 ____ _____ A. What are the next two numbers in the pattern? B. What is the relationship between the terms in the pattern? Answer: 60 and 600 , The terms relate to each other in that the next term is 10 times the previous term. Explanation: Relationship between term: The next term is ten times the previous term. 0.006 = 6/1000 0.06 = 6/100 0.6 = 6/10 6 = 6/1 This is a geometric sequence, where the common ratio is 10. 0.6/0.06 = 6/0.6 = 0.06/0.006 = 10 To get next number we simply multiply the previous number by 10. The terms relate to each other in that the next term is 10 times the previous term. Question 12. Kendra and her horse completed the barrel racing course in 15.839 seconds. What is this number rounded to the nearest tenth? Explain how you decided. Answer: The rounding number is 16. Explanation: Given, 15.839 5 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit. Drop the digits to the right of the decimal point, that is 16 The rounding number is 16. ### Topic 1 Performance Task Fruits and Vegetables Henry recorded how many pounds of fruits and vegetables his family bought during the past two months. Question 1. Pick four fruits and list them in the table below. Part A Round each fruit’s weight to the nearest 0.1 pound. Write the rounded weight in the next column. Answer: Part B Explain how you rounded the weights of the fruits. Answer: detailed explanation is given below. Explanation: For Apples, given 2.068 5 is the rounding place, the digit is less than 5, leave the rounding digit alone. Drop the digits to the right of the decimal point, that is 2 The rounding number is 2. For Blueberries 1.07 0 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit. Drop the digits to the right of the decimal point, that is 1.1 The rounding number is 1.1. For lemons 1.031 0 is the rounding place, the digit is less than 5, leave the rounding digit alone. Drop the digits to the right of the decimal point, that is 1 The rounding number is 1. For oranges 3.502 0 is the rounding place, the digit is less than 5, leave the rounding digit alone. Drop the digits to the right of the decimal point, that is 3.5 The rounding number is 3.5. For peaches 2.608 0 is the rounding place, the digit is less than 5, leave the rounding digit alone. Drop the digits to the right of the decimal point, that is 2.6 The rounding number is 2.6. For pears 3.592 5 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit. Drop the digits to the right of the decimal point, that is 3.6 The rounding number is 3.6. Question 2. Pick four vegetables and list them in the table below. Part A Round each vegetable’s weight to the nearest 0.01 pound. Write the rounded weight in the next column. Answer: Part B Explain how you rounded the weights of the vegetables. Answer: Detailed explanation is given below Explanation: For Beets 1.862 6 is the rounding place, the digit is less than 5, leave the rounding digit alone. Drop the digits to the right of the decimal point, that is 1.86 The rounding number is 1.86. For Celery 1.402 0 is the rounding place, the digit is less than 5, leave the rounding digit alone. Drop the digits to the right of the decimal point, that is 1.4 The rounding number is 1.4. For Corn 2.556 5 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit. Drop the digits to the right of the decimal point, that is 2.56 The rounding number is 2.56. For potatoes 3.441 4 is the rounding place, the digit is less than 5, leave the rounding digit alone. Drop the digits to the right of the decimal point, that is 3.44 The rounding number is 3.44. Question 3. Use <, >, or= to compare the weights of blueberries and lemons. Answer: Question 4. When rounded to the nearest hundredth, two items will round to the same decimal. What two items are they? Answer: Question 5. How does writing the weight for potatoes in expanded form show why the same digit can have different values? Answer: Because of the place value of the numbers in the decimals Explanation: For potatoes 3.441 4 is the rounding place, the digit is less than 5, leave the rounding digit alone. Drop the digits to the right of the decimal point, that is 3.44 The rounding number is 3.44. The first 4 has the place value of tenths and the last 4 has the place value of hundredths. Question 6. What is the relationship between the values of the two 4s in the weight of the potatoes? Answer: The first 4 has the place value of tenths and the last 4 has the place value of hundredths. and the first 4 is ten times the value of second 4. Explanation: For potatoes 3.441 4 is the rounding place, the digit is less than 5, leave the rounding digit alone. Drop the digits to the right of the decimal point, that is 3.44 The rounding number is 3.44. The first 4 has the place value of tenths and the last 4 has the place value of hundredths. and the first 4 is ten times the value of second 4. Question 7. Write the number of pounds of celery Henry’s family bought using number names and in expanded form. Answer: The number name is 1.4 pounds The expanded form is 14 × $$\frac{1}{10}$$ Explanation: For Celery 1.402 0 is the rounding place, the digit is less than 5, leave the rounding digit alone. Drop the digits to the right of the decimal point, that is 1.4 The rounding number is 1.4. The number name is 1.4 pounds The expanded form is 14 × $$\frac{1}{10}$$ Question 8. The store where Henry’s family shops sold 103 times as many pounds of corn as Henry’s family bought. Answer:263.68 pounds have sold. Explanation: For Corn 2.556 5 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit. Drop the digits to the right of the decimal point, that is 2.56 The rounding number is 2.56. Given, Henry’s family shops sold 103 times as many pounds of corn as Henry’s family bought. 2.56 × 103 = 263.68 Part A How many pounds of corn did the store sell? Write your answer in standard form and with number names. Answer: The number form is twenty six thousand and three hundred and sixty eight. Explanation: Store has sold 263.68 The number form is twenty six thousand and three hundred and sixty eight. Part B Explain how you found your answer. Answer: detailed explanation is given below Explanation: For Corn 2.556 5 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit. Drop the digits to the right of the decimal point, that is 2.56 The rounding number is 2.56. Given, Henry’s family shops sold 103 times as many pounds of corn as Henry’s family bought. 2.56 × 103 = 263.68 #### enVision Math Common Core Grade 5 Answer Key ## Envision Math Common Core Grade 5 Answer Key Topic 12 Convert Measurements ## Envision Math Common Core 5th Grade Answers Key Topic 12 Convert Measurements enVision STEM Project: Grand Canyon Do Research Use the Internet and other sources to learn about the Grand Canyon and the Colorado River. Where is the Grand Canyon? How was it formed? What do the different rock layers tell us? Predict how you think the canyon dimensions will change in a million years. Journal: Write a Report Include what you found. Also in your report: • Describe the canyon’s dimensions. • Describe the Colorado River’s dimensions. • Define erosion. • Make up and solve problems involving measurement units and conversions. Review What You Know A-Z Vocabulary Choose the best term from the box. Write it on the blank. • customary • multiplication • subtraction • exponent • metric Question 1. A meter is a unit of length in the ___ system of measurement. Answer: A meter is a unit of length in the Metric system of measurement. Question 2. A foot is a unit of length in the ____ system of measurement. Answer: A foot is a unit of length in the British imperial and United States customary system of measurement. Question 3. The division has an inverse relationship with _____ Answer: The division has an inverse relationship with Multiplication. Question 4. A(n) ____ shows the number of times a base is used as a factor. Answer: An Exponent shows the number of times a base is used as a factor. Multiplication Find each product. Question 5. 60 × 6 Answer: 60 × 6 = 360 Question 6. 24 × 103 Answer: 24 × 103 = 24000 Question 7 16 × 7 Answer: 16 × 7 = 112 Question 8. 102 × 1.6 Answer: 102 × 1.6 = 160 Question 9. 100 × 34 Answer: 100 x 34 = 3400 Question 10. 104 × 0.37 Answer: 104 × 0.37 = 3700 Question 11. 46.102 × 102 Answer: 46.102 × 102 = = 4702. 404 Question 12. 101 × 0.005 Answer: 0.05 Division Find each quotient. Question 13. 1,000 ÷ 100 Answer: 1,000 ÷ 100 Quotient = 10 Question 14. 176 ÷ 16 Answer: 176 ÷ 16 Quotient = 11 Question 15. 3,600 ÷ 60 Answer: 3,600 ÷ 60 Quotient = 60 Question 16. 120 ÷ 24 Answer: 120 ÷ 24 Quotient = 5 Measurement Circle the more appropriate unit of measure for each item. Question 17. The capacity of a swimming pool: liters or milliliters Answer: Liters Question 18. The length of an ear of corn: yards or inches Answer: Yards Question 19. The mass of a gorilla: grams or kilograms Answer: Kilograms Question 20. The weight of a tennis ball: ounces or pounds Answer: Ounces Question 21. Would you use more centimeters or meters to measure the length of car? Explain. Answer: meters is more useful to measure the length of the car. meters are used to measure the length of the car. Because centimeters are shorter than meters. Pick a Project PROJECT 12A What makes a treehouse so cool? Project: Build a Model of a Treehouse PROJECT 12B What would you weigh on Mars? Project: Make a Mobile Display of the Solar System PROJECT 12C Have you ever heard of National Punch Day? Project: Plan a Class Party PROJECT 12D What are the characteristics of Florida panthers? Project: Design a Zoo Space for Florida Panther Cubs ### Lesson 12.1 Convert Customary Units of Length Activity Solve & Share William has a piece of wire that measures 1 yard long. He will use wire to fix several electrical outlets in his house. How many inches long is the wire? Solve this problem by using bar diagrams. You can show the relationship between yards and inches in a bar diagram. Show your work! Look Back! Generalize How can you convert inches to yards? Would you multiply or divide when converting from a smaller unit to a larger unit? Explain. Visual Learning Bridge Essential Question How Do You Change from One Leon Unit of Length to Another? A. Some frogs can jump 11$$\frac{1}{4}$$ feet. What are some other ways to describe the same distance? The table shows equivalent measures. B. To change larger units to smaller units, multiply. You know 1 foot equals 12 inches. You know 3 feet is equal to 1 yard. C. To change smaller units to larger units, divide. Ed’s frog jumped 11 feet. How many yards is this? 11 ÷ 3 = 3 R2 So, 11 feet = 3 yards, 2 feet. Convince Me! Generalize In the example above, explain how you could use a mixed number to write 11 feet as an equivalent measure in yards. Guided Practice Do You Understand? Question 1. If you want to convert yards to feet, what operation would you use? Answer: To convert a yard measurement to a foot measurement, multiply the length by the conversion ratio. The length in feet is equal to the yards multiplied by 3. Question 2. If you want to convert feet to miles, what operation would you use? Answer: To convert a foot measurement to a mile measurement, divide the length by the conversion ratio. Question 3. What are some tools you could select to measure length? Explain when you would use them. Answer: The most common way to measure length is by using the scale on some sort of hand-held tool or implement, but you can also measure length — or distance — with radar, sonar, and laser beams. Do You Know How? In 4-8, convert each unit of length. Question 4. 9 ft = ___ yd Answer: 3 Question 5. 8 ft 7 in. = ___ in. Answer: 103 inches Question 6. 5$$\frac{1}{2}$$ ft = __in. Answer: 66 inches Question 7. 288 in. = __ yd Answer: 8 yards Question 8. 219 in. = ___ ft ___ in. or. ___ ft Answer: 18 feet 3 inches 18.25 ft Independent Practice In 9 and 10, complete the table to show equivalent measures. Will the number in your answer be greater than or less than the number in the given measurement? Question 9. Answer: 1 feet = 12 inches 2 feet = 24 inches 3 feet = 36 inches 4 feet = 48 inches Question 10. Answer: 1 yard = 3 feet 2 yard = 6 feet 3 yard = 9 feet 4 yard = 12 feet In 11-16, convert each unit of length. Question 11. 3 yd = ___ in. Answer: 3 yd = 108 inches Question 12. 324 ft =__ yd Answer: 324 ft = 108 yd Question 13. 2$$\frac{2}{3}$$ mi = ___ ft Answer: 2 2/3 miles = 14080 feet Question 14. 56 ft = ___ yd ___ ft Answer: 56 ft = 18 yd 2 ft Question 15. 12$$\frac{1}{2}$$ = ___ in. Answer: 12 1/2 feet = 150 inches Question 16. 6 in. = ___ ft Answer: 6 in = 0.5 feet In 17-19, compare lengths. Write >,<, or = for each Question 17. 100 ft 3 yd Answer: 100 ft >3 yd Question 18. 74 in. 2 yds 2 in. Answer: 74 in = 2 yd 2 in Question 19. 5,200 ft 145 in. 1 mi 40 in. Answer: 5200 ft 145 in > 1 mi 40 in. Problem Solving Question 20. Number Sense Which number would be greater, the height of a tree in feet or the height of the same tree in yards? Answer: 1 feet = 30.48 centimeters 1 yard = 91.44 centimeters. Question 21. Reasoning The dimensions of the nation’s smallest post office are 8 feet 4 inches by 7 feet 3 inches. Why would you use the measurement 8 feet 4 inches instead of 7 feet 16 inches? Answer: 16 inches = 1 foot 4 inches 1 foot is added to 7 inches Therefore, The measurement 8 feet 4 inches instead of 7 feet 16 inches Question 22. Roger earns$24 a week mowing lawns. He spends $$\frac{1}{6}$$ of his earnings on lunch and $$\frac{2}{3}$$ of his earnings on music. He saves the rest. How many dollars does Roger save? Tell me how you found the answer.

Given that, Roger earns $24 a week The amount he spends on his lunch = 1/6 of his earnings Which means, 1/6 x 24 = 4 Also given, he spends 2/3 of his earnings on music This means, 2/3 x 24 = 16 Total amount he spend = 16 + 4 = 20 Remaining amount = 24 – 20 =$4

Therefore, Roger saves 4 dollars.

Question 23.
Ariana has 144 peaches. She has to pack 9 boxes with an equal number of peaches. How many peaches should she pack in each box?

Total number of peaches = 144

The number of boxes she has to pack  = 9

Now,

144/9 = 16

Therefore, Ariana has to pack 15 peaches in each box.

Question 24.
Higher-Order Thinking How do you convert 108 inches to yards?

108 inches = 3 yards

1 inch = 0.02 yards

108 inches =

= 108 x 0.02

= 3 yards.

Question 25.
A-Z Vocabulary What is an appropriate customary unit to use when measuring the length of a driveway? Justify your answer.

The most appropriate units are either feet, yards, or meters.

Assessment Practice

Question 26.
Select all of the measurements greater than 7 feet.
2 yards
2 yards 2 inches
2 yards 2 feet
3 yards

c. 2 yards 2 feet

d. 3 yards.

Question 27.
Select all of the measurements less than 435 inches.
37 feet
36 feet 2 inches
12 yards 3 inches
12 feet 3 inches

36 feet 2 inches

12 feet 3 inches

### Lesson 12.2 Convert Customary Units of Capacity

Activity

Solve&Share

A recipe makes 16 cups of soup. How many quarts does the recipe make? Remember, there are 2 cups in a pint and 2 pints in a quart. Solve this problem any way you choose!

Given,

There are 16 cups of soup

4 cup =  1 quart

16 cups = 4 pints.

Look Back! Is the number of cups greater than or less than the number of quarts? Why do you think that is?

Visual Learning Bridge

Essential Question How Do You Convert Customary stion Units of Capacity?

A.
Sue is making punch. She needs 3$$\frac{3}{4}$$ cups of orange juice and 5 pints of lemonade. How many fluid ounces of orange juice and how many quarts of lemonade does she need?

1 gallon (gal) = 4 quarts (qt)
1 quart = 2 pints (pt)
1 pint = 2 cups (c)
1 cup = 8 fluid Ounces (fl oz)

You can multiply or divide to convert one unit of capacity to a different one.

B.
To change a larger unit to a smaller unit, multiply.

C.
To change a smaller unit to a larger unit, divide.

Find 5 ÷ 2.
5 ÷ 2 = $$\frac{5}{2}$$ = 2$$\frac{1}{2}$$
So, 5 pints = 2$$\frac{1}{2}$$ quarts.

Convince Me! Generalize When you convert from pints to quarts, why do you divide?

Guided Practice

Do You Understand?

Question 1.
Why would you change 4 gallons 5 quarts to 5 gallons 1 quart?

Given, 4 gallons 5 quarts

4 quarts = 1 gallon

So, 4 gallons 5 quarts can also be written as 5 gallons 1 quart.

Question 2.
Why is $$\frac{1}{8}$$ cup equal to 1 fluid ounce?

1 cup = 8 fluid ounces

1/8 cup = 2 tablespoon  = 1 fluid ounce

Do You Know How?

In 3-8, convert each unit of capacity.

Question 3.
32c = __ gal

2 gal

Question 4.
$$\frac{1}{4}$$qt = __ gal

64 gal

Question 5.
48 qt = __ pt

96 pt

Question 6.
6$$\frac{1}{8}$$ qt = __ c

1 qt = 4 cups

6 qt = 24 cups

1/8 qt = 0.5 cups

6 1/8 qt = 24.5 cups

Question 7.
73 qt 1 pt = __ pt

1 qt = 2 pints

73 qt= 73 x 2

= 146

146 pt + 1 pt

= 147 pints.

Question 8.
9 pt = __ qt__ pt or ___ qt

1 qt = 2 pt

9 pt =

4 qt 1 pt

or

4.5 qt.

Independent Practice

You may need to convert more than once.

In 9-20, convert each unit of capacity.

Question 9.
10 pt = __ qt

Question 10.
48 fl oz = ___ c

Question 11.
$$\frac{1}{2}$$c = __ pt

1  cup = 0.5 pt

So, 1/2 cup = 0.25 pt

Question 12.
9$$\frac{1}{4}$$ pt = ___ c

1 pt = 2 cups

9 pt = 9 x 2 = 18 cups

1/4 = 0.5 or 1/2 cups

Total = 18+0.5

= 18.5 cups

Question 13.
36 pt = ___ qt

1 pt = 0.5 qt

36 pt =

36 x 0.5

= 18 qt.

Question 14.
30 qt = __ gal __ qt

1 quart = 0.25 gal

30 qt = 7 gal 2 qt

Question 15.
1qt = ____ gal

1 qt = 0.25 gal

Question 16.
5 gal = ___ c

1 gal = 16 cups

So, 5 gal = 5 x 16 = 80 cups.

Question 17.
1 gal 1c = ___ fl oz

136 fl oz

1 gal = 16 c

16 c + 1 c = 17 c

1 c = 8 fl oz

17 c = 17 x 8 = 136 fl oz

Question 18.
7c = __ fl oz

1 cup = 8 fl oz

7 c = 7 x 8

= 56 fl oz

Question 19.
72 pt = ___ gal

Question 20.
$$\frac{1}{3}$$ pt = ___ c

1/3 pt = 0.66 c

Question 21.
Complete the table to show equivalent measures.

1 gallon = 4 quarts = 8 pints = 16 cups = 128 fl oz

2 gallon = 8 quarts = 16 pints = 36 cups = 256 fl oz.

Problem Solving

For 22-24, use the aquarium.

Question 22.
The class aquarium holds 2 gallons of water. How many cups is this? How many fluid ounces is this?

1 gallon = 16 cups

Which means, 2 gallon  = 2 x 16 = 32 cups

Therefore, there are 32 cups in 2 gallons

1 gallon = 128 fl oz

Which means, 2 gallons = 2 x 128 = 256 fl oz .

Therefore, 256 fl oz are there in 2 gallons.

Question 23.
Susan finds that 2 pints 1 cup of water have evaporated from the class aquarium. How many pints of water are left in the aquarium?

The number of cups in aquarium = 32 cups

2 pints 1 cup = 5 cups

Now, 32 – 5 = 27 cups

1 cup = 0.5 pints

So, 27 cups = 13.5 pints

Therefore, 13.5 pints of water are left in the aquarium.

Question 24.
If all of the dimensions of the aquarium were doubled, what would be the volume of the new aquarium?

Given,

the dimensions of the aquarium are doubles means

Volume = 20 in x 16 in x 18 in

= 4320 cubic inches.

Therefore, the volume of the new aquarium = 4320 cubic inches.

Question 25.
Carrie has 3 gallons of paint. Bryan has 10 quarts of paint. How many more pints of paint does Carrie have than Bryan?

Given Carrie has 3 gallons of paint

3 gallons = 24 pints

Bryan has 10 quarts of paint

10 quarts = 20 pints

Now,

24 pints – 20 pints

= 4 pints

There are 4 more pints of paint Carrie has than Bryan.

Question 26.
Reasoning Lorelei filled her 5-gallon jug with water. How many times could she fill her 2-quart canteen with water from the jug? Explain.

1 gallon = 4 quarts

5 gallons = 4 x 5 = 20 quarts

Now, 20/2 = 10

Therefore, she can fill the canteen by 10 times.

Question 27.
Higher-Order Thinking A recipe calls for 3 tablespoons of pineapple juice. A can of pineapple juice is 12 fluid ounces. How many teaspoons of juice are in the can?

Given, a can of pineapple juice = 12 fluid ounces

1 fl oz = 2 tablespoon

Now, 12 fl oz = 24 tablespoons

The amount of pineapple juice recipe contains = 3 tablespoon

Now, 3  x 24 tablespoons = 72 teaspoon

Therefore, there are 72 teaspoons of pineapple are there in the can.

Assessment Practice

Question 28.
Choose all the measurements that are greater than 4 cups.
30 fluid ounces
2 pints
3 pints
1 quart
1 gallon

1 gallon > 4 cups.

Question 29.
Choose all the statements that are true.
15 pt < 2 gal
1 gal < 5 qt
12 fl oz > 2c
2 qt 1 cup > 10 cups
20 pints = 10 quarts

20 pints = 10 quarts

### Lesson 12.3 Convert Customary Units of Weight

Activity

Solve & Share
Maria adopted 4 dogs. All together they eat 1$$\frac{3}{4}$$ pound of food each day. One pound is equal to 16 ounces. How many ounces of food will the dogs eat in 5 days? Solve this problem any way you choose.

1 pound = 16 ounces

1 3/4 pounds = 28 ounces

Number of days = 5

Now, 28 x 5

= 140 ounces

Therefore, the dogs can eat 140 ounces of food in 5 days.

A model with Math You can use drawings or equations to solve the problem. Show your work!

Look Back! Which is the larger unit of weight, an ounce or a pound? How can you use this relationship to find the number of ounces in 5 pounds?

1 pound = 16 ounces

5 pounds = 5 x 16 = 80 ounces

Visual Learning Bridge

Essential Question How Can You Convert Units of Weight?

A.
An adult African elephant weighs about 5 tons. A baby African elephant weighs about 250 pounds. How many pounds does the adult elephant weigh? How can you convert 250 pounds to tons?

To convert from one unit of weight to another, you can use multiplication or division.

B.
To convert from larger units to smaller units, multiply.

Find 5 × 2,000.
5 × 2,000 = 10,000
So, 5 tons = 10,000 pounds.

C.
To convert from smaller units to larger units, divide.

Convince Me! Generalize When you convert 16 pounds to ounces, do you multiply or divide? Explain.

To convert pounds to ounces we have to multiply.

16 pounds = 256 ounces

Guided Practice

Do You Understand?

Question 1.
Would it be best to measure the weight of an egg in tons, pounds, or ounces? Explain.

Ounces are the best way to measure the weight of an egg.

Question 2.
What types of tools do people select to measure weight? Explain your example.

A scale or a balance are the tools used to measure weight

A balance is used to determine an object’s mass.

A scale to measure ounces and pounds.

Do You Know How?

In 3-6, convert each unit of weight.

Question 3.
2,000 lb = __ T

Question 4.
48 oz = ___ lb

Question 5.
6,500 lb = ___oz

Question 6.
$$\frac{1}{2}$$ lb = __ oz

In 7 and 8, compare. Write >, <, or = for each .

Question 7.
2T 45,000 lb

2T < 45,000 lb

Question 8.
4 lb 64 oz

4 lb = 64 oz

Independent Practice

In 9-14, convert each unit of weight.

Will your answer be greater than or less than the number you started with?

Question 9.
240 oz = __ lb

Question 10.
7$$\frac{1}{10}$$T = ___ lb

Question 11.
8 lb = ___ oz

Question 12.
4 oz = ___ lb

Question 13.
250 lb = ___ T

Question 14.
1 T = ___ oz

In 15-17, compare. Write >, <, or = for each .

Question 15.
5,000 lb 3 T

5,000 lb > 3 T

Question 16.
24 lb 124 oz

24 lb > 124 oz

Question 17.
64,000 oz 2 T

64,000 oz < 2 T

In 18 and 19, complete each table to show equivalent measures.

Question 18.

1/2 lb = 8

2 lb = 32 oz

5 lb = 80 oz

Question 19.

1/2 t = 1000 pounds

2 pounds = 4000

6 tons = 12,000

Problem Solving

Question 20.
Be Precise The perimeter of the rectangular playground shown below is 160 feet. What is the area of the playground?

Given, perimeter = 160 feet

We know that, perimeter = 2 (l + b )

160 = 2 ( 50 + b )

80 = 50 + b

b = 80 – 50

b = 30 feet

We know that area of rectangle = length x breadth

A = 80 x 30

= 2400 square feet

Therefore, Area = 2400 square feet.

Question 21.
enVision® STEM Humans exploring space have left behind bags of trash, bolts, gloves, and pieces of satellites. There are currently about 4,000,000 pounds of litter in orbit around Earth. Julia says that this amount using number names is four billion. Do you agree? Explain your thinking.

No,  I do not agree because 4 billion > 4,000,000

In 22-25, use the table.

Question 22.
What would be the most appropriate unit to measure the combined weight of 4 horses?
Answer: Tons will be the most appropriate unit to measure the combined weight of 4 horses

Question 23.
About how much would 4 horses weigh? Write the weight in two different ways.
Answer: The horses weigh 1500 pounds

1500 pounds = 0.75 tons

1500 pounds = 24000 ounces

Question 24.
How many more ounces do the sheep weigh than the ape?

1600 ounces more the sheep weigh than the ape.

Question 25.
Higher-Order Thinking What is the difference in weight between the horse and the combined weight of the dolphin and the ape? Write your answer in tons.

The  weight of 4 horses = 0.75 tons

The combined weight of dolphin and ape = 0.3 tons

Now, the difference between the weights =

= 0.75 – 0.3

= 0.45 tons

Therefore, the weight difference = 0.445 tons

Assessment Practice

Question 26.
Part A
The world’s heaviest lobster weighed 44 pounds 6 ounces. Write the lobster’s weight in ounces below.
44 lb 6 oz = ___ ounces

44 lb 6 oz = 710 ounces
Part B

1 lb = 16 ounces

44 lb = 44 x 16

= 704 oz

704 oz + 6 oz

= 710 oz

### Lesson 12.4 Convert Metric Units of Length

Activity

Solve & Share
Measure the length of your book in centimeters. Then measure it in millimeters. What do you notice about the two measurements?

1 cm = 10 mm

length of book = 25 cm

length of book = 250 mm

You can select appropriate units and tools to measure the length of objects!

Look Back! Use Structure How many meters long is your textbook? How do you know?

Visual Learning Bridge

Essential Question How Do You Convert Metric question Units of Length?

A.
The most commonly used metric units of length are the kilometer (km), meter (m), centimeter (cm), and millimeter (mm).
1 km = 103 m = 1,000 m
1 m = 102 cm = 100 cm
1 m = 103 mm = 1,000 mm
1 cm = 10 mm

Every metric unit is 10 times as great as the next smaller unit.

B.
The distance between two towns is 3 kilometers. How many meters apart are they?
3 km = __ m
To change from larger units to smaller units, multiply.
Find 3 × 103.
3 km = 3,000 m
So, the towns are 3,000 meters apart.

One kilometer equals 1,000 meters.

To change from smaller units to larger units, divide.

C.
The distance between a kitchen and living room is 1,200 centimeters. How many meters apart are they?
1,200 cm = __ m
Find 1,200 ÷ 102
1,200 cm = 12 m
So, the kitchen and the living room are 12 meters apart.

Convince Me! Critique Reasoning Elena says that 25 cm is equal to 250 mm. Do you agree? Why or why not?

Yes, I agree because,

1 cm = 10 mm

Now,

25 cm = 25 x 10 = 250 mm

Guided Practice

Do You Understand?

Question 1.
To find the number of meters in six kilometers, why do you multiply 6 × 103?
Answer: 6000 meters = 6 kilometers

Because, 1 km = 1000 m

6 km = 6 x 1000 = 6000m

Question 2.
Convert 12.5 centimeters to millimeters. Explain.

12.5 cm = 125 mm

Explanation:

1 cm = 10 mm

12 cm = 120mm

0.5 cm = 5 mm

Total : 120 + 5 = 125 mm

Do You Know How?

In 3-6, convert each unit of length.

Question 3.
103 cm = __ m

Question 4.
58 m = ___ mm

Question 5.
1,000 mm = __ cm

Question 6.
3 km = ___ m

In 7 and 8, compare lengths. Write >,<, or = for each .

Question 7.
9,000 m 20 km

9,000 m < 20 km

Question 8.
400 cm 4m

400 cm = 4 m

Independent Practice

In 9-14, convert each unit of length.

Question 9.
7.5 cm = ___ mm

Question 10.
6m = __ сm

Question 11.
0.8 km = ___ cm

Question 12.
17,000 m = ___ km

Question 13.
48,000 mm = ___ m

Question 14.
4 km = ___ m

In 15-20, compare lengths. Write >, <, or = for each .

Question 15.
25,365 cm 30 m

25,365 cm > 30 m

Question 16.
3.6 km 3,600 m

3.6 km = 3600

Question 17.
1,200 mm 12 m

1200 mm< 12 m

Question 18.
52,800 cm 1 km

52,800 cm <  1 km

Question 19.
7,500,000 m 750 km

7,500,000 m > 750 km

Question 20.
800 m 799,999 mm

800 m > 799,999 mm

In 21 and 22, complete each table.

Question 21.

1 km = 1000 m

0.5 km = 500 m

0.1 km = 100 m

Question 22.

50 m = 5,000 cm

5 m = 500 cm

0.5 m = 50 cm

Problem Solving

Question 23.
Number Sense Let x = the length of an object in meters and y = the length of the same object in millimeters. Which is a smaller number, x or y?

1 meter = 1000 mm

1 mm = 0.001 mm

Therefore, Y is smaller than x

Question 24.
Higher-Order Thinking How many millimeters are equal to one kilometer? Show your work.

One kilometer is equal to 1000000 millimeters.

1 km = 1000000 mm

Question 25.
Reasoning Which fraction is greater: $$\frac{7}{8}$$ or $$\frac{9}{12}$$? Explain how you know.

7/8 >9/12

Explanation :

L.c.m of 8,12 = 24

Now,

7/8 x 3/3 = 21/24

9/12 x 2/2 = 18/24

21/24 > 18/24

Therefore,

7/8 >9/12

Question 26.
A week ago, Trudy bought the pencil shown. Now the pencil measures 12.7 centimeters.
How many centimeters of the pencil has been used?

The original length of the pencil = 18 cm

The present length of pencil = 12.7 cm

Length difference = 18 – 12.7

= 5.3 cm

Therefore, the length of pencil used = 5.3 cm

How do you compare fractions?

Question 27.
enVision® STEM Mount St. Helens, located in Washington, erupted on May 18, 1980. Before the eruption, the volcano was 2.95 kilometers high. After the eruption, the volcano was 2.55 kilometers high. Use the bar diagram to find the difference in height of Mount St. Helens before and after the eruption. Convert the difference to meters.

The original height of the volcano = 2.95 km

After the eruption, the height of the volcano = 2.55 km

Difference = 2.95 – 2.55

= 0.4 km

0.4 km = 400 meters.

Therefore, the difference in height of the volcano = 400 meters.

Assessment Practice

Question 28.
Eileen plants a tree that is 2 meters tall in her yard. Which of the following is equivalent to 2 meters?
A. 200 mm
B. 20 cm
C. 200 km
D. 2,000 mm

B. 2 meters = 20 cm

Question 29.
Which of these number sentences is NOT true?
A. 600 cm = 6 m
B. 1 m < 9,000 mm
C. 900 mm = 9 cm
D. 10 km > 5,000 m

B. 1 m < 9,000 mm

### Lesson 12.5 Convert Metric Units of Capacity

Solve & Share
A pitcher holds 4 liters of water. How many milliliters does the pitcher hold? Solve this problem any way you choose.

You can convert metric units of capacity using multiplication or division. Show your work!

1 liter = 1000000 milliliters

4 liters = 4 x 1000000 mL

= 4000000 mL

Look Back! Look for Relationships Juanita shares a one-liter bottle of water equally with 3 friends. How much water does each person get? Give your answer in liters and milliliters.

Visual Learning Bridge

Essential Question How Do You Convert Metric Units of Capacity?

A.
The most commonly used units of capacity in the metric system are the liter (L) and the milliliter (mL).

Can you find a liter or milliliter in the real world?

B.
Susan has 1.875 liters of water. How many milliliters is this?
1.875 L = __ mL
To change a larger unit to a smaller unit, multiply.
Find 1.875 × 103.
1.875 × 103 = 1,875
1.875 L = 1,875 mL
So, Susan has 1,875 milliliters of water.

C.
Jorge has 3,500 milliliters of water. How many liters is this?
3,500 mL = __ L
To change a smaller unit to a larger unit, divide.
Find 3,500 ÷ 103.
3,500 ÷ 103 = 3.5
3,500 mL = 3.5 L
So, Jorge has 3.5 liters of water.

Convince Me! Reasoning Order these measurements from greatest to least. Explain how you decided.

2,300 L >22L > 3000 mL > 2L > 500 mL

Guided Practice

Do You Understand?

Question 1.
Explain how you can convert milliliters to liters.

1 milliliters = 0.001 liters

To convert milliliters to liters, we divide by 1000.

Question 2.
What types of tools would you select to measure capacity? Give an example and explain how that tool could be used.

The most popular tool used to measure capacity is the measuring cup.

And also There are five basic units for measuring capacity in the U.S. customary measurement system. These are the fluid ounce, cup, pint, quart, and gallon.

Do You Know How?

In 3-8, convert each unit of capacity.

Question 3.
2.75 L = ___ mL

Question 4.
3,000 mL = ___L

Question 5.
5L = ___ mL

Question 6.
250 mL = __ L

Question 7.
0.027 L = __ mL

Question 8.
400 mL = __ L

Independent Practice

In 9-20, convert each unit of capacity.

Question 9.
5,000 mL = ___ L

5 L

Question 10.
45,000 mL = __ L

45

Question 11.
4.27 L = __ mL

4270 ml

Question 12.
13 L = ___ mL

13000

Question 13.
3,700 mL = __ L

3.7 L

Question 14.
0.35 L = __ mL

350 ml

Question 15.
2,640 mL = ___ L

2.64 L

Question 16.
314 mL = ___ L

0.314 L

Question 17.
0.06 L = __ mL

60

Question 18.
2,109 mL = ___ L

0.0021

Question 19.
85 mL = __ L

0.085 L

Question 20.
9.05 L = ___ mL

9050

In 21 and 22, complete each table to show equivalent measures.

Question 21.

0.1 L = 100 mL

1 L = 1000 mL

10 L = 10000 mL

Question 22.

500 mL = 0.5

5000 mL = 5

50,000 mL =50

Problem Solving

Question 23.
Reasoning Carla’s famous punch calls for 3 liters of mango juice. The only mango juice she can find is sold in 500-milliliter cartons. How many cartons of mango juice does Carla need to buy?

Amount of Carla’s Famous punch = 3 liters

We know that 1 liter = 1000 milliliters

Also given, 500 mL = 1 Carton

Now, Number of cartons =

3000/ 500 = 6

Therefore, Carla needs to buy 6 cartons of mango juice.

Question 24.
Carla makes 6 liters of punch. She pours the punch into 800 ml bottles. How many bottles can she fill?

6 liters of punch = 6000 milliliters

Now, 6000/800

=  7.5

Therefore, she can fill approximately 7.5 bottles.

Question 25.
Bobby filled the jug with water for soccer practice. If each player gets 250 milliliters of water, how many players will the water jug serve?

The capacity of water jug =  5 liters

1 litre = 1000 millilitres

Now, 5 litres = 5000 millilitres

Now, 5000/ 250

= 20

Therefore, the water jug will serve 20 people.

Question 26.
Higher-Order Thinking One cubic centimeter will hold 1 milliliter of water. How many milliliters will the aquarium below hold? How many liters will it hold?

Volume of the aquarium =

length x width x height

V =  40 x 20 x 30

V = 24000 cubic centimeters

We know that, 1 L = 1000 mL

Therefore, The aquarium holds 24 L .

Question 27.
Terry is buying juice. He needs 3 liters. A half-liter of juice costs $2.39. A 250-milliliter container of juice costs$1.69. What should Terry buy so he gets 3 liters at the lowest price? Explain.

He needs 3 L.

A half-liter of juice costs $2.39. A 250 mL container of juice costs$1.69.
One liter of juice in half-liter packs costs,

$2.39 x 2 =$4.78.
One liter of juice in 250 mL packs costs,

$1.69 x 4 =$6.76.
So, Terry should buy 6 half-liter packs of juice and spend
$2.39 x 6 =$9.56 to get 3 liters at the lowest price.

What steps do you need to do to solve this problem?

Assessment Practice

Question 28.
A birdbath holds 4 liters of water. How many milliliters of water does it hold?
A. 400 ml
B. 800 mL
C. 4,000 mL
D. 8,000 mL

C. 4,000 mL

1 litre = 1000 milliliters

So, 4 litres = 4000 millilitres.

Question 29.
You are filling a 2-liter bottle with liquid from full 80-milliliter containers. How many containers will it take to fill the
A. 400
B. 250
C. 40
D. 25

1 litre = 1000 millilitres

So, The number of  80 mL containers need are

2000/80

= 25

Therefore, 25 containers are required.

### Lesson 12.6 Convert Metric Units of Mass

Activity

Solve & Share
In chemistry class, Rhonda measured 10 grams of a substance. How many milligrams is this? Solve this problem any way you choose.

1 gram = 1000 mg

So,

10 grams = 10000 milligrams.

Look for Relationships You can use patterns to help you see a relationship between the units.

Look Back! How many kilograms did Rhonda measure? Write an equation to model your work.

Visual Learning Bridge

Essential Question How Do You Convert Metric Units of Mass?

A.
The three most commonly used units of mass are the milligram (mg), the gram (g), and the kilogram (kg).

Converting metric (units of mass is like converting other metric units.

B.
A whistle has a mass of about 5 grams. How many milligrams is this?
To change from a larger unit to a smaller unit, multiply.
Find 5 × 103
5 × 103 = 5 × 1,000 = 5,000
So, 5 g = 5,000 mg
So, a whistle has a mass of about 5,000 milligrams.

C.
How many kilograms is the whistle?
To change from a smaller unit to a larger unit, divide.
Find 5 ÷ 103
5 ÷ 103 = 5 ÷ 1,000 = 0.005
So, 5 g = 0.005 kg.
So, a whistle has a mass of about 0.005 kilograms.

Convince Me! Use Structure in the picture above, what is the football player’s mass in grams and in milligrams? How can you tell?

Guided Practice

Do You Understand?

Question 1.
A-Z Vocabulary How does the relationship between meters and millimeters help you understand the relationship between grams and milligrams?

1 meter = 1000 millimeters.

1 Gram = 1000 milligrams.

Question 2.
Which has the greater mass: 1 kilogram or 137,000 milligrams? Explain how you made your comparison.

137,000 is greater

Because, 1 kg = 1000000

Therefore, 137,000 is greater than 1 kg

Do You Know How?

In 3 and 4, convert each unit of mass.

Question 3.
9.25 g = ___ mg

9250 mg

Question 4.
190 g = __ kg

0.19 kg

In 5 and 6, compare. Write >,<, or = for each

Question 5.
7,000 mg 7,000 g

7,000 mg < 7,000 g

Question 6.
102 kg 104 g

102 kg > 104 g

Independent Practice

In 7-12, convert each unit of mass.

Question 7.
17,000 g = ___ kg

17,000 = 17 kg

Question 8.
18 kg = ___ g

18 kg = 18000 g

Question 9.
4,200 mg = ___ g

Question 10.
0.276 g = ____ mg

Question 11.
4.08 kg = ___ g

Question 12.
43 mg = ___ g

In 13-18, compare. Write >, <, or = for each

Question 13.
2,000 g 3 kg

2000 g < 3 kg

Question 14.
4 kg 4,000 g

4 kg = 4000 g

4 kg  = 4,000 g

Question 15.
104 mg 13 g

104 mg < 13 g

Question 16.
7 kg 7,000 g

7 kg < 7000 g

Question 17.
9,000 g 8 kg

9000 g > 8 kg

Question 18.
8,000 g 5 kg

8000 > 5 kg

In 19 and 20, complete each table.

Question 19.

1 grams = 1000 milligrams

10 grams = 10000 milligrams

100 grams = 100000 milligrams

Question 20.

500 grams = 0.5 kg

5000 grams = 5 kg

50000 grams = 50 kg

Problem Solving

Question 21.
Make Sense and Persevere Sheryl has a recipe for pasta with vegetables. The recipe calls for 130 grams of vegetables and twice as much pasta as vegetables. What is the total mass in grams of the recipe?

Given,

The mass of vegetables = 130 grams

Also given, the mass of pasta is twice as vegetables

Which means, 130 x 2 = 260

Total mass = 260 + 130

= 390 grams

Therefore, the total mass of the recipe = 390 grams.

Question 22.
Terri is beginning a science experiment in the lab. The instructions call for 227 milligrams of potassium. Calculate the difference between this amount and 1 gram.

We know that,

1 gram = 1000 milligrams

Given that, The weight of potassium = 227 milligrams

Now, The difference between the amount

= 1000 – 227

= 773 grams.

Therefore, the difference between the amount and 1 gram = 773 grams.

Question 23.
Number Sense One of the world’s heaviest hailstones weighed 2.2 pounds. Which is more appropriate to express its mass, 1 kilogram or 1 gram?

2.2 pounds = 0.997 kg

2.2 pounds = 997. 9 grams

Therefore, 1 kilogram is more appropriate to express the mass.

Question 24.
Higher Order Thinking A cook has 6 onions that have a total mass of 900 grams and 8 apples that have a total mass of 1 kilogram. All onions are the same size, and all apples are the same size. Which has the greater mass, an onion or an apple? Explain.

Given,

Number of onions = 6

Total mass = 900 grams

Weight of each onion = 900/6

= 150 grams.

Number of apples = 8

Total mass = 1 kg or 1000 grams

Weight of each apple = 1000/8

= 125 grams.

Therefore, Onion has a greater mass than apple.

In 25 and 26, use the given information and the picture.

enVision® STEM If a man weighs 198 pounds on Earth, his mass on Earth is 90 kilograms.

Question 25.
What is this man’s weight on the Moon?

Given,

The weight of the person on the moon is 1/6 weight on earth.

Also given, the mass on earth = 90 kgs

The weight of man’s weight on the moon =

90/6 =15 or 14.9 approx.

Question 26.
What is his mass in grams?

1 kg = 1000 grams.

15 kg = 15000 grams

Assessment Practice

Question 27.
Write the following masses on the lines from least to greatest.

5000 mg > 500 g > 50 kg

Question 28.
If you convert grams to milligrams, what operation would you use?
B. Subtraction
C. Multiplication
D. Division

Multiplication.

### Lesson 12.7 Convert Units of Time

Activity

Solve&Share
Emily played softball all weekend. She was wondering the difference in time between the shortest game and the longest game. Can you help her figure it out?

Select a common unit of time to help compare game times.

Look Back! Make Sense and Persevere Mateo saw a professional baseball game, which lasted 2$$\frac{1}{2}$$ hours. How many minutes longer was the professional game than Emily’s Game 3? Explain.

Visual Learning Bridge

Essential Question How Do You Solve Problems that Involve Different Units of Time?

A.
Kendall’s family is driving to the theater to see a 2-hour movie. Kendall notices this sign at the parking lot closest to the theater. Do you think they should park there?

You can convert one of these times so you are comparing like units.

B.
One Way:
Convert 2 hours to minutes. Then compare.
To change from larger units to smaller units, multiply.

Remember, 1 hour equals 60 minutes.

2 × 60 minutes = 120 minutes
120 minutes > 90 minutes, so Kendall’s family should not park in that lot.

C.
Another Way:
Convert 90 minutes to hours. Then compare.
To change from smaller units to larger units, divide.
90 ÷ 60 = $$\frac{90}{60}$$ = 1$$\frac{1}{2}$$ hours
1$$\frac{1}{2}$$ hours < 2 hours, so Kendall’s family should not park in that lot.

Convince Me! Make Sense and Persevere explain how to convert 4 hours, 15 minutes to minutes.

Another example!
There is often more than one way to show converted units of time. Find the missing numbers.
Divide. Write the quotient with a remainder.
210 ÷ 60 = 3 R 30
So, 210 seconds = 3 minutes, 30 seconds

Remember, 1 minute equals 60 seconds.

210 seconds = ___ minutes
Divide. Write the quotient as a mixed number.
$$\frac{210}{60}$$ = 3$$\frac{30}{60}$$ = 3$$\frac{1}{2}$$
So, 210 seconds = 3$$\frac{1}{2}$$ minutes

Guided Practices

Do You Understand?

Question 1.
Which is the longest time: 5 minutes, 25 seconds, or 315 seconds? Explain.

5 hours 25 seconds

Explanation :

1 minute = 60 seconds

So, 5 minutes = 300 seconds

Now, 300 +25 = 325 seconds

Therefore, 325 seconds is longer than 315

Question 2.
How many minutes are in a quarter-hour? How do you know?

Quarter hour = 15 minutes

We know that 15 minutes = quarter.

Do You Know How?

In 3-6, convert each time.

Question 3.
240 seconds = ___ minutes

4 Minutes.

Question 4.
2 hours, 18 minutes = ___ minutes

120 + 18

138 minutes

Question 5.
4$$\frac{1}{2}$$ minutes ___ seconds

4 minutes = 4 x 60 = 240 seconds

1/2 minute = 30 seconds

Now, 240 + 30 = 270 Seconds

Therefore, 4 1/2 Minute = 270 seconds.

Question 6.
80 minutes = ___ Seconds

1 minute = 60 seconds

80 minutes = 80 x 60 = 480 seconds.

Independent Practice

In 7-10, convert each time.

Question 7.
6 hours = ___ minutes

1 hour = 60 minutes

So, 6 hours = 6 x 60 =

= 360 minutes.

Question 8.
390 seconds = ___ minutes

60 seconds = 1 minute

Now, 360 seconds = 6 minutes

30 seconds = 1/2 minute

390 seconds = 6 1/2 minutes.

Question 9.
208 minutes = ___hours, ___ minutes

1 hour = 60 minutes

3 hours = 180 minutes And

208 – 180 = 28 minutes

Therefore, 208 minutes = 3 hours 28 minutes.

Question 10.
7 minutes, 12 seconds = ___ seconds

1 minute = 60 seconds

7 minutes = 7 x 60 = 420 seconds

420 + 12 = 432 seconds .

Therefore, 7 minutes, 12 seconds = 432 seconds

In 11-12, compare. Write >, <, or = for each

Question 11.
330 minutes 7.5 hours

330 minutes  < 7.5 hours

Question 12.
45 minutes $$\frac{3}{4}$$ hour

45 minutes  = 3/4  hour

Problem Solving

Question 13.
Brock spends 15 minutes walking to school and 15 minutes walking home each day. By the end of the school week (5 days) how many hours has Brock spent traveling between home and school?

Given,

Brock spends 15 minutes walking to school

15 minutes walking home

Total = 15 + 15

= 30

Number of days = 5

Now, 5 x 30 = 150 minutes

1 hour = 60 minutes

150 minutes = 2.5 hours

Therefore, Brock spent 2.5 hours traveling between home and school

Question 14.
A television station shows commercials for 7$$\frac{1}{2}$$ minutes each hour. How many 45-second commercials can it show per hour?

1 minute  = 60 seconds

7 minutes = 60 x 7 = 420 seconds

420 + 30 = 450

450/45 = 10

Therefore, 10 45 second commercials it can show per hour.

Question 15.
Leslie is making these two recipes. Which takes longer to make, the strawberry bread or the spaghetti sauce? How many minutes longer?

The time is taken to prepare spaghetti sauce is 1 hour 40 minutes

Also, the time taken to prepare strawberry bread is 1 hour 15 minutes

Therefore spaghetti sauce takes longer

Now,

1 hour 40 minutes – 1 hour 15 minutes

= 25 minutes

spaghetti sauce takes 25 minutes longer than strawberry bread.

Question 16.
Critique Reasoning The school day is 6 hours, 15 minutes long. Jenna says that it’s 6$$\frac{1}{4}$$ hours. Henry says it’s 6.25 hours. Can they both be correct? Explain.

Jenna is correct.

Given, the time of the school day = 6 hours 15 minutes long

1 hour = 60 minutes

1/4 hour = 15 minutes and 6 hours

Total: 6 1/4 hour

6.25 hours means 25 minutes more

But the school day is only 15 minutes long

Therefore, Jenna is correct.

Question 17.
Higher-Order Thinking How many seconds are there in 1 hour? In 10 hours? Explain.

1 hour = 60 minutes

1 minute = 60 seconds

60 minutes = 60 x 60 = 3600 seconds

10 hours = 10 x 60 = 600 minutes

600 minutes = 600 x 600 = 36000 seconds.

Assessment Practice

Question 18.
Three hikers reported how long it took to hike a trail. Write the names of the hikers from fastest to slowest.
___ ____ _____

Anita – First – 70 minutes

Brad – second – 75 minutes

Sanjay – Third – 90 minutes

### Lesson 12.8 Solve Word Problems Using Measurement Conversions

Activity

Solve&Share
Amy wants to frame a poster that has a width of 8 inches and a length of 1 foot. What is the perimeter of the poster? Solve this problem any way you choose.

Make Sense and Persevere
You can use measurement conversions in real-world situations. Show your work!

Look Back! Which measurement did you convert? Can you find the perimeter by converting to the other unit of measurement?

Visual Learning Bridge

Essential Question How Can You Convert Units of Session Measurement to Solve a Problem?

A.
A city pool is in the shape of a rectangle with the dimensions shown. What is the perimeter of the pool?

You can convert one of the measures so that you are adding like units.

B.
What do you know?
The dimensions of the pool:
l = 25 yards
w = 60 feet
What are you asked to find?
The perimeter of the pool

You can use feet for perimeter.

C.
Convert 25 yards to feet so you can add like units.
1 yard = 3 feet
To change from larger units to smaller units, multiply.
25 × 3 feet = 75 feet
So, 25 yards = 75 feet.

D.
Substitute like measurements into the perimeter formula.
Perimeter = (2 × length) + (2 × width)
P = (2 × l) + (2 × w)
P = (2 × 75) + (2 × 60)
P = 150 + 120
P = 270 feet
The perimeter of the pool is 270 feet.

Convince Me! Be Precise If the width of the pool is increased by 3 feet, what would be the new perimeter of the pool? Explain.

length = 75 feet

width = 63 feet

Perimeter = 2 ( l+b)

2 (75+63)

2 ( 138 )

= 276 feet.

Guided Practice

Do You Understand?

Question 1.
In the example on the previous page, how could you find the perimeter by converting all measurements to yards?

75 feet = 25 yards

60 feet = 20 yards

perimeter = 2 ( 25 + 20 )

= 2 ( 45)

perimeter = 90 feet

Question 2.
Write a real-world multiple-step problem that involves measurement.

Multiple-step problem :

Robert had 16 marbles. His brother gave him 3 more bags of marbles. If each bag contained 5 marbles, how many marbles does Robert have now?

Do You Know How?

Question 3.
Stacia needs enough ribbon to wrap around the length (l) and height (h) of a box. If the length is 2 feet and the height is 4 inches, how much ribbon will she need?

Given length = 2 feet

And height = 4 inches

Perimeter = 2 (l+b)

= 2 ( 2 + 4)

= 2 (6)

= 12 feet

1 feet = 12 inches

Now, 12 feet = 12 x 12

= 144 inches.

Question 4.
If ribbon is sold in whole number yards and costs $1.50 per yard, how much will it cost Stacia to buy the ribbon? Answer: 144 x$1.05

= 151.2

Therefore it costs approx $151 to buy the ribbon. Independent Practice In 5-7, use conversions to solve each problem. Edging means she will put bricks around the perimeter of the hexagon. Question 5. Becca wants to edge her hexagonal garden with brick. All sides are equal. The brick costs$6 per yard. What is the perimeter of the garden? How much will it cost to buy the edging she needs?

The perimeter of the hexagon =

Number of sides = 6

Now, 12 x 6 = 72 feet

The perimeter of hexagon = 72 feet or 24 yards

Given, the cost of the brick = $6 Now, 24 x$6

= $144 Therefore, it costs$144 to buy the edging.

Question 6.
Isaac buys milk to make milkshakes for his friends. He buys 1 quart of milk and $$\frac{1}{2}$$ gallon of milk. How many cups of milk does he buy?

1 quart = 4 cups

1/2 gallon = 8 cups

Total = 4 + 8

= 12 cups

Therefore, he bought 12 cups of milk.

Question 7.
Maggie buys 1$$\frac{1}{2}$$ pounds of walnuts, 8 ounces of pecans, and pound of almonds. How much do the nuts weigh in all?

1 1/2 pounds of walnuts = 24 ounces

8 ounces of pecans

1 pound of walnuts = 16 ounces

Total : 24 + 8 + 16

= 48 ounces

Therefore, the nuts weigh 48 ounces

Problem Solving

Question 8.
Reasoning Matt’s family is thinking about buying a family pass to the city pool. The pass is $80 for a family of 4. Individual passes are$25 each. How much money can Matt’s family save by purchasing a family pass instead of 4 individual passes?

Given, the amount of family pass = $80 Also given, the price of an individual pass =$25

Total number of family members = 4

Now, $25 x 4 =$100

So, $100 –$80

=$20 Therefore, Matt’s family saves$20 by purchasing a family pass instead of 4 individual passes

Question 9.
Marcia walked 900 meters on Friday. On Saturday, she walked 4 kilometers. On Sunday, she walked 3 kilometers, 600 meters. How many kilometers did Marcia walk over all three days?

Given,

On Monday she walked 900 meters

On Tuesday she walked 4 kilometers

1 km = 1000 m

4 km = 4000 m

On Sunday, she walked 3 km,600 m

3 km =3000 m + 600 m

= 3600 m

Total : 900 + 4000 + 3600

= 8500 meters.

8500 meters = 8.5 kilometers

Therefore, Marica walked 8.5 kilometers.

Question 10.
Higher-Order Thinking

Raul wants to put wood shavings in his rabbit’s cage. The floor of the cage measures 1 yard wide by 5 feet long. One bag of shavings covers 10 square feet. How many bags will Raul have to buy to cover the floor of the cage? Explain.

1.5 bags

Explanation:

Width = 1 yard = 3 feet

Length = 5 feet

Area of the floor = 5ft × 3ft = 15sqft

1 bag = 10sqft

x bag = 15sqft

x bags = (15 × 1)/10

= 1.5 bags

Therefore, Raul needs 1.5 bags.

Question 11.
Cheryl’s fish tank is 2 yards long by 24 inches wide by 3 feet high. What is the volume of Cheryl’s tank in cubic inches?

Given,

length = 2 yards

2 yards = 72 inches

wide = 24 inches

Height = 3 feet

3 feet = 36 inches

Volume of the tank = length x width x height

Question 12.
Some statistics about a typical adult Royal antelope are shown in the data table.
a What is a typical Royal antelope’s tail length in millimeters?
b. How many centimeters high can a typical Royal antelope jump?
c. What is the mass of a typical Royal antelope in grams?

a.

Given,

tail length =  6 cm

1 cm = 10 mm

6 cm = 6 x 10 = 60 mm

b.

Given, vertical leap = 2 meters

1 m = 1000 cm

Now, 2 meters = 2 x 1000 = 2000 cm

Therefore, it can jump 2000 cm

c.

2.4 kg = mass of antelope

1 kg = 1000 grams

2.4 kg = 2000 + 400

= 2400 grams.

Therefore, the mass of a typical Royal antelope in grams= 2400 grams

Assessment Practice

Question 13.
Joann wants to put a wallpaper border around her room. The border costs $3 per foot. The diagram shows Joann’s room. How much money will the border cost? A.$120
B. $102 C.$84
D. $60 Answer: Given, The cost of border per foot =$3

Now, 11 x 3 = $33 3 yards = 9 feet 9 feet = 9 x 3 =$27

Total : $33 +$27

= $60 The border cost$60.

### Lesson 12.9 Precision

Problem Solving

Solve & Share
Beth wants to make a picture frame like the one pictured below. She recorded the outside dimensions as 5 cm by 7 cm. Measure the outside dimensions of the frame in millimeters. Compare your measurements to Beth’s. Do you think her measurements are precise enough? Explain.

Thinking Habits
• Am I using numbers, units, and symbols appropriately?
• Am I using the correct definitions?
• Am I calculating accurately?

Look Back! Be Precise What is the difference between the perimeter based on the measurements Beth made and the perimeter based on the measurements you made? Explain how you found the answer.

Visual Learning Bridge

Essential Question How Can You Be Precise When Solving Math Problems?

A.
Chad and Rhoda are hanging a swing. Chad cut a piece of chain 6 feet 2 inches long. Rhoda cut a piece of chain 72 inches long. When they hung the swing, it was crooked.
Use precise language to explain why.

Be Precise means that you use appropriate math words, symbols, and units as well as accurate calculations when you solve problems.

B.
How can I be precise in solving this problem?
I can
• calculate accurately.
• use the correct units.

Here’s my thinking.

C.
Convert 6 ft 2 inches to inches to see if Chad and Rhoda cut equal lengths of chain.
6 ft 2 in. = ___ in.
6 × 12 = 72, so 6ft = 72 in.
6 ft 2 in. = 72 +2 = 74 in.
Chad’s chain is 74 inches long, but Rhoda’s chain is only 72 inches long. Since Chad and Rhoda used unequal lengths of chain, the swing is crooked.

Convince Me! Be Precise What recommendations would you make to Chad and Rhoda so that the swing hangs level?

Guided Practice
Mary needs a board 4 feet 8 inches long. She cut a board 56 inches long.

Remember to be precise by converting measurements accurately.

Question 1.
What measurements are given? Are the same units used for each measurement? Explain.

The measurements are mentioned with different units.

Question 2.
Explain how you can convert one of the measurements so that both use the same unit.

4 feet 8 inches can be converted into 56 inches as

1 foot = 12 inches

Now, 4 feet = 4 x 12 = 48 inches

48 + 8 = 56 inches.

Question 3.
Is the board Mary cut the right length? Give a clear and appropriate answer.

Yes, mary cut the right length.

4 feet 8 inches can be shown as 56 inches

Therefore, mary cut the right length.

Independent Practice

Be Precise
Sean is making meat loaf. He used the amount of catsup shown in the measuring cup.

Question 4.
Are the units that Sean used to measure the catsup s the same as those given in the recipe? Explain.

No, the units that Sean used to measure the catsup s the same as those given in the recipe.

Question 5.
How can you convert one of the measurements so that both use the same unit?

1 fl oz = 0.125 cups

So, 6 fl oz = 0.75 cups.

Question 6.
Did Sean use the right amount of catsup? Give a clear and appropriate answer.

No,

6 fl oz = 0.75 cups

0.75 cups = 3/4

But sean used 2/3 cup of catsup.

Problem Solving

Shipping a Package
A customer is using regular delivery to ship a package. Northside Shipping Company discovered that its old scale is not very accurate. It registers a weight that is 2 ounces too heavy. A new, accurate scale shows that the actual weight of the customer’s package is 2 pounds 11 ounces.

Question 7.
Make Sense and Persevere Which information do you need to determine the total shipping cost using either scale?

We need the weight of the package and delivery charges to the total shipping cost using either scale

Question 8
Be Precise Why do you need to convert measurements to determine total shipping costs?

Because To make the answer accurate we have to convert the measurements.

To be precise, you need to check that the words, numbers, symbols, and units you use are correct and that your calculations are accurate.

Question 9.
Model with Math Show how to convert the measurements you described in exercise 8.

1 pound =16 ounces.

2 pounds 11 ounces = 43 ounces.

Question 10.
Be Precise What would the total cost be if the package is weighed on the new scale? What would the total cost be if the package is weighed on the old scale? Show your work.

The weight on the new scale = $25. 95 The weight on the old scale =$27.15

### Topic 12 Fluency Practice

Activity

Point&Tally

Find a partner. Get paper and a pencil. Each partner chooses light blue or dark blue.
At the same time, Partner 1 and Partner 2 each point to one of their black numbers. Both partners find the product of the two numbers.
The partner who chose the color where the product appears gets a tally mark. Work until one partner has seven tally marks.

Topic 12 Vocabulary Review

Glossary

Word List

• capacity
• centimeter
• cup
• fluid ounce
• foot
• gallon
• gram
• inch
• kilogram
• kilometer
• liter
• mass
• meter
• mile
• milligram
• milliliter
• millimeter
• ounce
• pint
• pound
• quart
• ton
• weight
• yard

Understand Vocabulary

Choose the best term from the Word List. Write it on the blank.

Question 1.
One ___ is equivalent to twelve ___

one foot is equivalent to 12 inches

Question 2.
The measure of the amount of matter in an object is known as ___

Question 3.
The volume of a container measured in liquid units is its ____

Question 4.
There are 1,000 meters in one ____

Question 5.
Finding how light or how heavy an object is means measuring its _____

Question 6.
There are 2 cups in one _____

For each of these objects, give an example and a non-example of a unit of measure that could be used to describe it.

Milk is measured in liters

person’s height is measured in feet

shoe’s length is measured in centimeters or inches

Use Vocabulary in Writing

Question 10.
Explain the relationship among the metric units of mass in the Word List.

Length = meter, kilometer

Mass = gram, kilogram

volume = liter, milliliter.

### Topic 12 Reteaching

Set A
pages 489-492, 517-520

Convert 3 yards to inches.
1 foot (ft) = 12 inches (in.)
1 yard (yd) = 3 ft = 36 in.
1 mile (mi) = 1,760 yd = 5,280 ft
1 yard = 36 inches. To change larger units to smaller units, multiply: 3 × 36 = 108.
So, 3 yards = 108 inches.

Remember to divide when changing smaller units to larger units.
Convert.

Question 1.
7 ft = ___ in.

Question 2.
7,920 ft = ___ mi

Question 3.
Max wants to put a fence around his triangular garden. If each side is 6 yards, how many feet of fencing does Max need?

Given, the side of the triangle = 6 yards

Perimeter = 6 x 6 x 6

= 18 yards

1yd/3 ft x 18/18

Now, 18/ 54

Therefore, Max have 54 feet of fencing.

Set B
pages 493-496
Convert 16 cups to pints.
2 cups = 1 pint. To change smaller units to larger units, divide: 16 ÷ 2 = 8.
So, 16 cups = 8 pints.

Remember that 1 gal = 4 qt, 1 qt = 2 pt, 1 pt = 2 c, and 1 c= 8 fl oz.

Convert.

Question 1.
36C = ___ gal

Question 2.
7 pt = __qt

Question 3.
1$$\frac{1}{2}[latex] gal = ___ fl oz Answer: 1 1/2 gal = 192 fl oz Question 4. 6 pt = ___ c Answer: 1 pt = 2 cups So, 6 pt = 6 x 2 = 12 cups. Set C pages 497-500 Convert 6 pounds to ounces. 1 pound = 16 ounces. To change larger units to smaller units, multiply: 6 × 16 = 96. So, 6 pounds = 96 ounces. Remember that 2,000 pounds = 1 ton. Convert. Question 1. 2[latex]\frac{3}{4}$$ lb = __oz

Question 2.
56 oz = __ lb

Question 3.
4,000 lb = ___ T

Question 4.
6$$\frac{1}{2}$$T = __ lb

Set D
pages 501-504
Convert 2 meters to centimeters.
1 km = 1,000 m
1 m= 100 cm
1 m = 1,000 mm
1 cm = 10 mm
1 meter = 100 centimeters. To change larger units to smaller units, multiply: 2 × 100 = 200.
So, 2 meters = 200 centimeters.

Remember to multiply or divide by a power of 10 to convert metric measurements.

Convert.

Question 1.
5.4 m = ___ cm

Question 2.
2.7 km = ___ m

Question 3.
0.02 km = __ cm

Question 4.
0.025 m = ___ mm

Question 5.
675 mm = ___ m

Question 6.
7,435 cm = ___ m

Set E
pages 505-508
Convert 6,000 milliliters to liters.
1,000 milliliters = 1 liter. To change milliliters to larger units, divide: 6,000 ÷ 1,000 = 6.
So, 6,000 milliliters = 6 liters.

Remember that the most commonly used metric units of capacity are the liter and milliliter.

Convert.

Question 1.
6L = ___ mL

Question 2.
0.15 L = ___mL

Question 3.
2,000 mL = __ L

Question 4.
900 mL = ___ L

Set F
pages 509-512
Convert 6 kilograms (kg) to grams (g).
1 kilogram = 1,000 grams. To change larger units to smaller units, multiply:
6 × 1,000 = 6,000.
So, 6 kg = 6,000 g.

Remember that to convert metric units, you can annex zeros and move the decimal point.

Convert.

Question 1.
30 kg = ___ g

Question 2.
3,000 mg = ___ g

Question 3.
560 g = ___ kg

Question 4.
0.17g = __mg

Set G
pages 513-516
The choir concert is scheduled to last 90 minutes. The band concert is scheduled from 7:00-8:45. Which concert is scheduled to be longer? By how many minutes?
The choir concert will last 90 minutes = 1 hour, 30 minutes. The band concert will last 1 hour, 45 minutes. The band concert will be 15 minutes longer.

Remember to check if the units in the problem are the same.

Convert.

Question 1.
8 minutes = ___ seconds

1 minute = 60 seconds

8 minutes = 8 x 60 = 480 seconds

Question 2.
86 minutes = ___ hour, ___ minutes

1 hour = 60 minutes

86 minutes = 1 hour 26 minutes

Question 3.
A movie starts at 7:10 and ends at 9:03. How long does the movie last? __ hour, ___ minutes

The movie lasts 1 hour 53 minutes

Set H
pages 521-524

Thinking Habits
• Am I using numbers, units, and symbols appropriately?
• Am I using the correct definitions?
• Am I calculating accurately?

Remember that the problem might have more than one step.

Question 1.
Monica bought a 40-pound bag of dog food. Twice a day, she gives her dog 6 ounces of food. How many pounds of dog food will she use in 1 week? Explain.

Given, The amount of dog food Monica bought = 40 pounds

40 pounds = 640 ounces

The amount of food Monica gives dog = 6 ounces

Twice = 6 x 2 = 12 oz

In one week = 12 x 7 = 84 oz

Therefore, Monica gives 84 oz of dog food will she use in 1 week

### Topic 12 Assessment Practice

Question 1.
Which of the following are equivalent to 7 grams? Select all that apply.
0.007 kilogram
70 milligrams
7,000 kilograms
7,000 milligrams
0.007 milligram

0.007 kilogram

7,000 milligrams

Question 2.
Justin’s garden is shown below.

A. How can you convert the dimensions of Justin’s garden from yards to inches?
B. What is the perimeter of Justin’s garden in inches?

A.

Given, Length = 8 yards

8 yards = 8 x 36 = 288 inches

6 yards = 6 x 36 = 216 inches.

B.

Perimeter = length x width

288 x 216 = P

62208 = Perimeter

Therefore, The perimeter of Justin’s Garden = 62208  inches.

Question 3.
Which of the following equations can be used to find how many kilograms are in 2,000 grams?
A. 1,000 ÷ 2,000 = 0.5 kilogram
B. 2,000 ÷ 1,000 = 2 kilograms
C. 2,000 × 1,000 = 2,000,000 kilograms
D. 2,000 × 100 = 200,000 kilograms

2,000 ÷ 1,000 = 2 kilograms

Question 4.
A. 10 bales of cotton weigh approximately 5,000 pounds. How can you convert 5,000 pounds to tons?
B. Which comparison is true?
A. 5,000 pounds > 10,000 tons
B. 5,000 pounds = 3 tons
C. 5,000 pounds < 3 tons
D. 5,000 pounds > 3 tons

A.

1 pound = 0.0005 tons

Now, 5000 pounds = 2.5 tons.

B.

5,000 pounds > 3 tons

Question 5.
Tyrell bought 4 liters of fruit punch for a party. He will serve the punch in glasses that can hold 200 milliliters. How many full glasses of fruit punch can he serve?

1 litre = 1000 millilitres

Now, 4 litres = 4000 milliliters.

4000/200 = 20

Therefore, Tyrell can fill 20 glasses of fruit punch.

Question 6.
Select each equation that the number 103 will make true.
? km = 1 mm
? mm = 1 m
? cm = 1 m
?m = 1 km
? dm = 1 m

1000000 km = 1 mm

1000 mm = 1 m

100 cm = 1 m

1000 m = 1 km

10 dm = 1m

Question 7.
Match each measurement on the left to its equivalent measurement.

1 gallon = 4 quarts

1 cup = 8 fl oz

1 quart = 2 pints

1 pint = 2 cups.

Question 8.
Select all lengths that are equal to 6 feet 12 inches.
3 yd 1 ft
7 ft
7 ft 2 in.
2 yd 1 ft
1 yd 4 ft

a. 7 feet

b. 2 yd 1 ft

c. 1 yd 4 ft

Question 9.
Write and solve an equation to find how many milliliters are in 3.4 liters.

1 liter = 1000 milliliters

3.4 liters = 3400 milliliters.

Question 10.
Mason made 5 quarts of salsa. Which of the following can be used to find the number of cups of salsa Mason made?
A. 5 × 2 × 2
B. 5 × 4 × 4
C. 5 ÷ 2 ÷ 2
D. 5 × 4 ÷ 2

1 quart = 4 cups

The answer is 5 x 2 x 2.

Question 11.
Alicia bought 5 pounds of potting soil. She wants to put 10 ounces of soil in each flower pot.
A. How can she convert 5 pounds to ounces?
B. How many flower pots can she fill?

A.

1 pound = 16 ounces

So, 5 pounds = 5 x 16 = 80 ounces

B.

Now, 80/10 = 8

Therefore, she can make 8 flower pots.

Question 12.
The tail of a Boeing 747 is 63 feet 8 inches tall. How many inches tall is the tail?

Given,

The length of tail = 63 feet

1 feet = 12 inches

Now, 63 x 12

= 756

Total = 756 + 8

= 764 inches

Question 13.
Write and solve an equation to convert 0.38 meters to centimeters.

1 meter = 100 centimeters

Now, 0.38 meters = 38 cm.

Orange Juice
Heidi sells freshly-squeezed orange juice in Heidi’s Orange Juice cups.

Question 1.
Use the Information About Oranges. Answer the questions below to find how many pounds of oranges Heidi needs for her orange juice.
Part A
How many oranges does Heidi need to make one large orange juice? Show your work.

Part B
How many pounds of oranges does Heidi need to make one large orange juice? Show your work.

2.5 cups = 20 fluid ounces.

20 fl oz = 1.5 pounds

Question 2.
Answer the following to find the area of Heidi’s Display Shelf.
Part A
What units can you use for the area? Explain.

Part B
What is the area of Heidi’s Display Shelf? Show your work.

4 feet = 48 inches

Area = length x width

= 48 x 15

= 720 sq. inches

therefore, the area of hiedi’s shelf = 720 inches

Question 3.
The Orange Nutrition table shows nutrients in one medium-sized orange that weighs 5 ounces or 140 grams. All the nutrients in the orange are also in Heidi’s orange juice.

Part A
How many grams of potassium are in one large cup of Heidi’s orange juice? Explain how you solved.

Given,

250 milligrams of potassium is there in the juice cup

1 milligram = 0.001

250 mg = 0.25 grams

therefore, there are 0.25 grams of potassium in orange juice.

Part B
How many milligrams of fiber is in one large cup of Heidi’s orange juice? Use an exponent when you explain the computation you used to solve.

1 gram = 1000 milligrams

3.5 grams = 3500 milligrams

Therefore, there is 3500 mg of fiber.

Question 4.
Heidi also sells cartons of orange juice. Use the picture of Heidi’s Orange Juice Carton. Find the volume of the carton in cubic centimeters. Explain.

The measurements of the carton are

10 cm, 50 mm = 5 cm, 0.2 m = 20 cm

Volume = l x w x h

= 50 x 5 x 20

= 500 Cubic centimeters

therefore, volume = 500 cubic centimeters.

## Envision Math Common Core 5th Grade Answers Key Topic 5 Use Models and Strategies to Divide Whole Numbers

Envision STEM Project: Average Temperature

Does Research Use a weather site from the Internet or another source of daily weather reports to find the average daily temperature for your city or town for every day of one month? The average daily temperature is the average temperature for a whole 24-hour period.
Journal: Write a Report that Includes what you found about daily temperatures. Also in your report:
• Find the average daily high temperature for the month. Which day had the greatest high temperature?
• Find the average daily low temperature for the month. Which day had the least low temperature?
• Makeup and solve division problems based on your data.

Review What You Know

VOC

• dividend
• quotient
• divisor
• remainder

Choose the best term from the box. Write it in the blank.

Question 1.
In the equation 80 ÷ 10 = 8, the number 80 is the ___.

80 is called dividend

Question 2.
The number used to divide another number is the ___.

The number used to divide another number is the Divisor

Question 3.
The result of dividing two numbers is the ___.

The result of dividing two numbers is the quotient.

Multiplication and Division

Multiply or divide.

Question 4.
630 ÷ 9

70

Question 5.
480 ÷ 6

80

Question 6.
755 ÷ 5

151

Question 7.
657 ÷ 9

73

Question 8.
57 × 13

741

Question 9.
71 × 109

7739

Question 10.
For the state fair next month, 132 people volunteered to plan the fair’s activities. The volunteers formed 12 equal groups. How many volunteers were in each group?

Number of volunteers = 132

Number of groups formed = 12

Now, the number of volunteers in each group =

132/12

= 11

Therefore, the number of volunteers in each group = 11

Question 11.
A town is holding a competition for various athletic games. Each community has 14 players. 112 communities are competing in the games. How many players are competing?
A. 1,676
B. 1,568
C. 126
D. 98

1568

Estimate

Question 12.
A county has a goal to build 12,000 bus stop shelters in 48 months. If the county builds 215 bus shelters each month, will it reach its goal? Explain one way to estimate the answer.

No

Explanation:

Because, if 215 bus shelters were built each month, in 48 months only 10320 bus shelters were built. 1680 bus shelters will remain.

215×48 = 10320.

12000-10320 = 1680.

Pick a Project

PROJECT 5A
How much does a field trip cost?
Project: Plan an Educational Field Trip

PROJECT 5B
How does an assembly line work?
Project: Design an Assembly Line for Toy Vehicles

PROJECT 5C
How do marathon runners get enough water?
Project: Position Water Stations

3-ACT MATH PREVIEW

Math Modeling

Flapjack Stack

Before watching the video, think:
Pancakes have been around for thousands of years and are popular all over the world. Some pancakes are made from potatoes and served with applesauce and sour cream. Other pancakes are sweet and served with blueberries and maple syrup.
Maybe I should plant a maple tree.

### Lesson 5.1 Use Patterns and Mental Math to Divide

Solve & Share
A bakery sells muffins to local grocery stores in boxes that hold 20 muffins each. How many boxes are used if 60 muffins are sold? 600 muffins? 6,000 muffins? Solve this problem any way you choose.

Find the answer for 60 muffins. Then you can look for relationships to help find the answers for 600 and 6,000 muffins. Show your work!

Look Back! How can you use multiplication to help you divide 6,000 by 202?

Visual Learning Bridge

Essential Question How Can Patterns Help You Question Divide Multiples of 10?

A.
A jet carries 18,000 passengers in 90 trips. The plane is full for each trip. How many passengers does the plane hold?

Find 18,000 ÷ 90, the number of passengers on each trip.

B.
Think of a basic fact to help you. 18 ÷ 9 = 2
Think about patterns in place value and multiples of 10:
180 ÷ 90 = 18 tens ÷ 9 tens = 2
1,800 ÷ 90 = 180 tens ÷ 9 tens = 20
18,000 ÷ 90 = 1,800 tens ÷ 9 tens = 200
So, the jet can hold 200 passengers.

C.
Or, use multiplication.
90 × 2 = 180
90 × 20 = 1,800
90 × 200 = 18,000
So 18,000 = 90 = 200
The jet can hold 200 passengers.

Convince Me! Look for Relationships If the jet above carried 10,000 people in 50 trips, how many people did it carry each trip? The jet carried the same number of people each trip.

What basic fact helped you find the answer?

Guided Practice

Do You Understand?

Question 1.
Why is 210 ÷ 30 the same as 21 tens ÷ 3 tens?

Because 21 tens are 210

3 tens are 30

so, 210 ÷ 30 is the same as 21 tens ÷ 3 tens.

Question 2.
A jet carried 12,000 people in 40 trips. If the jet was full each trip, how many people did it carry for each trip?

Do You Know How?

Given,

The jet carried 12,000 people in 40 trips.

12000 ÷ 40 = 300

Therefore, 300 people carried in each trip.

In 3-9, find each quotient. Use mental math.

Question 3.
210 ÷ 30 = 21 tens ÷ 3 tens = ____

210 ÷ 30 = 21 tens ÷ 3 tens =

21 ÷ 3  = 7

Question 4.
480 ÷ 60 = 48 tens ÷ 6 tens = ___

480 ÷ 60 = 48 tens ÷ 6 tens =

48 ÷ 6 = 8.

Question 5.
15,000 ÷ 30 = 1,500 tens ÷ 3 tens = ___

15,000 ÷ 30 = 1,500 tens ÷ 3 tens =

1500 ÷ 3 = 500.

Question 6.
___ = 8,100 ÷ 90

8,100 ÷ 90 =

810 ÷ 9 = 90.

Question 7.
2,800 ÷ 70 = ___

2,800 ÷ 70 = 40

Question 8.
30,000 ÷ 50 = ___

30,000 ÷ 50 =

3000 ÷ 5 = 600.

Question 9.
__= 1,800 ÷ 60

1,800 ÷ 60 =

180 ÷ 6 = 30.

Independent Practice

Leveled Practice In 10-25, use mental math to find the missing numbers.

Question 10.
560 ÷ 70 = 56 tens ÷ 7 tens = ___

560 ÷ 70 = 56 tens ÷ 7 tens =

56 ÷ 7 =

8.

Question 11.
360 ÷ 60 = 36 tens ÷ 6 tens = ___

360 ÷ 60 = 36 tens ÷ 6 tens =

36 ÷ 6

= 6.

Question 12.
6,000 ÷ 50 = 600 tens ÷ 5 tens = ___

6,000 ÷ 50 = 600 tens ÷ 5 tens =

600 ÷ 5

= 120.

Question 13.
24,000 ÷ 60 = 2,400 tens ÷ 6 tens = ___

24,000 ÷ 60 = 2,400 tens ÷ 6 tens =

2400 ÷ 6 =

400.

Question 14.
___= 2,000 ÷ 20

2,000 ÷ 20 =

200 ÷ 2 =

100.

Question 15.
6,300 ÷ 90 = ____

630 ÷ 9 =

70.

Question 16.
___ ÷ 10 = 24

___ ÷ 10 = 24

Let the number be x

x ÷ 10 = 24

x= 10×24 = 240

Therefore, number is 240.

Question 17.
21,000 ÷ ___ = 700

21,000 ÷ ___ = 700

Let the number be x

21000 ÷ x = 700

x = 21000 ÷ 700

x= 30

Therefore, number is 30.

Question 18.
2,500 ÷ 50 = ___

2,500 ÷ 50 =

250 ÷ 5 = 50.

Question 19.
72,000 ÷ ____ = 800

72,000 ÷ ____ = 800

Let the number be x

72000 ÷ x = 800

x = 72000 ÷ 800

x = 90

Therefore, number is 90.

Question 20.
56,000 ÷ ___ = 800

56,000 ÷ ___ = 800

Let the number be x

56,000 ÷ x = 800

x = 56,000 ÷ 800

x = 560 ÷ 8 = 70

Therefore, number is 70.

Question 21.
___ ÷ 10 = 100

___ ÷ 10 = 100

Let the number be x

x ÷ 10 = 100

x = 10 x 100

x = 1000

Therefore, number is 1000.

Question 22.
45,000 ÷ 90 = ____

45,000 ÷ 90 = ____

4500 ÷ 9 = 500.

Question 23.
___ = 42,000 ÷ 70

___ = 42,000 ÷ 70

4200 ÷ 7 = 600.

Question 24.
64,000 ÷ ___ = 800

64,000 ÷ ___ = 800

Let the number be x

64,000 ÷ x = 800

x = 64000 ÷ 800

x = 640 ÷ 8 = 80

Therefore, number is 80.

Question 25.
32,000 ÷ ___ = 400

32,000 ÷ ___ = 400

Let the number be x

32,000 ÷ x = 400

x = 32000 ÷ 400

x = 320 ÷ 4 = 80

Therefore, the number is 80.

Problem Solving

Question 26.
The table shows the number of passengers who flew on airplane flights in or out of one airport. Each flight had the same number of passengers. How many passengers were on each flight?

Given,

Total passengers = 27000

Number of flights = 90

Let the number of passengers were on each flight be x

Then,

x = 27000 ÷ 90

x = 2700 ÷ 9

x = 300

Therefore, the number of passengers were on each flight are 300.

Question 27.
Algebra A truck delivers 478 dozen eggs to stores in one day. Write and solve an equation to find n, the number of eggs the truck delivers in one day.

1 dozen = 12 eggs

Number of dozens = 478

Equation = 478 x 12

n = 478 x 12

n = 5736 eggs

Therefore, the truck driver delivers 5736 eggs in one day.

Question 28.
Paula wants to divide 480 tomatoes equally among 80 baskets. How many tomatoes will Paula put in each basket?

Number of Tomatoes = 480

Now,

= 480/80

= 6

Therefore, paul keeps 6 tomatoes in each basket.

Question 29.
Be Precise Ernesto measured the width of each of the three coins shown below.

What is the difference in width between the widest coin and the least wide coin?

The width of the widest coin = 0.84 inch

The width of the least wide coin = 0.7 inch

Difference between them

= 0.84 – 0.7

= 0.14 inch

Therefore, the difference in width between the widest coin and the least wide coin = 0.14 inches.

Question 30.
Higher-Order Thinking A baker uses 30 grams of sea salt for each batch of bread. Sea salt comes in an 18-kilogram package or an 800-gram package. Which size package should the baker buy so that no sea salt is left after all of the batches are made? Explain.
1 kilogram equals 1,000 grams

Salt for one batch of bread = 30 grams

For 800 grams package,

Now,

= 800/30

= 26 batches.

For 18 kg salt,

1 kg = 1000 grams

18 kg = 18 x 1000 = 18000

Now, 18000/30

= 600 batches.

For 18 kg divided by 30 then, 600 batches no salt will be left.

But in, 800 grams pack 26 batches are left

Therefore, The baker should use 18 kg so that no salt will be left.

Assessment Practice

Question 31.
Which is 2,400 divided by 80?
A. 3
B. 4
C. 30
D. 40

2400 ÷ 80

240 ÷ 8 = 30

Therefore, C is the correct option.

Question 32.
Which expression has a quotient of 70?
A. 420 ÷ 60
B. 4,200 ÷ 6
C. 4,200 ÷ 60
D. 4,200 ÷ 600

4,200 ÷ 60 = 420 ÷ 6 = 70

Therefore, C is correct option.

### Lesson 5.2 Estimate Quotients with 2-Digit Divisors

Solve & Share
Kyle’s school needs to buy posters for a fundraiser. The school has a budget of $147. Each poster costs$13. About how many posters can his school buy? Solve this problem any way you choose.

You can find compatible numbers to estimate quotients. Show your work!

Look Back! Make Sense and Persevere What numbers are close to 147 and 13 that would be easy to divide using mental math?

Visual Learning Bridge.

Essential Question
How Can You Use Compatible Numbers to Estimate Quotients?

A.
Ella earned $159 by selling bracelets. Each bracelet was the same price. About how much did each bracelet cost? You can use division to find the price. You know the total amount earned and the number of bracelets. B. The question asks, “About how much?” So, an estimate is enough. Use compatible numbers to estimate 159 ÷ 75. Think: 159 is close to 160. Is there a number close to 75 that divides 160 evenly? Try 80. 160 ÷ 80 = 2 So, 160 and 80 are compatible numbers. 16 can be divided evenly by 8. C. Since 160 ÷ 80 = 2, 159 ÷ 75 is about 2. Ella charged about$2 for each bracelet.
Use multiplication to check for reasonableness:
2 × 80 = 160.

Convince Me! Make Sense and Persevere Suppose Ella earned $230 selling the 75 bracelets. Estimate the price of each bracelet. What compatible numbers did you use? Guided Practice Do You Understand? Question 1. Ella has 425 more bracelets to sell. She wants to store these in bags that hold 20 bracelets each. She estimates she will need about 25 bags. Do you agree? Why or why not? Answer: Given, Number of bracelets = 425 Number of bracelets in each = 20 Therefore, the Number of bags = 425 ÷ 20 = 21.25 which is approximately equal to 22. Hence I agree with Ella because the number of bags is approximately equal to 25. Do You Know How? In 2-7, estimate using compatible numbers. Question 2. 287 ÷ 42 Answer: 287 is approximately equal to 280 42 is approximately equal to 40 280 ÷ 40 = 7 Therefore, the estimation of 287 ÷ 42 is 7. Question 3. 320 ÷ 11 Answer: 11 approximately equals 10 320 ÷ 10 = 32 Therefore, the estimation of 320 ÷ 11 is 32 Question 4. 208 ÷ 72 Answer: 208 approximately equals 210 72 approximately equals 70 210 ÷ 70 = 3 Therefore, the estimation of 208 ÷ 72 is 3 Question 5. 554 ÷ 62 Answer: 554 approximately equals 540 62 approximately equals 60 540 ÷ 60 = 9 Therefore, the estimation of 554 ÷ 62 is 9. Question 6. 815 ÷ 23 Answer: 815 approximately equals 820 22 approximately eq820uals to 20 820 ÷ 20 = 41 Therefore, the estimation of 815 ÷ 23 is 41 Question 7. 2,491 ÷ 48 Answer: 2491 approximately equals 2500 48 approximately equals 50 2500 ÷ 50 = 50 Therefore, the estimation of 2,491 ÷ 48 is 50. Independent Practice Leveled Practice In 8-10, fill in the blanks to find each estimate. Question 8. Answer: 80 and the estimate is 5 Question 9. Answer: 40 and the estimate is 7 Question 10. Answer: 70 and the estimate is 40 In 11-22, estimate using compatible numbers. Question 11. 228 ÷ 19 Answer: 220 approximately equals 228 20 approximately equals 19 220 ÷ 20 = 11 Therefore, the estimation of 228 ÷ 19 is 11. Question 12. 1,784 ÷ 64 Answer: 1800 approximately equals 1784 60 approximately equals 64 1800 ÷ 60 = 30 Therefore, the estimation of 1784 ÷ 64 is 30. Question 13. 7,260 ÷ 83 Answer: 7200 approximately equals 7260 80 approximately equals 83 7200 ÷ 80 = 90 Therefore, the estimation of 7260 ÷ 83 is 90. Question 14. 2,280 ÷ 12 Answer: 2400 approximately equals 2280 2400 ÷ 12 = 200 Therefore, the estimation of 2280 ÷ 12 is 20. Question 15. 485 ÷ 92 Answer: 450 approximately equals 485 90 approximately equals 92 450 ÷ 90 = 5 Therefore, the estimation of 485 ÷ 92 is 5. Question 16. 540 ÷ 61 Answer: 60 approximately equals 61 540 ÷ 60 = 9 Therefore, the estimation of 540 ÷ 61 is 9. Question 17. 1,710 ÷ 32 Answer: 1800 approximately equals 1710 30 approximately equals 32 1800 ÷ 30 = 60 Therefore, the estimation of 1710 ÷ 32 is 60. Question 18. 2,740 ÷ 67 Answer: 2800 approximately equals 2740 70 approximately equals 67 2800 ÷ 70 = 40 Therefore, the estimation of 2740 ÷ 67 is 40. Question 19. 4,322 ÷ 81 Answer: 4400 approximately equals 4322 80 approximately equals 81 4400 ÷ 80 = 55 Therefore, the estimation of 4322 ÷ 81 is 55. Question 20. 5,700 ÷ 58 Answer: 60 approximately equals 58 5700 ÷ 60 = 95 Therefore, the estimation of 5700 ÷ 60 is 95. Question 21. 7,810 ÷ 44 Answer: 7875 approximately equals 7810 45 approximately equals 44 7875 ÷ 45 = 175 Therefore, the estimation of 7810 ÷ 44 is 175. Question 22. 6,395 ÷ 84 Answer: 6400 approximately equals 6395 80 approximately equals 84 6400 ÷ 80 = 80 Therefore, the estimation of 6395 ÷ 84 is 80. Problem Solving Question 23. A model with Math The sign shows the price of baseball caps for different pack sizes. Coach Lewis will buy the medium-size pack of caps. About how much will each cap cost? Write an equation to model the problem. Answer: Given, Number of medium size caps = 32 Total cost =$270

30 approximately equals 32

Cost of each cap = $270 ÷ 30 =$9.

The cost of each cap approximately equals $9. Question 24. There are 91 days until the craft sale. Autumn needs to make 817 rings before the sale. She wants to make about the same number of rings each day. About how many rings should she make each day? Explain how Autumn can use compatible numbers to estimate. Answer: Number of days left = 91 which approximately equals 90 Number of rings to be made = 817 which approximately equals 810 Therefore, rings should she make each day are 810 ÷ 90 = 9 Approximately, she should make 9 rings each day. Question 25. Higher-Order Thinking A company purchased 3,128 bottles of water. Each department needs 55 bottles. Find compatible numbers to estimate the number of departments that can get the bottles they need. Explain. Answer: Given, Number of bottles = 3128 which approximately equals 3080 Number of bottles each department need = 55 The number of departments that can get the bottles they need is 3080 ÷ 55 = 56 Therefore, the number of departments that can get the bottles they need is approximately equaled to 56. Question 26. Rita had$20. Then, she saved $5.85 each week for 8 weeks. How much money does she have now? Use the bar diagram to solve the problem. Show your work. Answer: Given, The amount Rita have =$20

The amount she saved = $5.85 Total number of weeks = 8 Now,$5.85 x 8

= 44.64

Now, $44.64 +$20

= $66.80. Assessment Practice Question 27. Lea bought 225 flowers and 12 vases. She put about the same number of flowers in each vase. Which is the best estimate for the number of flowers in each vase? A. 40 flowers B. 30 flowers C. 20 flowers D. 10 flowers Answer: C. 20 flowers Question 28. A school has 617 students. Each class has between 28 and 32 students. Which is the best estimate of the number of classes in the school? A. 14 classes B. 20 classes C. 30 classes D. 60 classes Answer: B. 20 classes ### Lesson 5.3 Use Models and Properties to Divide with 2-Digit Divisors Solve & Share A parking lot has 270 parking spaces. Each row has 18 parking spaces. How many rows are in this parking lot? Solve this problem any way you choose. You can use appropriate tools, such as grid paper, to solve the problem. Show your work! Look Back! How can you use estimation to check that your answer to the problem above is reasonable? Answer: Total number of parking spaces = 270 Number of parking spaces in each row = 18 Therefore, the total number of parking rows = 270 ÷ 18 =15. Visual Learning Bridge Essential Question How Can You Use Area Models and Properties to Find Quotients? A. Emily has a rectangular garden with an area of 360 square feet. The length of her garden measures 20 feet. How many feet wide is her garden? Think 20 × w = 360 or 360 ÷ 20 = w. You can have w stand for the unknown side. Use place value and the Distributive Property to find the unknown side length. B. 20 × 10 = 200 and 20 × 20 = 400, so, w is between 10 and 20. That is, W= 10 + ? 20 × ? = 160 C. 20 × 8 = 160 So w = 10 + 8 = 18. Multiply to check: 20 × 18 = 20 (10 + 8) = 200 + 160 = 360 So 360 ÷ 20 = 18. The garden is 18 feet wide. Convince Me! Make Sense and Persevere Use the diagram, place value, and the Distributive Property to find the quotient 408 ÷ 12. Hint: Find the value of x and solve. Answer: 12(30 + x) = 360+48 12(30 + x) = 408 30 + x = 408 ÷ 12 30 + x = 34 x = 34 – 30 = 4 Therefore value of x is 4. Guided Practice Do You Understand? Question 1. Write the missing numbers to find 154 ÷ 11. 154 = 11 × __ + 11 × ___ = 11 × (___ + ___) = 11 × ___ So, 154 ÷ 11 = ___ Answer: 154 = 11 × 10 + 11 × 4 = 11 × (10 + 4) = 11 × 14 So, 154 ÷ 11 = 14 Do You Know How? Question 2. Use the diagram to find 156 ÷ 12. So, 156 ÷ 12 = ___ Answer: 156 = 12 × 10 + 12 × 3 = 12 × (10 + 3) = 12 × 13 So, 156 ÷ 12 = 13 In 3 and 4, use grid paper or draw a picture to find each quotient. Question 3. 682 ÷ 22 Answer: 682 = 22 × 30 + 22 × 1 = 22 × (30 + 1) = 22 × 31 So, 682 ÷ 22 = 31 Question 4. 143 ÷ 11 Answer: 143 = 11 × 10 + 11 × 3 = 11 × (10 + 3) = 11 × 13 So, 143 ÷ 11 = 13 Start by estimating how many tens will be in the quotient. Independent Practice Leveled Practice In 5-11, use grid paper or draw a picture to find each quotient. Question 5. Use the diagram to find 182 ÷ 13. So, 182 ÷ 13 = ___ Answer: Question 6. 342 ÷ 38 Answer: Question 7. 720 ÷ 16 Answer: Question 8. 608 ÷ 19 Answer: Question 9. 752 ÷ 47 Answer: Question 10. 375 ÷ 25 Answer: Question 11. 576 ÷ 24 Answer: Problem Solving Question 12. Angelo is training for a long-distance bicycle ride. He travels 15 miles each hour. How many hours will it take him to ride 210 miles? Answer: Total Number of miles = 210 Number of miles each hour = 15 The number of hours it will take to ride 210 miles = 210 ÷ 15 = 14. Question 13. Higher-Order Thinking A rectangular doormat is 21 inches long and has an area of 714 square inches. Find its width. Will the doormat fit in an entryway that is 36 inches wide? Show your work. Answer: Given, Total area = 714 square inches Length = 21 inches Then, length x width = area 21 x width = 714 width = 714 ÷ 21 = 34 Therefore, the Width of the doormat is 34 inches. Yes, the doormat fits in an entryway that is 36 inches wide. Question 14. Use the map. How much longer is the distance from the library to the park to the train station than the distance from the library straight to the train station? Answer: Distance from the library to the park to the train station is 2.14 + 2.96 = 5.1 mi Distance from the library straight to the train station is 3.82 mi Difference = 5.1 – 3.82 = 1.28. Question 15. Algebra If you walk from the train station to the library, then to the park, and then back to the train station, how many miles would you walk in all? Write an equation to model your work. Answer: Distance from the library straight to the train station is 3.82 mi Distance from the library to the park to the train station is 2.14 + 2.96 = 5.1 mi Total distance = 3.82 + 5.1 = 8.92 mi Question 16. Make Sense and Persevere Explain how you can use the picture to show that 391 ÷ 23 = 17. Answer: 391 = 23 × 10 + 23 × 7 = 23 × (10 + 7) = 23 × 17 So, 391 ÷ 23 = 17 Assessment Practice Question 17. There are 16 rows of chairs in the auditorium. Each row has the same number of chairs. There are 512 chairs in all. How many chairs are in each row? A. 22 chairs B. 30 chairs C. 32 chairs D. 33 chairs Answer: C. 32 chairs Question 18. A patio has an area of 286 square feet. If the length of the patio is 22 feet, what is the width? A. 10 feet B. 13 feet C. 14 feet D. 144 feet Answer: B. 13 feet ### Lesson 5.4 Use Partial Quotients to Divide Activity Solve & Share A hotel sets up tables for a conference for 156 people. If each table seats 12 people, how many tables will be needed? Solve this problem any way you choose. You can use estimation to help solve this problem. Think about how many groups of 12 you can take away from 156. Show your work! Look Back! Make Sense and Persevere How can you check that the answer to a division problem is correct? Visual Learning Bridge Essential Question How Can You Use Partial Quotients to Solve Division Problems? A. A theater has 375 seats arranged in rows with 15 seats in each row. How many rows are in this theater? Let r equal the number of rows. Think: 15 × r = 375 or 375 ÷ 15 = r. An area model can help you find how many 15s are in 375. B. 375 ÷ 15 = 25 because 15 × 25 = 375. So, there are 25 rows and 0 additional seats in the theater. Convince Me! Critique Reasoning Yimil’s solution to the problem above is shown on the right. Is his solution correct? Explain. Guided Practice Do You Understand? Question 1. Show one way of using partial quotients to find 233 ÷ 11. Answer: Partial quotients are 20,1 Question 2. How can you use estimation to check that your answer to Problem 1 is correct? Answer: 240 approximately equals 233 12 approximately equals 11 240 ÷ 12 = 20. It approximately equals 21. Do You Know How? In 3-6, use partial quotients to divide. Show your work. Question 3. Answer: partial quotients are 10,4 Question 4. Answer: partial quotients are 20,2 Question 5. Answer: Question 6. Answer: partial quotients are 30. Independent Practice Leveled Practice In 7-16, use partial quotients to divide. Show your work. Question 7. Answer: Question 8. Answer: Question 9. Answer: partial quotients are 10,2 Question 10. Answer: partial quotients are 40, and 5 leftover Question 11. Answer: partial quotients are 10,9, and 0 leftover Question 12. Answer: partial quotients are 10,5, and 0 leftover Question 13. Answer: partial quotients are 30,1, and 2 leftover Question 14. Answer: partial quotients are 20,2, and o left over Question 15. Answer: partial quotients are 10,7, and 0 leftovers. Question 16. Answer: partial quotients are 30,8, and 2 leftovers. Problem Solving Question 17. A 969-acre wildlife preserve has 19 cheetahs. About how many acres does each cheetah have to itself, if each cheetah roams the same number of acres? Answer: Number of cheetahs = 19 Number of acres = 969 Number of acres does each cheetah have to itself = 969 ÷ 19 = 51 Question 18. A factory produces 272 chairs in an 8-hour shift. If the factory produces the same number of chairs each hour, how many chairs does it produce in 30 minutes? Answer: Number of chairs = 272 Number of hours per shift = 8 Number of chairs in 1 hour = 272 ÷ 8 = 34 Therefore, the number of chairs it produces in 30 minutes = 34/ 2 = 17. Question 19. A cafeteria can seat 5 × 102 students. Each table has 2 × 101 seats. How many tables are in the cafeteria? Answer: Number of students in cafeteria = 500 Number of seats in a table = 20 Number of tables = 500 ÷ 20 = 25. Question 20. A model with Math Peter is driving 992 miles from Chicago to Dallas. His sister Anna is driving 1,068 miles from Phoenix to Dallas. Write and solve an equation to find how much farther Anna drives than Peter drives. Answer: Peter is driving 992 miles from Chicago to Dallas His sister Anna is driving 1,068 miles from Phoenix to Dallas Difference in Distance = 1068 – 992 = 78 miles. Question 21. Write a multiplication equation and a division equation that represent the model shown below. Answer: Question 22. Higher-Order Thinking How can you use partial quotients to find 325 ÷ 13? Explain. Answer: Assessment Practice Question 23. Which expressions are equivalent to 35? 1,400 ÷ 4 420 ÷ 12 875 ÷ 25 7,700 ÷ 22 14,000 ÷ 40 Answer: 420 ÷ 12 = 35 875 ÷ 25 = 35 Question 24. Which expressions are equivalent to 22? 704 ÷ 32 1,078 ÷ 49 1,890 ÷ 30 1,430 ÷ 65 4,500 ÷ 50 Answer: 704 ÷ 32 1,078 ÷ 49 1,430 ÷ 65 ### Lesson 5.5 Use Sharing to Divide: Two-Digit Divisors Solve & Share The Recycling Club has$294 to purchase one set of recycling bins for each of the 14 members. Each of the 14 sets of bins will be identical to the others and cost the same amount. What is the greatest amount they can spend on one set of bins? Use objects or draw pictures to help solve this problem. Explain how you found your answer.

Using appropriate tools like play money or place-value blocks can help you divide.

Look Back! Why can you use division to answer this question?

Visual Learning Bridge

Essential Question How Can You Record Division with a Two-Digit Divisor?

A.
Orchard workers have 258 grapefruit seedlings to plant in 12 equal rows. How many seedlings will be in each row?

Estimate: 258 ÷ 12 is close to 250 ÷ 10 = 25.

You can think about place-value and area models to solve the problem.

B.
Regroup the blocks to fill the 12 rows.

258 ÷ 12 = 21 R6 because 12 × 21 + 6 = 258.
There will be 21 seedlings in each row with 6 seedlings leftover.

Convince Me! Reasoning What does the remainder mean in the problem above?

Guided Practice

Do You Understand?

Question 1.
If the orchard has 200 seedlings and 12 are planted in each row, how many rows will be filled? Draw place-value blocks to show your answer.

Question 2.
In Problem 1, what does the remainder represent?

The remainder represents the remaining orchards after planting 12 in a row.

Do You Know How?

In 3 and 4, divide. Write the missing numbers.

Question 3.

Question 4.

Independent Practice

Leveled Practice In 5-13, divide. Write the missing numbers.

Question 5.

Question 6.

Question 7.

Question 8.

Question 9.

Question 10.

Question 11.

Question 12.

Question 13.

Problem Solving

Question 14.
Rita’s family is moving from Grand Junction to Dallas. The moving van averages 60 miles each hour. About how many hours does the van take to reach Dallas? Explain your work.

Total miles = 980

Average speed = 60 miles each hour

Time it will take = 980 ÷ 60 = 16.33 hours

Question 15.
Due to construction delays on the trip from Little Rock to Chicago, a van driver averaged 48 miles each hour. About how long did that trip take?

Total miles = 660

Number of miles each hour = 48

Time taken = 660 ÷ 48 = 13.75 hours

Question 16.
Higher-Order Thinking A scientist needs 72 milliliters of distilled water for each of 15 experiments. She has a bottle that contains 975 milliliters of distilled water. Is there enough water in the bottle for all 15 experiments? Explain.

Total distilled water = 975 milliliters

Milliliters needed for  experiments =72

Number of experiments can be done = 972 ÷ 72 = 13.5

Therefore, there is no enough water in the bottle for 15 experiments.

Question 17.
A model with Math The Port Lavaca fishing pier is 3,200 feet long. One person is fishing for every ten feet of length. Write and solve an equation to find how many people are fishing from the pier.

Total length = 3200

One person fishing for every ten feet of length

Therefore, number of people = 3200 ÷ 10 = 320.

Question 18.
Todd made a table to show different plans he can use to save $500. Complete the table. Which plan can Todd use to save$500 in less than 16 weeks and have $20 extra? Explain how you found your answer. Answer: Assessment Practice Question 19. Find an expression that produces a quotient of 9 R15. Write the expression in the box. Answer: 375 ÷ 40 ### Lesson 5.6 Use Sharing to Divide: Greater Dividends Activity Solve & Share The city built a skate park that cost$3,240 and will be paid for over two years in equal monthly payments. How much is each monthly payment? Use objects or draw pictures to help solve this problem. Explain how you found your answer.

To make sense of the problem, you need to read carefully to find all the important information.

Look Back! Use Structure How did you use what you know about place value to find the answer to the problem?

Visual Learning Bridge

Essential Question
How Can You Record Division with a Two-Digit Divisor and a Four-Digit Dividend?

A.
Jake works at a flower shop. The shop just received a delivery of 1,830 roses. If the roses are distributed evenly among 15 coolers, how many roses should Jake put in each cooler?

You can use place-value and area models to solve the problem.

B.
There are not enough thousands to put one thousand in each group, so regroup the thousands into hundreds.

1,500 + 300 + 30 = 1,830
So 1,850 ÷ 15 = 122.

1,830 ÷ 15 = 122 because 15 × 122 = 1,830.
There should be 122 roses in each cooler with no roses left over.

Convince Me! Reasoning Why is 122 a reasonable answer for the problem?

Another Example
Divide 4,108 ÷ 85

Think 41 hundred divided into 82 equal groups.

4,108 ÷ 82 = 50 R8 because 82 × 50 + 8 = 4,108.

Guided Practice

Do You Understand?

Question 1.
Use place-value blocks to model
3,710 ÷ 18.

Do You Know How?

In 2-5, divide. Use place-value blocks to help.

Question 2.
4,632 ÷ 15

Question 3.
3,332 ÷ 30

Question 4.

Question 5.

Independent Practice

In 6, draw an area model for the division problem.

Question 6.

In 7-12, divide. Use place-value blocks or area models to help.

Question 7.
7,905 ÷ 35

Question 8.
5,500 ÷ 90

Question 9.
2,838 ÷ 11

Question 10.

Question 11.

Question 12.

Problem Solving

Question 13.
Number Sense The booster club picked 1,370 apples. They plan to sell bags of apples with 15 apples in each bag. How many bags can they make? Explain.

Number of apples = 1370

Number of apples in each bag = 15

Number of bags= 1370 ÷ 15 =91.33 approximately equals to 91.

Question 14.
Mason teaches ice skating. He earns $24.50 per lesson. How much does he earn in 5 days if he gives 6 lessons per day? Answer: Money earned per lesson=$24.50

Number of lessons per day = 6

Total earned in a day = $24.50 x 6 =$147

In 5 days, she will earn = 147 x 5 = $735 Question 15. Reasoning A delivery to the flower shop is recorded at the right. The shop makes centerpiece arrangements using 36 flowers that are all the same type. Will they be able to make at least 10 arrangements using each type of flower? At least 100 arrangements? Explain. Answer: Question 16. Amelia and Ben have two different answers for 1,955 ÷ 85. Without dividing, how can you tell who might be correct? Amelia: 1,955 ÷ 85 = 23 Ben: 1,955 ÷ 85 = 203 Answer: I can tell by estimating the values. 1955 approximately equals 2000 85 approximately equals 80 2000 ÷ 80 = 25 which is close to 23. so, amelia is correct Question 17. Higher-Order Thinking Estimate the quotient of 4,839 ÷ 15 to the nearest hundred. Explain how you found the estimate. Answer: 4839 is close to 4800 Now, 4800 ÷15 = 320. so, the estimation of the quotient of 4,839 ÷ 15 is close to 300. Assessment Practice Question 18. Find 5,092 ÷ 38. How can you check the reasonableness of your answer? Answer: 5,092 ÷ 38 = 134 I can check reasonableness by estimating the values 5000 close to 5092 40 is close to 38 5000 ÷ 40 = 125 which is close to 134. ### Lesson 5.7 Choose a Strategy to Divide Activity Solve & Share Choose a strategy to solve each problem. Explain your solutions. Problem 1: Bob’s Citrus and Nursery sells cartons of citrus fruits. There are 24 oranges in each carton. They have 5,643 oranges to pack into cartons. How many cartons can they fill? Problem 2: This year, 4,338 grapefruits have been harvested so far. Bob’s has 18 storage bins for grapefruits. If the grapefruits are distributed evenly among the 18 bins, how many grapefruits are in each bin? The numbers and the situations in the problems can help you choose strategies. Show your work! Look Back! Make Sense and Persevere How are your two strategies alike? How are they different? Visual Learning Bridge Essential Question What Are Some Different Strategies I Can Use to Solve a Division Problem? A. A company has three printers. The printer in Room 102 is for all of their 15 employees to use. If the available pages are distributed evenly among the employees, how many pages can each employee use? Answer: Number of pages available to print = 3720 Number of employees = 15 Number of pages can each employee use = 3720 ÷ 15 = 248 B. You can first estimate. 3,720 ÷ 15 is between 3,000 ÷ 15 = 200 and 4,500 ÷ 15 = 300. Each employee can print 248 pages. Convince Me! Reasoning How can you check your answer? Another Example How many 32-page brochures can the 40 144 printers in Room 101 print? The printer in Room 101 can print 144 brochures. You need to find how many 32s are in 4,618. You can use division and show partial quotients. You can estimate 4,618 ÷ 32 using 4,500 ÷ 30 = 150. Guided Practice Do You Understand? Question 1. Can the remainder be greater than the divisor? Why or why not? Answer: No. the Remainder can not be greater than the divisor. If the remainder is greater than the divisor, the division is incomplete. Do You Know How? Question 2. Estimate 452 ÷ 21. Answer: 452 is close to 460 21 is close to 20 460 ÷ 20 = 23. Question 3. Divide. Answer: Yes, the answer is reasonable. Remember to check that your answer is reasonable. Independent Practice In 4-11, estimate and then find the quotient. Use your estimate to check for reasonableness. Question 4. Answer: Estimation is 400 ÷ 50 = 8 Yes, the estimation is reasonableness. Question 5. Answer: Estimation is 640 ÷ 80 = 8 Yes, the estimation is reasonableness. Question 6. 761 ÷ 5 Answer: Estimation is 750 ÷ 5 = 150 Yes, the estimation is reasonableness. Question 7. 510 ÷ 30 Answer: Estimation is 510 ÷ 30 = 17 Yes, the estimation is reasonableness. Question 8. 7,704 ÷ 24 Answer: Estimation is 7,680 ÷ 24 = 320 Yes, the estimation is reasonableness. Question 9. 7,830 ÷ 33 Answer: Estimation is 7821 ÷ 33 = 237 Yes, the estimation is reasonableness. Question 10. 3,136 ÷ 64 Answer: Estimation is 3,200 ÷ 64 = 50 Yes, the estimation is reasonableness. Question 11. 6,253 ÷ 71 Answer: Estimation is 6,300 ÷ 70 = 90 Yes, the estimation is reasonableness. Problem Solving For 12, use the table at right. Question 12. Make Sense and Persevere Bob’s sells tangelo gift cartons each December. Last year, they shipped a total of 3,300 tangelos. If each carton sells for$28, how much money did Bob’s earn from the tangelo gift cartons sold?

Number of tangelos = 3300

Number of tangelos per carton = 12

Number of cartons = 3300 ÷ 12 = 275

Money earned = $28 x 275 =$7700.

Question 13.
Higher-Order Thinking A score is a group of 20 things. For example, a period of 20 years is called a score of years. The Statue of Liberty was dedicated in 1886. About how many scores of years ago was that?

The score for every 20 years

The Statue of Liberty was dedicated in 1886.

1886 ÷ 20 = 94.3 approximately equals to 94.

Question 14.
At an automobile plant, each car is inspected by 34 different workers before it is shipped to a dealer. One day, workers performed 9,690 inspections. How many cars were shipped? Explain.

workers performed 9,690 inspections

each car is inspected by 34 different workers

Therefore, number of cars were inspected = 9,690 ÷ 34 = 285

Assessment Practice

Question 15.
For which division problems is 46 the quotient? Write those division problems in the box.

### Lesson 5.8 Make Sense and Persevere

Problem Solving

Activity

Solve & Share

Write a word problem for the equation: 2,530 ÷ 23 = q

Solve your problem any way you choose.

Thinking Habits
• What information is provided?
• How are the quantities related to each other?
• What strategies do I know for solving this type of problem?
• What tools might help me?
• How can I check that my solution makes sense?

Look Back! Make Sense and Persevere Does your word problem ask you to find the equal number in each group, or the number of equal groups?

Visual Learning Bridge

Essential Question How Can You Make Sense of Problems and Persevere in Solving Them?

A.
For 3 months, a fifth-grade class raised money for charities. If the class divides the money equally among 32 different organizations, how many dollars will each organization receive, and how many dollars will be leftover?

You can make sense of the problem by answering these questions. How much money was raised in all?
How much should each organization get?

Here’s my thinking.

B.
How can I make sense of and solve the problem?
I can
• identify the quantities given.
• understand how the quantities are related.
• choose and implement an appropriate strategy.
• check to be sure my work and answer make sense.

C.
First, I can write an addition equation to find the total amount raised:
1,104 + 2,117 + 3,275 = 6,496
Next, I can write a division equation to model equal sharing:
6,496 ÷ 32 =
(6400 + 96) ÷ 32 =
(6400 ÷ 32) + (96 ÷ 32) =
200 + 3 = 203
So, each organization will receive $203. Convince Me! Critique Reasoning Julio says that you can solve this problem by dividing each month’s total by 32 and then adding the three quotients together. Do you agree? Do you think his approach is easier or harder? Justify your answer. Stuck? Try solving a simpler problem. Guided Practice Make Sense and Persevere Dana starts with 875 stamps in her stamp collection. Her grandparents give her 332 stamps. Then, she buys 72 more. How many pages in her scrapbook can she fill? Question 1. What do you know? Answer: Dana starts with 875 stamps in her stamp collection. Her grandparents give her 332 stamps. Then, she buys 72 more Question 2. What are you trying to find? Answer: How many pages in her scrapbook can she fill? Question 3. How are the quantities related? What is the answer to the problem? Write equations to model your work. Answer: 875 + 332 + 72 = 1279 Number of stamps in a page = 24 The number of pages in her scrapbook she can fill = 1279 ÷ 24 = 53.2 approximately equals 53. Independent Practice Make Sense and Persevere Tanya is saving for a vacation. She wants to have at least$75 for each of the 12 days of her trip. If she saves $85 each month for 10 months, will she save enough money? Question 4. Use the strategy of mental math to find the total amount she will save. Then, write a division equation to see if she will save enough. Answer: Amount for 12 days =$75 x 12 = $900 Amount for 10 months =$85 x 10 = $850 She can not save enough money. Question 5. Jorge says he can solve this problem differently. He says that he can compare 85 × 10 and 75 × 12. Do you agree? Explain your thinking. Answer: Yes. I agree with Jorge. Because it is easy to estimate and compare both values of amounts. Problem Solving Performance Task Pumpkin Patch Farms The table shows the number of seeds the owners of Pumpkin Patch Farms received from different seed suppliers. Each of the pumpkins they harvest usually weighs between 10 and 12 pounds. There are 60 rows, and the farmers will plant the same number of seeds in each row. How many seeds will they plant in each row? Question 6. Make Sense and Persevere What do you know? What are you trying to find? Answer: Question 7. Reasoning How are the quantities in the problem-related? What steps are needed to solve the problem? Answer: Question 8. A model with Math Writes equations with variables to represent the steps needed to solve the problem. Think about the problem-solving strategies to help you! Question 9. Be Precise Solve the equations and answer the question. Answer: Question 10. Reasoning What strategy can you use to check that your answer makes sense? Answer: ### Topic 5 Fluency Practice Activity Point & Tally Work with a partner. Get paper and a pencil. Each partner chooses light blue or dark blue. At the same time, Partner 1 and Partner 2 each point to one of their black numbers. Both partners find the product of the two numbers. The partner who chose the color where the product appears gets a tally mark. Work until one partner has seven tally marks. Topic 5 Vocabulary Review Glossary Understand Vocabulary Word List • compatible numbers • dividend • divisor • estimate • multiple • product • quotient • remainder Choose the best term from the Word List. Write it on the blank. Question 1. One way to estimate the answer to a division problem is to replace the divisor and dividend with ___ Answer: Estimate and multiple Question 2. The part that is left when you divide into equal groups is called the ____ Answer: remainder Question 3. To decide where to place the first digit of a quotient, ____ is the number of digits in the answer. Answer: The first digit of the quotient will be in the Hundreds Place, If the first digit is less than the divisor, If the first digit is greater than or equal to the divisor, the first digit of the quotient will be in the thousand’s place. Question 4. The answer to a division problem is the ______ Answer: Remainder. For each of these terms, give an example and a non-example. Use Vocabulary in Writing Question 8. Write a division problem with a 3-digit dividend, a divisor of 20, and the remainder of 10. Use at least three of the terms in the Word List to explain how you chose the numbers for your example. Answer: ### Topic 5 Reteaching Set A pages 181-184 Find 32,000 ÷ 80 using mental math. Use basic facts and place-value patterns to help. 32 ÷ 8 = 4 320 ÷ 80 = 4 3,200 ÷ 80 = 40 32,000 ÷ 80 = 400 Remember to look for a basic division fact in the numbers. Check your answer by multiplying. Find each quotient. Use mental math. Question 1. 360 ÷ 40 Answer: 360 ÷ 40 36 ÷ 4 = 9. Question 2. 270 ÷ 90 Answer: 270 ÷ 90 27 ÷ 9 = 3 Question 3. 2,100 ÷ 30 Answer: 2,100 ÷ 30 210 ÷ 3 = 70 Question 4. 4,800 ÷ 80 Answer: 4,800 ÷ 80 480 ÷ 8 = 60 Question 5. 72,000 ÷ 80 Answer: 72,000 ÷ 80 7200 ÷ 8 = 900 Question 6. 81,000 ÷ 90 Answer: 81,000 ÷ 90 8100 ÷ 9 900 Set B pages 185-188 Estimate 364 ÷ 57. Use compatible numbers and patterns to divide. So, 364 ÷ 57 is about 6. Remember that compatible numbers are numbers that are easy to compute mentally. Estimate using compatible numbers. Question 1. 168 ÷ 45 Answer: 168 ÷ 45 160 ÷ 40 4 Question 2. 525 ÷ 96 Answer: 525 ÷ 96 500 ÷ 100 5 Question 3. 379 ÷ 63 Answer: 379 ÷ 63 360 ÷ 60 6 Question 4. 234 ÷ 72 Answer: 234 ÷ 72 210 ÷ 70 3 Question 5.$613 ÷ 93

$613 ÷ 93$630 ÷ 90

$7 Question 6.$748 ÷ 92

$748 ÷ 92$720 ÷ 90

$8 Set C pages 189-192 Find 195 ÷ 13. Draw a model to help you find 131 the number of tens and ones in the quotient. 1 ten + 5 ones = 15. So, 195 ÷ 13 = 15 Remember to find the number of tens first, then find the number of ones. Use a model to find each quotient. Question 1. 180 ÷ 15 Answer: Question 2. 154 ÷ 14 Answer: Question 3. 351 ÷ 27 Answer: Question 4. 192 ÷ 16 Answer: Question 5. 143 ÷ 11 Answer: Question 6. 217 ÷ 31 Answer: Question 7. 130 ÷ 26 Answer: Question 8. 270 ÷ 18 Answer: Set D pages 193–196 Find 336 ÷ 21 using partial quotients. Add the partial quotients: 10 + 6 = 16. So, 336 = 21 = 16. Remember to add the partial quotients to find the actual quotient. Use partial quotients to divide. Question 1. Answer: Partial quotients are 10, 9 and sum is 19 Question 2. Answer: Partial quotients are 40,2 and sum is 42 Question 3. Answer: Partial quotients are 40,1 and sum is 41 Question 4. Answer: partial quotients are 20,3 and sum is 23. Question 5. Answer: PArtial quotients are 60,6 and sum is 66. Question 6. Answer: partial quotients are 30,4 and sum is 34. Question 7. Answer: partial quotients are 20,8 and sum is 28 Question 8. Answer: partial quotients are 10,1 and sum is 11. Set E pages 197-200 Find 461 ÷ 50. Remember that you can check your answer by multiplying the quotient by the divisor, and then adding any remainder. Question 1. Answer: Question 2. Answer: Question 3. Answer: Question 4. Answer: Question 5. Answer: Question 6. Answer: Question 7. Answer: Question 8. Answer: Question 9. Ivan uses 30 craft sticks to make each toy cabin. He has a box of 342 craft sticks. How many toy cabins can Ivan make? How many sticks will be left? Answer: Set F pages 201-204 Find 3,657 ÷ 23. Remember you can multiply the divisor by powers of 10 to estimate the quotient. Divide. Use place-value blocks to help. Question 1. Answer: Question 2. Answer: Question 3. Answer: Question 4. Answer: Question 5. Answer: Question 6. Answer: Question 7. Answer: Question 8. Answer: Set G pages 205-208 Find 789 ÷ 19. Remember that you can check your answer by multiplying the quotient by the divisor, and then adding any remainder. Question 1. Answer: Question 2. Answer: Question 3. Answer: Question 4. Answer: Question 5. Answer: Question 6. Answer: Question 7. Answer: Question 8. Answer: Question 9. Answer: Question 10. Answer: Set H pages 209-212 Think about these questions to help you make sense and persevere in solving problems. Thinking Habits • What do I know? • What do I need to find? • What’s my plan for solving the problem? • What else can I try if I get stuck? • How can I check that my solution makes sense? Selena is planning to visit her aunt in 5 weeks. She has saved$365 but thinks the trip will cost $500. She plans to save the same amount each week so she has$500 for the trip. How much does she need to save each week?
I can write an equation to find how much more money Selena needs:
500 – 365 = 135
Then divide the amount she needs by 5 weeks: 135 ÷ 5 = 27
Selena needs to save $27 each week. My answer is reasonable because 365 + 27 + 27 + 27 + 27 + 27 = 500. Remember to think about what steps are needed to solve each problem. Solve. Show your work. Question 1. The football coach spent a total of$890 including $50 in tax for 35 shirts for the team. Each shirt cost the same amount. What was the price of one shirt before tax was added? Answer: Total cost =$890 including tax

Total cost before tax added = $890 –$50 = $840 Total number of shirts = 35 Therefore, Price of one shirt =$840 ÷ 35 = $24 Question 2. A gymnast practices 6 days each week. She practices the same number of hours each day. If she practices a total of 120 hours in a 4-week period, how many hours each day does she practice? Answer: Number of days in the 4-week period = 4×7 = 28 But, she practices only 6 days, Therefore, 6 days in a 4-week period = 6×4 = 24 Total number of hours in 4-week period = 120 Therefore, the number of hours each day she practices = 120 ÷ 24 =5. Question 3. Nathan works the same number of hours each day, 5 days each week. He earns$12 per hour. Last week he earned $420. How many hours did he work each day last week? Write equations to model your work. Answer: Total earned =$420

Number of days in each week = 5

Total earned in each day = $420 ÷ 5 =$84

Total earned per hour = $12 Therefore, the number of hours he works each day =$84 ÷$12 = 7. Question 4. A high-rise apartment building has 15 floors with 26 apartments on each floor. There are 3 kinds of apartments in the building: 1-, 2-, and 3-bedroom. The building has the same number of each kind of apartment. How many of each kind of apartment is in the building? Show your work. Answer: The number of floors = 15 The number of Apartments = 26 Now, 15 x 26 = 390 Apartments. Given that, There are three kinds of apartments = 390/3 = 130 Therefore, there are 130 apartments in the building. ### Topic 5 Assessment Practice Question 1. Select all of the following equations the number 60 will make true. 420 ÷ = 70 1,800 ÷ = 300 5,400 ÷ = 90 2,400 ÷ = 40 500 ÷ = 10 Answer: 5,400 ÷ = 90 2,400 ÷ = 40 Question 2. Which of the following is the best estimate of 487 ÷ 67? A. 80 B. 70 C. 10 D. 7 Answer: B. 70 Question 3. The carnival committee has purchased 985 small prizes. The prizes are to be divided equally among the 20 game booths. A. In what place will the first digit of the quotient be? B. How many prizes will each booth have? C. How many prizes will be left? Answer: A. The first digit of the quotient be in Ten’s place. B. 985 /20 = 49 as Quotient Therefore, each booth have 49 prizes in each booth C. 985 /20 = 5 as remainder Therefore, 5 prizes will be left. Question 4. A rectangular living room has an area of 425 square feet. The width of the room is 17 feet. Write a number in the box to show the missing dimension. What is the length of the room? ____ feet Answer: Given, The Area of the living room = 425 square feet. Now, 85/17 = 5 Therefore, the length of the room = 20 + 5 = 25 feet. Question 5. A. Divide. 2,700 ÷ 30 = ____ B. Select all the expressions that are equal to 2,700 ÷ 30. 270 ÷ 3 270 tens ÷ 3 tens 2,700 ÷ 3 tens 2,700 ÷ 3 2,700 tens ÷ 30 Answer: A. 90 B. 270 ÷ 3 270 tens ÷ 3 tens 2,700 ÷ 3 tens Question 6. Select the quotient for each expression. Answer: Question 7. Use the table. A. Using Plan B, how many weeks will it take Althea to reach her savings goal? Write the missing number in the table. B. Show how you found your answer to A. Answer: Althea’s plans for savings =$384

It approximately equals $390 Amount to save each week =$30 according to plan B

Therefore, Number of weeks needed = $390 ÷$30  = 13

Question 8.
Five Star Farm purchased 2,400 apple trees. If 80 trees can be planted on each acre of land, how many acres will be needed to plant all the trees?
A. Identify which expression represents the problem.
A. 2,400 × 80
B. 80 × 2,400
C. 80 ÷ 2,400
D. 2,400 ÷ 80
B. How many acres will be needed to plant all the trees?

A.

D. 2,400 ÷ 80

B.

Total number of apple trees = 2400

Number of trees planted on each acre land = 80

Number of acres needed to plant all the trees = 2400 ÷ 80 = 30

Question 9.
Mrs. Reiss has 264 crayons for her art class of 22 students. How many crayons will each student get if the crayons are divided equally? Use the model.

Number of students = 22

Number of Crayons = 264

Now,

264/22

= 12

Therefore, Each Student gets 12 crayons.

Question 10.
Select all the expressions that have the value of 9.
270 ÷ 3
250 ÷ 25
270 ÷ 30
207 ÷ 23
189 ÷ 21

1. 270 ÷ 3
2. 270 ÷ 30
3. 207 ÷ 23
4. 189 ÷ 21

Question 11.
Kari wants to find 3,277 ÷ 29.
A. Without doing the division, which number will the quotient be closest to?
A. 1
B. 10
C. 100
D. 1,000
B. What is the exact quotient?

A.

The quotient be closest to 100

B. The exact quotient =

3,277 ÷ 29.

= 113

Question 12.
The cost to rent a lodge for a family reunion is $975. If 65 people attend and pay the same price, how much does each person pay? A. Which of the following expressions represents the problem? A. 975 + 65 B. 975 ÷ 65 C. 975 × 65 D. 975 ÷ 2 B. How much does each person pay? A.$16
B. $15 C.$14
D. $13 Answer:$975 ÷ 65

= $15 Therefore, Each person pays$15.

Question 13.
Shady Rivers summer camp has 188 campers this week. If there are 22 campers to each cabin, what is the least number of cabins needed?
A. 7 cabins
B. 8 cabins
C. 9 cabins
D. 10 cabins

188/22

= 8 cabins.

Question 14.
The area of a rectangular banquet hall is 7,400 square feet. The length of one side of the hall is 82 feet. Explain how you can use compatible numbers to estimate the width of the hall.

Given,

The area of the rectangle = 7,400 square feet.

The length of one side = 82 feet.

We know that, The area of the rectangle =

Area = length x width

7400 = 82 x n

n = 7400/82

= 90.2 feet

Therefore, the width of the rectangular banquet hall = 90.2 feet.

Question 15.
The cost of renting a bus is $1,344. Tony wants to find how much each person will pay if 32 people ride the bus and share the cost equally. Fill in the partial quotients that are missing from Tony’s work below. Answer: Question 16. Jessie made 312 mini energy bars. She puts 24 bars in each bag. She plans to sell each bag for$6.
A. Write two equations with variables that Jessie can use to find the amount of money she will earn if she sells all of the bags.
B. How much will she earn if she sells all of the bags?

Question 17.
Select all of the following equations the number 40 will make true.
280 ÷ = 7
800 ÷ = 20
4,000 ÷ = 10
3,200 ÷ = 80
800 ÷ = 200

1. 280 ÷ 40  =  7
2. 800 ÷ 40 = 20
3. 3200 ÷ 40 = 80

Question 18.
Select the quotient for each expression.

1. 2700 ÷ 30 = 90
2. 270 ÷ 30 = 9
3. 2400 ÷ 30 = 80
4. 240 ÷ 30 = 8

Question 19.
Charles burns 4,350 calories hiking 15 miles of the Appalachian Trail. How many calories does he burn each mile?
A. Identify which expression represents the problem.
A. 4,350 ÷ 15
B. 4,350 × 15
C. 4,350 – 15
D. 4,350 ÷ 10
B. How many calories does he burn each mile?

A. 4,350 ÷ 15

B. In each mile he burns 290 calories.

Question 20.
Find the quotient 432 ÷ 48.

432 ÷ 48 = 9

Question 21.
Which partial quotients could be added to find 465 ÷ 15?
A. 20 and 1
B. 30 and 1
C. 30 and 9
D. 30 and 10

B. 30 + 1

465 ÷ 15 = 31

30 + 1 = 31

Question 22.
The table shows the number of students going on field trips. For each trip, one adult is needed for every 15 students.

Given,

Number of students in seventh grade = 225

Condition is one adult is needed for every 15 students.

Number of adults are needed to go on the seventh-grade trip = 225 ÷ 15 = 15

Therefore, the number of adults are needed to go on the seventh-grade trip is 15.

School Supplies
A store had a sale on school supplies in August. The store manager recorded how many of several types of items were sold. Each of the same types of item costs the same amount. Use the information in the table to answer the questions.

Question 1.
Backpack sales totaled $1,200. How much did each backpack cost? Write an equation to model your work. Answer: Given, Total sales =$1200

Total number of backpacks = 60

cost of each backpack = $1200 ÷ 60 =$120 ÷ 6 = $20. Therefore, the cost of each backpack is$20.

Question 2.
The store sold 71 packages of pens. Use compatible numbers to estimate how many pens were in each package. Show your work.

Given,

Number of packages = 71

Number of pens sold = 568

number of pens in each package is 568 ÷ 71 = 8

Therefore, the number of pens in each package is 8

Question 3.
There were 16 pencils in each box. Olivia wants to find how many boxes of pencils were sold.
Part A
When Olivia divides 784 by 16, in which place should she write the first digit of the quotient? Tell how you know without dividing.
Part B
How many boxes of pencils were sold?

Number of pencils sold = 784

Number of pencils in each box = 16

Number of boxes of pencils sold = 784 ÷ 16 = 49

Therefore, the number of boxes of pencils sold is 49.

Question 4.
The store manager has ordered the calculators shown, but the shipment has been delayed.
Part A

If all the calculators ordered are sold, the total sales would be $2,014. Was the number of calculators ordered less than or greater than 100? How do you know without dividing? Part B How many calculators were ordered? Write an equation to model your work. Answer: Given, Total sales =$2,014

Cost for each calculator = $19 Therefore number of calculators =$2,014 ÷ $19 = 106 So, the number of calculators ordered was 106 and it is greater than 100. we can say it not by diving. 19 x 100 = 1900 and the total sales given is$2014. So it must be greater than 100.

Question 5.
The manager wants to order 408 more notebooks. The notebooks are shipped in packages of 12. He used partial quotients to find the number of packages to order. His work is shown on the right. Is his solution correct? Explain.

No. The model to find the number of packages to order is not correct.

Explanation:

408 ÷ 12 = 34

But his work shown at right is showing the value is 40

So, it is not correct and the correct value is 34.

Question 6.
An additional 40 packages of paper were ordered at a total cost of $520. How much did each package of paper cost? Write an equation to model your work. Answer: Each package of paper cost$13

Because,

$520 ÷ 40$52 ÷ 4 = $13. #### enVision Math Common Core Grade 5 Answer Key ## Envision Math Common Core Grade 5 Answer Key Topic 6 Use Model Strategies to Divide Decimals ## Envision Math Common Core 5th Grade Answers Key Topic 6 Use Model Strategies to Divide Decimals enVision STEM Project: States of Water Do Research Use the Internet or other sources to learn about the states of water. Find at least 5 examples of water in nature as a solid, as a liquid, and as a gas. At what temperature does liquid water change to ice? At what temperature does liquid water change to water vapor? Journal: Write a Report Include what you found. Also in your report: • Explain how liquid water changes to ice and to water vapor. • At 23°F, 1 inch of rain equals 10 inches of snow. Convert 2 inches of rainfall to snowfall. • Make up and solve division problems that involve decimals. Review What You Know Vocabulary Choose the best term from the box. Write it on the blank. • decimal • dividend • divisor • quotient Question 1. _____ is the name for the answer to a division problem. Answer: Quotient is the name for the answer to a division problem. Question 2. A number that is being divided by another number is called the _____ Answer: A number that is being divided by another number is called the dividend. Whole Number Operations Find each value. Question 3. 9,007 – 3,128 Answer: 9,007 – 3,128 = 5,879 Explanation: The properties of whole numbers are based on arithmetic operations such as addition, subtraction, division and multiplication. Subtraction of two whole numbers may not result in whole numbers. It can be an integer too. Whole numbers are positive integers along with zero. They are all the natural numbers including zero. Subtract 3,128 from 9,007 then the difference is 5,879. Question 4. 725,864 + 39,798 Answer: 725,864 + 39,798 = 765,662 Explanation: The properties of whole numbers are based on arithmetic operations such as addition, subtraction, division and multiplication. Two whole numbers if added or multiplied will give a whole number itself. Whole numbers are positive integers along with zero. They are all the natural numbers including zero. Perform addition operation on these two numbers 725,864 and 39,798 then the sum is 765,662. Question 5. 35 × 17 Answer: 35 × 17 = 595 Explanation: The properties of whole numbers are based on arithmetic operations such as addition, subtraction, division and multiplication. Two whole numbers if added or multiplied will give a whole number itself. Whole numbers are positive integers along with zero. They are all the natural numbers including zero. Perform multiplication operation on these two numbers 35 and 17 then the result is 595. Question 6. 181 × 42 Answer: 181 × 42 = 7,602 Explanation: The properties of whole numbers are based on arithmetic operations such as addition, subtraction, division and multiplication. Two whole numbers if added or multiplied will give a whole number itself. Whole numbers are positive integers along with zero. They are all the natural numbers including zero. Perform multiplication operation on these two numbers 181 and 42 then the result is 7,602. Question 7. 768 ÷ 6 Answer: 768 ÷ 6 = 128 Explanation: The properties of whole numbers are based on arithmetic operations such as addition, subtraction, division and multiplication. The inverse operation of multiplication is division. Whole numbers are positive integers along with zero. They are all the natural numbers including zero. Perform division operation on these two numbers 768 and 6. Here 768 is dividend and 6 is divisor then the quotient is 128. Question 8. 506 ÷ 22 Answer: 506 ÷ 22 = 23 The properties of whole numbers are based on arithmetic operations such as addition, subtraction, division and multiplication. The inverse operation of multiplication is division. Whole numbers are positive integers along with zero. They are all the natural numbers including zero. Perform division operation on these two numbers 506 and 22. Here 506 is dividend and 22 is divisor then the quotient is 23. Question 9. 6,357 ÷ 60 Answer: 6,357 ÷ 60 = 105.95 Explanation: The properties of whole numbers are based on arithmetic operations such as addition, subtraction, division and multiplication. The inverse operation of multiplication is division. Whole numbers are positive integers along with zero. They are all the natural numbers including zero. Perform division operation on these two numbers 6357 and 60. Here 6,357 is dividend and 60 is divisor then the quotient is 105.95. Question 10. 3,320 ÷ 89 Answer: 3,320 ÷ 89 = 37.30 Explanation: The properties of whole numbers are based on arithmetic operations such as addition, subtraction, division and multiplication. The inverse operation of multiplication is division. Whole numbers are positive integers along with zero. They are all the natural numbers including zero. Perform division operation on these two numbers 3,320 and 89. Here 3,320 is dividend and 89 is divisor then the quotient is 37.30. Question 11. 88,888 ÷ 20 Answer: 88,888 ÷ 20 = 4,444.4 Explanation: The properties of whole numbers are based on arithmetic operations such as addition, subtraction, division and multiplication. The inverse operation of multiplication is division. Whole numbers are positive integers along with zero. They are all the natural numbers including zero. Perform division operation on these two numbers 88,888 and 20. Here 88,888 is dividend and 20 is divisor then the quotient is 4,444.4. Rounding Decimals Round each number to the place of the underlined digit. Question 12. 0.34 Answer: 0.3 Explanation: Rounding is a process to estimate a particular number in a context. To round a number look at the next digit in the right place, if the digit is less than 5, round down and if the digit is 5 or more than 5, round up. The number 0.34 is rounded to 0.3. Question 13. 96.5 Answer: 97 Explanation: Rounding is a process to estimate a particular number in a context. To round a number look at the next digit in the right place, if the digit is less than 5, round down and if the digit is 5 or more than 5, round up. The number 96.5 is rounded to 97. Question 14. 81.27 Answer: 81.3 Explanation: Rounding is a process to estimate a particular number in a context. To round a number look at the next digit in the right place, if the digit is less than 5, round down and if the digit is 5 or more than 5, round up. The number 81.27 is rounded to 81.3. Question 15. 205.3 Answer: 205 Explanation: Rounding is a process to estimate a particular number in a context. To round a number look at the next digit in the right place, if the digit is less than 5, round down and if the digit is 5 or more than 5, round up. The number 205.3 is rounded to 205. Decimals Question 16. An insect measured 1.25 cm long. Which number is less than 1.25? A. 1.35 B. 1.3 C. 1.26 D. 1.2 Answer: Option D is correct. Explanation: An insect measured 1.25 cm long. We have to find out the number less than 1.25. A. 1.35 is greater than 1.25. So option A is not correct. B. 1.3 is greater than 1.25. So option B is not correct. C. 1.26 is greater than 1.25 . So option C is not correct. D. 1.2 is less than 1.25. So option D is correct. Question 17. Explain The grid in this model represents 1. What decimal does the shaded part represent? Explain. Answer: 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.04 = 0.64 The shaded part represents the decimal 0.64. Explanation: The above grid represents 1. In that grid 1 column is shaded. The shaded part decimal value is 0.1. The second column shaded part decimal value is 0.1. The third column shaded part decimal value is 0.1. The fourth column shaded part decimal value is 0.1. The fifth column shaded part decimal value is 0.1. The sixth column shaded part decimal value is 0.1. The seventh column shaded part decimal value is 0.04. By adding these columns decimal values the sum is 0.64. Decimal Operations Find each value. Question 18. 23.7 – 11.82 Answer: 23.7 – 11.82 = 11.88 Explanation: Four basic operations on decimals are addition, subtraction, multiplication and division. To add decimals numbers we follow this step. Step 1: Write numbers under each other and line up vertically the decimal points. Perform subtraction operation on these two numbers 23.7 and 11.82. Subtract 11.82 from 23.7 then the difference is 11.88. Question 19. 66.8 + 3.64 Answer: 66.8 + 3.64 = 70.44 Explanation: Four basic operations on decimals are addition, subtraction, multiplication and division. To add decimals numbers we follow this step. Step 1: Write numbers under each other and line up vertically the decimal points. Perform addition operation on these two numbers 66.8 and 3.64. Add 66.8 with 3.64 then the sum is 70.44. Question 20. 9 × 1.4 Answer: 9 x 1. 4 = 12. 6 Explanation: Four basic operations on decimals are addition, subtraction, multiplication and division. To add decimals numbers we follow this step. Step 1: Write numbers under each other and line up vertically the decimal points. Perform multiplication operation on these two numbers 9 and 1.4. Multiply 9 with 1.4 then the result is 12.6. Question 21. 3.2 × 7.6 Answer: 3.2 x 7.6 = 24.32 Explanation: Four basic operations on decimals are addition, subtraction, multiplication and division. To add decimals numbers we follow this step. Step 1: Write numbers under each other and line up vertically the decimal points. Perform multiplication operation on these two numbers 3.2 and 7.6. Multiply 3.2 with 7.6 then the result is 24.32. Pick a Project PROJECT 6A Can you throw a dinner party? Project: Plan a Party PROJECT 6B How much does it cost to run a company? Project: Build a Company PROJECT 6C How do you organize food? Project: Open Your Own Fruit Stand PROJECT 6D Would you like to build a house? Project: Draw Plans for a Doll House ### Lesson 6.1 Patterns for Dividing with Decimals Activity Solve & Share An object is 279.4 centimeters wide. If you divide the object into 10 equal parts, how wide will each part be? Solve this problem any way you choose. How can you use structure and the relationship between multiplication and division to help you? Look Back! What do you notice about the width of the object and the width of each part? Answer: The width of the object is 279.4 279.4/10 = 27.94 The width of each part is 27.94 Explanation: An object is 279.4 centimeters wide. If we divide the object into 10 equal parts. The wide of the each part is 27.94. Multiplication and division are the reverse or opposite of each other in that when we divide, we break apart, and when we multiply, we put together. Visual Learning Bridge Essential Question How Can You Divide Question Decimals by Powers of 10? A. Shondra wants to cut a cloth into 10 strips. All the strips should be exactly the same size. You can use place value and what you know about whole numbers to divide decimals by powers of 10. How long will each strip be? You can divide to find equal parts of a whole. Remember that 10 = 10! B. Find 89.5 ÷ 10. Place value is based on 10. The value of each place is $$\frac{1}{10}$$ the value of the place to the left. Dividing by 10 results in moving each digit one place to the right. This looks the same as moving the decimal point one place to the left. 89.5 ÷ 101 = 8.95 Each cloth strip will be 8.95 cm long. Convince Me! Use Structure Celinda thought of 89.5 in parts, 80 + 9 + 0.5, and divided each part: 80 ÷ 10 = 8; 9 ÷ 10 = $$\frac{9}{10}$$ or 0.9; 0.5 ÷ 10 = 0.05. Then she added the parts to get 8.95. What do you notice? Answer: Celinda divided 89.5 into parts as 80 + 9 + 0.5, and divided each part 80 ÷ 10 = 8; 9 ÷ 10 = 0.9; 0.5 ÷ 10 = 0.05. Then she added the parts to get 8.95. I notice when we divide the number 89.5 with 10 then the result is 8.95 and dividing 89.5 into parts also we get same result as 8.95. Guided Practice999 Do You Understand? Question 1. Suppose Shondra wanted to cut the cloth into 102 strips. How long would each strip be? Answer: 89.5 ÷ 102 89.5 ÷ 100 = 0.895 Each cloth strip will be 0.895 cm long. Explanation: Shondra wanted to cut the cloth into 102 strips. Place value is based on 102. The value of each place is 1/100 the value of the place to the left. Dividing by 102 results in moving each digit two places to the right. This looks the same as moving the decimal point two places to the left. Each cloth strip will be 0.895 cm long. Question 2. Krista divides a number by 10. Then she divides the same number by 50. Which quotient is greater? How can you tell? Answer: The quotient 10 is greater. Explanation: In the above image we can observe the quotient 10 and 2. Krista divides the number 100 by 10 then the quotient is 10. Then she divides the same number 100 by 50 then the quotient is 2. The quotient 10 is greater. Do You Know How? In 3-10, use mental math to find each quotient. Question 3. 370.2 ÷ 102 Answer: 370.2 ÷ 102 370.2 ÷ 100 = 3.702 Explanation: In the above division method the place value is based on 102. The value of each place is 1/100 the value of the place to the left. Dividing the number 370.2 by 102 results in moving each digit two places to the right. This looks the same as moving the decimal point two places to the left. The quotient for 370.2 ÷ 100 = 3.702. Question 4. 126.4 ÷ 101 Answer: 126.4 ÷ 101 126.4 ÷ 10 = 12.64 Explanation: In the above division method the place value is based on 101. The value of each place is 1/10 the value of the place to the left. Dividing the number 126.4 by 101 results in moving each digit one place to the right. This looks the same as moving the decimal point one place to the left. The quotient for 126.4 ÷ 10 = 12.64. Question 5. 7.25 ÷ 10 Answer: 7.25 ÷ 10 = 0.725 Explanation: In the above division method the place value is based on 10. The value of each place is 1/10 the value of the place to the left. Dividing the number 7.25 by 10 results in moving each digit one place to the right. This looks the same as moving the decimal point one place to the left. The quotient for 7.25 ÷ 10 = 0.725. Question 6. 72.5 ÷ 103 Answer: 72.5 ÷ 103 72.5 ÷ 1000 = 0.0725 Explanation: In the above division method the place value is based on 103. The value of each place is 1/1000 the value of the place to the left. Dividing the number 72.5 by 103 results in moving each digit three places to the right. This looks the same as moving the decimal point three places to the left. The quotient for 72.5 ÷ 1000 = 0.0725. Question 7. 281.4 ÷ 100 Answer: 281.4 ÷ 100 281.4 ÷ 1 = 281.4 Explanation: In the above division method the place value is based on 100. The value of each place is 1 the value of the place to the left. Dividing the number 281.4 by 1 results in moving each digit zero places to the right. This looks the same as moving the decimal point zero places to the left. The quotient for 281.4 ÷ 1 = 281.4. Question 8. 2,810 ÷ 104 Answer: 2,810 ÷ 104 2,810 ÷ 10000 = 0.2810 Explanation: In the above division method the place value is based on 104. The value of each place is 1/10000 the value of the place to the left. Dividing the number 2,810 by 104 results in moving each digit four places to the right. This looks the same as moving the decimal point four places to the left. The quotient for 2,810 ÷ 10000 = 0.2810. Question 9. 3,642.4 ÷ 102 Answer: 3,642.4 ÷ 102 3,642.4 ÷ 100 = 36.424 Explanation: In the above division method the place value is based on 102. The value of each place is 1/100 the value of the place to the left. Dividing the number 3,642.4 by 102 results in moving each digit two places to the right. This looks the same as moving the decimal point two places to the left. The quotient for 3,642.4 ÷ 100 = 36.424. Question 10. 364.24 ÷ 101 Answer: 364.24 ÷ 101 364.24 ÷ 10 = 36.424 Explanation: In the above division method the place value is based on 101. The value of each place is 1/10 the value of the place to the left. Dividing the number 364.24 by 101 results in moving each digit one place to the right. This looks the same as moving the decimal point one place to the left. The quotient for 364.24 ÷ 10 = 36.424. Independent Practice Leveled Practice in 11-25, find each quotient. Use mental math. Question 11. 4,600 ÷ 10 460 ÷ 10 46 ÷ 10 4.6 ÷ 10 Answer: 4,600 ÷ 10 = 460 460 ÷ 10 = 46 46 ÷ 10 = 4.6 4.6 ÷ 10 = 0.46 Explanation: In the above division method the place value is based on 10. The value of each place is 1/10 the value of the place to the left. Dividing the number 4,600 by 10 results in moving each digit one place to the right. This looks the same as moving the decimal point one place to the left. The quotient for 4,600 ÷ 10 = 460. Dividing the number 460 by 10 results in moving each digit one place to the right. This looks the same as moving the decimal point one place to the left. The quotient for 460 ÷ 10 = 46. Dividing the number 46 by 10 results in moving each digit one place to the right. This looks the same as moving the decimal point one place to the left. The quotient for 46 ÷ 10 = 4.6. Dividing the number 4.6 by 10 results in moving each digit one place to the right. This looks the same as moving the decimal point one place to the left. The quotient for 4.6 ÷ 10 = 0.46. Question 12. 134.4 ÷ 103 134.4 ÷ 102 134.4 ÷ 101 134.4 ÷ 100 Answer: 134.4 ÷ 10 134.4 ÷ 1000 = 0.1344 134.4 ÷ 102 134.4 ÷ 100 = 1.344 134.4 ÷ 101 134.4 ÷ 10 = 13.44 134.4 ÷ 100 134.4 ÷ 1 = 134.4 Explanation: Dividing the number 134.4 by 103 results in moving each digit three places to the right. This looks the same as moving the decimal point three places to the left. The quotient for 134.4 ÷ 1000 = 0.1344. Dividing the number 134.4 by 102 results in moving each digit two places to the right. This looks the same as moving the decimal point two places to the left. The quotient for 134.4 ÷ 100 = 1.344. Dividing the number 134.4 by 101 results in moving each digit one two place to the right. This looks the same as moving the decimal point one place to the left. The quotient for 134.4 ÷ 10 = 13.44. Dividing the number 134.4 by 100 results in moving each digit zero places to the right. This looks the same as moving the decimal point zero places to the left. The quotient for 134.4 ÷ 1 = 134.4. Question 13. 98.6 ÷ 1 98.6 ÷ 100 98.6 ÷ 10 98.6 ÷ 1,000 Answer: 98.6 ÷ 1 98.6 ÷ 1 = 98.6 98.6 ÷ 100 98.6 ÷ 100 = 0.986 98.6 ÷ 10 98.6 ÷ 10 = 9.86 98.6 ÷ 1,000 98.6 ÷ 1,000 = 0.0986 Explanation: Dividing the number 98.6 by 1 results the quotient for 98.6 ÷ 1 = 98.6. Dividing the number 98.6 by 100 results in moving each digit two places to the right. This looks the same as moving the decimal point two places to the left. The quotient for 98.6 ÷ 100 = 0.986. Dividing the number 98.6 by 10 results in moving each digit one place to the right. This looks the same as moving the decimal point one place to the left. The quotient for 98.6 ÷ 10 = 9.86. Dividing the number 98.6 by 1000 results in moving each digit three places to the right. This looks the same as moving the decimal point three places to the left. The quotient for 98.6 ÷ 1,000 = 0.0986. Question 14. 136.5 ÷ 10 Answer: 136.5 ÷ 10 = 13.65 Explanation: In the above division method the place value is based on 10. The value of each place is 1/10 the value of the place to the left. Dividing the number 136.5 by 10 results in moving each digit one place to the right. This looks the same as moving the decimal point one place to the left. The quotient for 136.5 ÷ 10 = 13.65. Question 15. 753 ÷ 100 Answer: 753 ÷ 100 = 7.53 Explanation: In the above division method the place value is based on 100. The value of each place is 1/100 the value of the place to the left. Dividing the number 753 by 100 results in moving each digit two place to the right. This looks the same as moving the decimal point two places to the left. The quotient for 753 ÷ 100 = 7.53. Question 16. 890.1 ÷ 100 Answer: 890.1 ÷ 1 = 890.1 Explanation: In the above division method the place value is based on 100. Dividing the number 890.1 by 1 results in moving each digit zero places to the right. This looks the same as moving the decimal point zero places to the left. The quotient for 890.1 ÷ 1 = 890.1. Question 17. 3.71 ÷ 102 Answer: 3.71 ÷ 100 = 0.0371 Explanation: In the above division method the place value is based on 102. The value of each place is 1/100 the value of the place to the left. Dividing the number 3.71 by 102results in moving each digit two places to the right. This looks the same as moving the decimal point two places to the left. The quotient for 3.71 ÷ 100 = 0.0371. Question 18. 8,100 ÷ 104 Answer: 8,100 ÷ 10000 = 0.81 Explanation: In the above division method the place value is based on 104. The value of each place is 1/10000 the value of the place to the left. Dividing the number 8100 by 104 results in moving each digit four places to the right. This looks the same as moving the decimal point four places to the left. The quotient for 8,100 ÷ 10000 = 0.81. Question 19. 864 ÷ 103 Answer: 864 ÷ 1000 = 0.864 Explanation: In the above division method the place value is based on 103. The value of each place is 1/1000 the value of the place to the left. Dividing the number 864 by 103 results in moving each digit three places to the right. This looks the same as moving the decimal point three places to the left. The quotient for 864 ÷ 1000 = 0.864. Question 20. 0.52 ÷ 101 Answer: 0.52 ÷ 10 = 0.052 Explanation: In the above division method the place value is based on 101. The value of each place is 1/10 the value of the place to the left. Dividing the number 0.52 by 101 results in moving each digit one place to the right. This looks the same as moving the decimal point one place to the left. The quotient for 0.52 ÷ 10 = 0.052. Question 21. 15.7 ÷ 1,000 Answer: 15.7 ÷ 1,000 = 0.157 Explanation: In the above division method the place value is based on 1000. Dividing the number 15.7 by 1000 results in moving each digit three places to the right. This looks the same as moving the decimal point three places to the left. The quotient for 15.7 ÷ 1,000 = 0.157. Question 22. 7,700 ÷ 102 Answer: 7,700 ÷ 100 = 77 Explanation: In the above division method the place value is based on 100. Dividing the number 7,700 by 100 results in moving each digit two places to the right. This looks the same as moving the decimal point two places to the left. The quotient for 7,700 ÷ 100 = 77. Question 23. 770 ÷ 102 Answer: 770 ÷ 100 = 7.7 Explanation: In the above division method the place value is based on 102. The value of each place is 1/100 the value of the place to the left. Dividing the number 770 by 100 results in moving each digit two places to the right. This looks the same as moving the decimal point two places to the left. The quotient for 770 ÷ 100 = 7.7. Question 24. 77 ÷ 101 Answer: 77 ÷ 10 = 7.7 Explanation: In the above division method the place value is based on 101. The value of each place is 1/10 the value of the place to the left. Dividing the number 77 by 10 results in moving each digit one place to the right. This looks the same as moving the decimal point one place to the left. The quotient for 77 ÷ 10 = 7.7. Question 25. 7.7 ÷ 101 Answer: 7.7 ÷ 10 = 0.77 Explanation: In the above division method the place value is based on 101. The value of each place is 1/10 the value of the place to the left. Dividing the number 7.7 by 10 results in moving each digit one place to the right. This looks the same as moving the decimal point one place to the left. The quotient for 7.7 ÷ 10 = 0.77. Problem Solving For 26-28, use the table that shows the winning times at the Pacific Middle School swim meet. Question 26. What was the difference between the winning butterfly time and the winning backstroke time? Answer: The difference between the winning butterfly time and the winning backstroke time is 4.66 seconds. Explanation: The winning butterfly time is 58.49 seconds. The winning backstroke time is 53.83 seconds. Subtract the winning backstroke time from winning butterfly time then the difference is 4.66 seconds. Question 27. The winning time for the 100-yard freestyle was twice the time for the 50-yard freestyle. What was the winning time for the 100-yard freestyle? Answer: The winning time for the 100-yard freestyle is 44.34 seconds. Explanation: The winning time for the 100-yard freestyle was twice the time for the 50-yard freestyle. The winning time for 50-yard freestyle is 22.17 seconds. So add 22.17 seconds with 22.17 seconds then the sum is 44.34 seconds. The winning time for the 100-yard freestyle is 44. 34 seconds. Question 28. What was the difference between the winning 100-yard freestyle time and the winning butterfly time? Answer: The difference between the winning 100-yard freestyle time and the winning butterfly time is 14.15 seconds. Explanation: The winning 100-yard freestyle time is 44.34 seconds. The winning 100-yard butterfly time is 58.49 seconds. Subtract 100-yard freestyle winning time from 100-yard butterfly winning time then the difference is 14.15 seconds. Question 29. Reasoning A pickup truck carrying 103 identical bricks weighs 6,755 pounds. If the empty truck weighs 6,240 pounds, what is the weight of each brick? Explain how to solve the problem. Answer: Question 30. Higher Order Thinking Katie noticed a pattern in the answers for each of the expressions below. What do you notice? Answer: 14.6 x 0.1 = 1.46 14.6 ÷ 10 = 1.46 146 x 0.01 = 1.46 146 ÷ 100 = 1.46 146 x 0.001 = 0.146 146 ÷ 1,000 = 0.146 Explanation: In first expressions I noticed that if we are multiplying 14.6 x 0.1 results the product as 1.46. When we are dividing 14.6 ÷ 10 results the quotient as 1.46. In second expressions I noticed that if we are multiplying 146 x 0.01 results the product as 1.46. When we are dividing 146 ÷ 100 results the quotient as 1.46. In third expressions I noticed that if we are multiplying 146 x 0.001 results the product as 0.146. When we are dividing 146 ÷ 1,000 results the quotient as 0.146. Assessment Practice Question 31. Choose the equations in which n = 1,000 makes the equation true. 0 2.5 ÷ n = 0.025 947.5 ÷ n = 0.9475 8,350 ÷ n = 8.35 16.4 ÷ n = 0.0164 0.57 ÷ n = 0.0057 Answer: Explanation: If we put n = 1,000 in first equation 0.25 ÷ 1,000 = 0.0025. The First equation is not true. If we put n = 1,000 in second equation 947.5 ÷ 1,000 = 0.9475. The Second equation is true. If we put n = 1,000 in third equation 8,350 ÷ 1,000 = 8.35. The third equation is true. If we put n = 1,000 in fourth equation 16.4 ÷ 1,000 = 0.0164. The fourth equation is true. If we put n = 1,000 in fifth equation 0.57 ÷ 1,000 = 0.00057. The fifth equation is not true. Question 32. Choose the equations in which d = 102 makes the equation true. 386.2 ÷ d = 3.862 4,963.6 ÷ d = 4.9636 0.6 ÷ d = 0.006 5.8 ÷ d = 0.58 15.3 ÷ d = 0.153 Answer: Explanation: If we put d = 100 in first equation 386.2 ÷ 100 = 3.862. The First equation is true. If we put d = 100 in second equation 4,963.6 ÷ 100 = 49.636. The Second equation is not true. If we put d = 100 in third equation 0.6 ÷ 100 = 0.006. The third equation is true. If we put d = 100 in fourth equation 5.8 ÷ 100 = 0.058. The fourth equation is not true. If we put d = 100 in fifth equation 15.3 ÷ 100 = 0.153. The fifth equation is true. ### Lesson 6.2 Estimate Decimal Quotients Activity Slove&Share A 135.8-foot piece of construction material needs to be cut into pieces that are each 16 feet long. About how many pieces can be cut? Solve this problem any way you choose. Answer: 135 ÷ 15 = 9 The construction material can be cut into 9 pieces. Explanation: A 135.8-foot piece of construction material is about 135. We need to cut the material into pieces that are each 16 feet long. The 16 pieces material is rounded to 15. Apply division method to solve the problem. Divide 135 with 15 then the result is 9. The construction material can be cut into 9 pieces. You can use reasoning to estimate decimal quotients. Look Back! Reasoning Can you find a different way to estimate the answer for the problem above? Explain. Visual Learning Bridge Essential Question How Can You Use Estimation to Find Quotients? A. Diego borrowed money from his parents to purchase a video gaming system for$473.89 (including tax). About how much are his monthly payments to his parents if he wants to pay this off in one year?

You can use division to find equal groups.

B.
One Way
Estimate $473.89 ÷ 12. Use rounding. Round to the nearest ten: 473.89 rounds to 470; 12 rounds to 10$473.89 ÷ 12 is about $470 ÷ 10 =$47.
Each monthly payment will be about $47. C. Another Way Estimate$473.89 ÷ 12. Use compatible numbers.
Look for compatible numbers.
$473.89 ÷ 12 is close to$480 ÷ 12 = $40. You know 48 ÷ 12 = 4. Each monthly payment will be about$40.

Convince Me! Construct Arguments in the example above, which estimate is closer to the exact answer? Tell how you decided.
We use compatible numbers to make the problem easier to solve in our head by rounding each number to the nearest ten, twenty, fifty or hundred. But if we make the numbers compatible and round up to the nearest hundred or ten spot, 300 and 350 are much easier to compute in our heads.

Guided Practice

Do You Understand?

Question 1.
Number Sense Leo is estimating 53.1 ÷ 8.4. Do you think he should use 53 ÷ 8 or 54 ÷ 9 to estimate? Why?
Estimate 53.1 ÷ 8.4. Use compatible numbers.
Look for compatible numbers.
53.1 ÷ 8.4 is close to 54 ÷ 9 = 6.
53 ÷ 8 is not easy to find and 54 ÷ 9 is easy to find because 54 is multiple of 9.
Explanation:
Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. In the above division problem 53.1 ÷ 8.4 is closes to 54 ÷ 9. The actual dividend 53.1 is compatible to 54. Perform division operation 54 ÷ 9 = 6. The Estimated quotient is 6. Here 54 is multiple of 9 so we can easily find out the quotient. 53 ÷ 8 = 6.625 so it is difficult to find.

Question 2.
Is each quotient greater than or less than 1?
A. 0.2 ÷ 4
B. 1.35 ÷ 0.6
A. 0.2 ÷ 4 = 0.05
0.05 < 1
B. 1.35 ÷ 0.6 = 2.25
2.25 > 1
Explanation:
A. Divide 0.2 by 4 then the quotient is 0.05. 0.05 is less than 1.
B. Divide 1.35 by 0.6 then the quotient is 2.25. 2.25 is greater than 1.

How do you know?

In 3-8, estimate each quotient. Use rounding or compatible numbers.

Question 3.
42 ÷ 6.8
Estimate 42 ÷ 6.8. Use compatible numbers.
Look for compatible numbers.
42 ÷ 6.8 is closes to 42 ÷ 7 = 6.
The Estimated quotient is 6.
Explanation:
Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. In the above division problem 42 ÷ 6.8 is closes to 42 ÷ 7. The actual divisor 6.8 is compatible to 7. Perform division operation 42 ÷ 7 = 6. The Estimated quotient is 6.

Question 4.
102 ÷ 9.6
Estimate 102 ÷ 9.6. Use rounding.
Round to the nearest ten: 102 rounds to 100; 9.6 rounds to 10
102 ÷ 9.6 is about 100 ÷ 10 = 10.
Explanation:
Rounding means replacing a number with an approximate value. In the above division method 102 ÷ 9.6. Round the numbers to the nearest ten or hundreds. Here 102 is rounded to 100 and 9.6 is rounded to 10. Now perform division operation on 100 ÷ 10 = 10. The estimated quotient is 10.

Question 5.
48.9 ÷ 4
Estimate 48.9 ÷ 4. Use compatible numbers.
Look for compatible numbers.
48.9 ÷ 4 is closes to 50 ÷ 5 = 10.
The Estimated quotient is 10.
Explanation:
Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. In the above division problem 48.9 ÷ 4 is closes to 50 ÷ 5. The actual dividend is 48.9 is compatible to 50. The actual divisor 4 is compatible to 5. Perform division operation 50 ÷ 5 =10. The Estimated quotient is 10.

Question 6.
72.59 ÷ 7
Estimate 72.59 ÷ 7. Use rounding.
Round to the nearest ten: 72.59 rounds to 70; 7 rounds to 10
72.59 ÷ 7 is about 70 ÷ 10 = 7.
Explanation:
Rounding means replacing a number with an approximate value. In the above division method 72.59 ÷ 7. Round the numbers to the nearest ten or hundreds. Here 72.59 is rounded to 70 and 7 is rounded to 10. Now perform division operation on 70 ÷ 10 = 7. The estimated quotient is 7.

Question 7.
15.4 ÷ 1.9
Estimate 15.4 ÷ 1.9. Use compatible numbers.
Look for compatible numbers.
15.4 ÷ 1.9 is closes to 16 ÷ 2 = 8.
The Estimated quotient is 8.
Explanation:
Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. In the above division problem 15.4 ÷ 1.9 is closes to 16 ÷ 2. The actual dividend is 15.4 is compatible to 16. The actual divisor 1.9 is compatible to 2. Perform division operation 16 ÷ 2 =8. The Estimated quotient is 8.

Question 8.
44.07 ÷ 6.3
Estimate 44.07 ÷ 6.3. Use compatible numbers.
Look for compatible numbers.
44.07 ÷ 6.3 is closes to 42 ÷ 6 = 7.
The Estimated quotient is 7.
Explanation:
Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. In the above division problem 44.07 ÷ 6.3 is closes to 42 ÷ 6. The actual dividend is 44.07 is compatible to 42. The actual divisor 6.3 is compatible to 6. Perform division operation 42 ÷ 6 =7. The Estimated quotient is 7.

Independent Practice

Leveled Practice In 9 and 10, complete the work to estimate each quotient.

Question 9.

Estimate 64.5 ÷ 12.3. Use rounding.
Round to the nearest ten: 64.5 rounds to 65 ; 12.3 rounds to 10
64.5 ÷ 12.3 is about 65 ÷ 10 = 6.5.
Explanation:
Rounding means replacing a number with an approximate value. In the above division method 64.5 ÷ 12.3. Round the numbers to the nearest ten or hundreds. Here 64.5 is rounded to 65 and 12.3 is rounded to 10. Now perform division operation on 65 ÷ 10 = 6.5. The estimated quotient is 6.5.

Question 10.
Estimate 64.5 ÷ 12.3 using compatible numbers.
65 ÷ 10 = _____
Estimate 64.5 ÷ 12.3. Use compatible numbers.
Look for compatible numbers.
64.5 ÷ 12.3 is closes to 65 ÷ 10 = 6.5.
The Estimated quotient is 6.5.
Explanation:
Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. In the above division problem 64.5 ÷ 12.3 is closes to 65 ÷ 10. The actual dividend is 64.5 is compatible to 65. The actual divisor 12.3 is compatible to 10. Perform division operation 65 ÷ 10 =6.5. The Estimated quotient is 6.5.

In 11-19, estimate each quotient.

Question 11.
7 ÷ 0.85
Estimate 7 ÷ 0.85. Use compatible numbers.
Look for compatible numbers.
7 ÷ 0.85 is closes to 7 ÷ 1 = 7.
The Estimated quotient is 7.
Explanation:
Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. In the above division problem 7 ÷ 0.85 is closes to 7 ÷ 1. The actual divisor 0.85 is compatible to 1. Perform division operation 7 ÷ 1 = 7. The Estimated quotient is 7.

Question 12.
9.6 ÷ 0.91
Estimate 9.6 ÷ 0.91. Use compatible numbers.
Look for compatible numbers.
9.6 ÷ 0.91 is closes to 10 ÷ 1 = 10.
The Estimated quotient is 10.
Explanation:
Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. In the above division problem 9.6 ÷ 0.91 is closes to 10 ÷ 1. The actual dividend 9.6 is compatible to 10.The actual divisor 0.91 is compatible to 1. Perform division operation 10 ÷ 1 = 10. The Estimated quotient is 10.

Question 13.
17.7 ÷ 3.2
Estimate 17.7 ÷ 3.2. Use compatible numbers.
Look for compatible numbers.
17.7 ÷ 3.2 is closes to 18 ÷ 3 = 6.
The Estimated quotient is 6.
Explanation:
Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. In the above division problem 17.7 ÷ 3.2 is closes to 18 ÷ 3. The actual dividend 17.7 is compatible to 18.The actual divisor 3.2 is compatible to 3. Perform division operation 18 ÷ 3 = 6. The Estimated quotient is 6.

Question 14.
91.02 ÷ 4.9
Estimate 91.02 ÷ 4.9. Use compatible numbers.
Look for compatible numbers.
91.02 ÷ 4.9 is closes to 90 ÷ 5 = 18.
The Estimated quotient is 18.
Explanation:
Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. In the above division problem 91.02 ÷ 4.9 is closes to 90 ÷ 5. The actual dividend 91.02 is compatible to 90.The actual divisor 4.9 is compatible to 5. Perform division operation 90 ÷ 5 = 18. The Estimated quotient is 18.

Question 15.
45.64 ÷ 6.87
Estimate 45.64 ÷ 6.87. Use compatible numbers.
Look for compatible numbers.
45.64 ÷ 6.87 is closes to 49 ÷ 7 = 7.
The Estimated quotient is 7.
Explanation:
Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. In the above division problem 45.64 ÷ 6.87 is closes to 49 ÷ 7. The actual dividend 45.64 is compatible to 49.The actual divisor 6.87 is compatible to 7. Perform division operation 49 ÷ 7 = 7. The Estimated quotient is 7.

Question 16.
821.22 ÷ 79.4
Estimate 821.22 ÷ 79.4. Use rounding.
Round to the nearest ten: 821.22 rounds to 800 ; 79.4 rounds to 80
821.22 ÷ 79.4 is about 800 ÷ 80 = 10.
Explanation:
Rounding means replacing a number with an approximate value. In the above division method 821.22 ÷ 79.4. Round the numbers to the nearest ten or hundreds. Here 821.22 is rounded to 800 and 79.4 is rounded to 80. Now perform division operation on 800 ÷ 80 = 10. The estimated quotient is 10.

Question 17.
22.5 ÷ 3
Estimate 22.5 ÷ 3. Use compatible numbers.
Look for compatible numbers.
22.5 ÷ 3 is closes to 24 ÷ 3 = 8.
The Estimated quotient is 8.
Explanation:
Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. In the above division problem 22.5 ÷ 3 is closes to 24 ÷ 3. The actual dividend 22.5 is compatible to 24. Perform division operation 24 ÷ 3 = 8. The Estimated quotient is 8.

Question 18.
15.66 ÷ 9.3
Estimate 15.66 ÷ 9.3. Use rounding.
Round to the nearest ten: 15.66 rounds to 20 ; 9.3 rounds to 10.
15.66 ÷ 9.3 is about 20 ÷ 10 = 2.
Explanation:
Rounding means replacing a number with an approximate value. In the above division method 15.66 ÷ 9.3. Round the numbers to the nearest ten or hundreds. Here 15.66 is rounded to 20 and 9.3 is rounded to 10. Now perform division operation on 20 ÷ 10 = 2. The estimated quotient is 2.

Question 19.
156.3 ÷ 14.5
Estimate 156.3 ÷ 14.5. Use rounding.
Round to the nearest ten: 156.3 rounds to 160 ; 14.5 rounds to 20.
156.3 ÷ 14.5 is about 160 ÷ 20 = 8.
Explanation:
Rounding means replacing a number with an approximate value. In the above division method 156.3 ÷ 14.5. Round the numbers to the nearest ten or hundreds. Here 156.3 is rounded to 160 and 14.5 is rounded to 20. Now perform division operation on 160 ÷ 20 = 8. The estimated quotient is 8.

Problem Solving

Question 20.
Luci’s mother gave her $7.50 to buy 8 spiral notebooks. With tax, the cost of each notebook is$1.05. Does Luci have enough money? Use compatible numbers and estimation to help you decide.

Question 21.
Critique Reasoning Kerri said that the quotient of 4.2 ÷ 5 is about 8 tenths because 4.2 ÷ 5 is close to 40 tenths ÷ 5. Do you agree with Kerri’s reasoning? Explain.

Question 22.
Higher Order Thinking Write a decimal division problem that has an estimated quotient of 4. Explain how to get that estimate.

Question 23.
Lia’s car averages 14.5 miles per gallon while Roman’s car averages 28.5 miles per gallon. Use estimation to find how many times as many miles per gallon Roman’s car gets compared to Lia’s car.

In 24-26, use the table.

Question 24.
enVision® STEM Which sample from the experiment had the least mass? Which had the lowest temperature?
In the above table we can observe Sample 3 had the least mass 0.058 g from the experiment. Sample 1 had the lowest temperature as 37.57°C.

Question 25.
Sample 3 was used in another experiment. A temperature of 82.14°C was recorded. How many degrees did the temperature change?
In the above table we can observe sample 3 had the temperature 75.50°C. Sample 3 was used in another experiment. A temperature of 82.14°C was recorded. The difference of these two temperatures are 6.64°C

Question 26.
What is the difference in mass between Sample 1 and Sample 2?
Sample 1 has the mass 0.98 g.
Sample 2 has the mass 0.58 g.
The difference in mass between Sample 1 and Sample 2 is 0.4.

Assessment Practice

Question 27.
Mauricio scored a total of 34.42 points in five gymnastic events. Which equation shows the best way to estimate Mauricio’s score for each event?
A. 35 ÷ 5 = 7
B. 35 ÷ 7 = 5
C. 30 ÷ 10 = 3
D. 40 ÷ 10 = 4
Option A 35 ÷ 5 = 7 is correct.
35 ÷ 5 = 7 is the best way to estimate Mauricio’s score for each event.
Explanation:
Mauricio scored a total of 34.42 points in five gymnastic events. Use compatible method. In the above options option A is correct. The equation 34.42 ÷ 5 is closes to 35 ÷ 5. The actual dividend 34.42 is compatible to 35. Perform division operation 35 ÷ 5 = 7. The Estimated score for each event is 8.

Question 28.
Terry paid $117.50 for 18 identical flash drives. Which is the best estimate for the cost of each flash drive? A.$6
B. $10 C.$12
D. $60 Answer: Option A$6 is correct.
$117.50 ÷ 18$120 ÷ 20 = $6. The estimated cost of each flash drive is 6. Explanation: Terry paid$117.50 for 18 identical flash drives. Use rounding method. Rounding means replacing a number with an approximate value. In the above division method $117.50 ÷ 18. Round the numbers to the nearest ten or hundreds. Here$117.50 is rounded to $120 and 18 is rounded to 20. Now perform division operation on$120 ÷ 20 = $6. The estimated cost of each flash drive is$6.

### Lesson 6.3 Use Models to Divide by a 1-Digit Whole Number

Activity

Solve&Share
Chris paid $3.60 for 3 colored pens. Each pen costs the same amount. How much did each pen cost? Solve this problem any way you choose. You can use appropriate tools such as drawings, money, or place-value blocks to help you divide. Show your work! Answer: Each pen cost is$1.20.
Explanation:
Chris paid $3.60 for 3 colored pens. Each pen costs the same amount. Divide$3.60 with 3 colored pens then the quotient is $1.20. Each pen cost is$1.20.

Look Back! Without dividing, how do you know that the answer to the problem above must be greater than 1?
The answer to the above problem must be greater than 1. Because each pen costs the same amount. we can clearly see that Chris paid $3.60 for 3 colored pens. So we can say without dividing method the answer to the above problem is greater than 1 Visual Learning Bridge Essential Question How Can You Use Models to Find a Decimal Quotient? A. Three friends received$2.58 for aluminum cans they recycled. They decided to share the money equally. How much will each friend get?

You can use place value blocks. Let a 100 square = $1.00, a tenth bar =$0.10, and a hundredth cube = $0.01. Find 2.58 ÷ 3. B. There are not enough ones to put 1 in each group, so regroup the 2 ones into 20 tenths. You can see that there are 25 tenths in 2.58. Divide the 25 tenths into 3 equal groups. C. Trade the one extra tenth for 10 hundredths to get 18 hundredths. Divide the 18 hundredths into 3 equal groups. Each group gets 6 hundredths. Each of the 3 friends will get$0.86.

Convince Me! Reasoning The next week, 4 friends got $8.24 for the cans they collected. How much money will each friend get? Estimate using compatible numbers and then use a strategy to find the answer. Guided Practice Do You Understand? Question 1. What is a reasonable estimate for 8.24 ÷ 4? Explain. Answer: Estimate$8.24 ÷ 4. Use compatible numbers.
Look for compatible numbers.
$8.24 ÷ 4 is closes to$8 ÷ 4 = $2. The Estimated quotient is$2.
Each friend get $2. Explanation: Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. The next week, 4 friends got$8.24 for the cans they collected. In the above division problem $8.24 ÷ 4 is closes to 8 ÷ 4. The actual dividend 8.24 is compatible to 8. Perform division operation 8 ÷ 4 = 2. The Estimated quotient is 2. Each friend get$2.

Question 2.
How is dividing a decimal by a whole number similar to dividing a whole number by a whole number? Explain.

Do You Know How?

Question 3.
Use models to help you divide 2.16 ÷ 4. Complete the division calculation.

2.16 ÷ 4 = 0.54.
Explanation:
In the above image we can observe the division of 2.16 ÷ 4 = 0.54. There are not enough ones to put 1 in each group, so regroup the 2 ones into 20 tenths. Divide 20 tenths into 4 equal groups. Each group gets 5 tenths. Four groups of 0.5 = 2.0.
Trade the one extra tenth for 10 hundredths to get 16 hundredths. Divide the 16 hundredths into 4 equal groups. Each group gets 4 hundredths. Four groups of 0.04 = 0.16.

Independent Practice

Leveled Practice In 4-9, divide. Use or draw models to help.

Question 4.

1.35 ÷ 3 = 0.45
Explanation:
In the above image we can observe the division of 1.35 ÷ 3 = 0.45. There are not enough ones to put 1 in each group, so regroup the 1 ones into 10 tenths. We can see that there are 13 tenths in 1.35. Divide the 13 tenths into 3 equal groups. Each group gets 4 tenths. Three groups of 0.4 = 1.2.
Trade the one extra tenth for 10 hundredths to get 15 hundredths. Divide the 15 hundredths into 3 equal groups. Each group gets 5 hundredths. Three groups of 0.05 = 0.15.

Question 5.

5.72 ÷ 4 = 1.43
Explanation:
In the above image we can observe the division of 5.72 ÷ 4 = 1.43. There are enough ones to put 1 in each group, and extra ones is regrouped. Divide the 5 ones into 4 equal groups. Each group gets 1 ones. Four groups of 1 = 4. Regroup of the 1 ones into 10 tenths. We can see that there are 17 tenths in 5.72. Divide the 17 tenths into 4 equal groups. Each group gets 4 tenths. Four groups of 0.4 = 1.6.
Trade the one extra tenth for 10 hundredths to get 12 hundredths. Divide the 12 hundredths into 4 equal groups. Each group gets 3 hundredths. Four groups of 0.03 = 0.12.

Question 6.
2.38 ÷ 7

2.38 ÷ 7 = 0.34.
Explanation:
In the above image we can observe the division of 2.38 ÷ 7 = 0.34. There are not enough ones to put 1 in each group, so regroup the 2 ones into 20 tenths. We can see that there are 23 tenths in 2.38. Divide the 23 tenths into 7 equal groups. Each group gets 3 tenths. Seven groups of 0.3 = 2.1.
Trade the two extra tenth for 20 hundredths to get 28 hundredths. Divide the 28 hundredths into 7 equal groups. Each group gets 4 hundredths. Seven groups of 0.04 = 0.28.

Question 7.
4.71 ÷ 3

4.71 ÷ 3 = 1.57.
Explanation:
In the above image we can observe the division of 4.71 ÷ 3 = 1.57. There are enough ones to put 1 in each group, and extra ones is regrouped. Divide the 3 ones into 3 equal groups. Each group gets 1 ones. Three groups of 1 = 3. Regroup of the 1 ones into 10 tenths. We can see that there are 17 tenths in 4.71. Divide the 17 tenths into 3 equal groups. Each group gets 5 tenths. Three groups of 0.5 = 1.5.
Trade the two extra tenth for 20 hundredths to get 21 hundredths. Divide the 21 hundredths into 3 equal groups. Each group gets 7 hundredths. Three groups of 0.07 = 0.21.

Question 8.
1.76 ÷ 8

1.76 ÷ 8 = 0.22.
Explanation:
In the above image we can observe the division of 1.76 ÷ 8 = 0.22. There are not enough ones to put 1 in each group, so regroup the 1 ones into 10 tenths. We can see that there are 17 tenths in 1.76. Divide the 17 tenths into 8 equal groups. Each group gets 2 tenths. Eight groups of 0.2 = 1.6.
Trade the one extra tenth for 10 hundredths to get 16 hundredths. Divide the 16 hundredths into 8 equal groups. Each group gets 2 hundredths. Eight groups of 0.02 = 0.16.

Question 9.
5.36 ÷ 2

5.36 ÷ 2 = 2.68.
Explanation:
In the above image we can observe the division of 5.36 ÷ 2 = 2.68. There are enough ones to put 2 in each group, and extra ones is regrouped. Divide the 5 ones into 2 equal groups. Each group gets 2 ones. Two groups of 2 = 4. Regroup of the 1 ones into 10 tenths. We can see that there are 13 tenths in 5.36. Divide the 13 tenths into 2 equal groups. Each group gets 6 tenths. Two groups of 0.6 = 1.2.
Trade the one extra tenth for 10 hundredths to get 16 hundredths. Divide the 16 hundredths into 2 equal groups. Each group gets 8 hundredths. Two groups of 0.08 = 0.16.

Problem Solving

Question 10.
Reasoning Alan is modeling 2.65 ÷ 5. How should he exchange the place-value blocks so he can make 5 equal shares?

2.65 ÷ 5 = 0.53
Explanation:
In the above image we can observe the division of 2.65 ÷ 5 = 0.53. There are not enough ones to put 1 in each group, so regroup the 2 ones into 20 tenths. We can see that there are 26 tenths in 2.65. Divide the 26 tenths into 5 equal groups. Each group gets 5 tenths. Five groups of 0.5 = 2.5.
Trade the one extra tenth for 10 hundredths to get 15 hundredths. Divide the 15 hundredths into 5 equal groups. Each group gets 3 hundredths. Five groups of 0.03 = 0.15.

Question 11.
Algebra Abby wants to know the value of n in the equation 7.913 × n = 791.3. What value for n makes the equation true?
7.913 × n = 791.3
7.913 × 100 = 791.3
If n= 100 makes the equation true.

Question 12.
To find 5.16 ÷ 6, should you divide the ones first or the tenths first? Why?

First we have to divide the ones first.
5.16 ÷ 6 = 0.86
Explanation:
In the above image we can observe the division of 5.16 ÷ 6 = 0.86. There are not enough ones to put 1 in each group, so regroup the 5 ones into 50 tenths. We can see that there are 51 tenths in 5.16. Divide the 51 tenths into 6 equal groups. Each group gets 8 tenths. Six groups of 0.8 = 4.8.
Trade the one extra tenth for 10 hundredths to get 36 hundredths. Divide the 36 hundredths into 6 equal groups. Each group gets 6 hundredths. Six groups of 0.06 = 0.36.

Question 13.
There are 264 children going on a field trip. Are 5 buses enough if each bus holds 52 children? Tell how you decided.
264 ÷ 5 = 52.8
5 buses are not enough to hold 52 children in each bus.
Explanation:
There are 264 children going on a field trip. Divide 264 by 5 then the quotient is 52.8. 5 buses are not enough to hold 52 children in each bus.

Question 14.
Higher Order Thinking Ginny earned $49.50 for 6 hours of gardening and$38.60 for 4 hours of babysitting. For which job did she earn more money per hour? How much more per hour did she earn? Explain how you found the answers.

Think about what information in the problem you need to compare.

Ginny earned $49.50 for 6 hours of gardening.$49.50 ÷ 6 = $8.25. Ginny earns$8.25 per hour.
Ginny earned $38.60 for 4 hours of babysitting.$38.60 ÷ 4 = $9.65. Ginny earns$9.65 per hour.
$9.65 –$8.25 = $1.4. She earns more$1.4 per hour.
Explanation:
Ginny earned $49.50 for 6 hours of gardening. Perform division operation on$49.50 ÷ 6 = $8.25. Ginny earns$8.25 per hour. Ginny earned $38.60 for 4 hours of babysitting. Perform division operation on$38.60 ÷ 4 = $9.65. Ginny earns$9.65 per hour. Now we have to calculate how much more money she is earning. Subtract
$9.65 –$8.25 = $1.4. She earns more$1.4 per hour.

Assessment Practice

Question 15.
Tia drew the model below for 1.35 ÷ 3.

Part A
1.35 ÷ 3 = 0.45.
Explanation:
In the above image we can observe the 4 groups. Tia made a mistake of drawing the model above. She has to draw 3 groups. She drew one extra group of 4 tenths and 5 hundredths.
Part B
Draw the correct model and find the quotient.

1.35 ÷ 3 = 0.45.
The quotient is 0.45.
Explanation:
In the above image we can observe the division of 1.35 ÷ 3 = 0.45. There are not enough ones to put 1 in each group, so regroup the 1 ones into 10 tenths. We can see that there are 13 tenths in 1.35. Divide the 13 tenths into 3 equal groups. Each group gets 4 tenths. Three groups of 0.4 = 1.2.
Trade the one extra tenth for 10 hundredths to get 15 hundredths. Divide the 15 hundredths into 3 equal groups. Each group gets 5 hundredths. Three groups of 0.05 = 0.15.

### Lesson 6.4 Divide by a 2-Digit Whole Number

Activity

Solve&Share

Stan has a rectangular piece of carpet with an area of 23.4 square meters. The piece of carpet is 13 meters long. What is the width of the piece of carpet? Solve this problem any way you choose.

Model with Math You can write an equation to model the problem.

Look Back! How could you estimate the width of the piece of carpet?
23.4 ÷ 13 = ?
23.4 ÷ 13 = 1.8
The width of the piece of carpet is 1.8 meters.
Explanation:
Stan has a rectangular piece of carpet with an area of 23.4 square meters. The piece of carpet is 13 meters long. Perform division operation 23.4 ÷ 13 = 1.8. The width of the piece of carpet is 1.8 meters.

Visual Learning Bridge

Essential Question How Do You Divide Decimals Question by 2-Digit Numbers?

A.
Erin’s garden has an area of 84.8 square feet. She knows the length is 16 feet. What is the width of Erin’s garden? How can you solve 84.8 ÷ 16 = w?

You can use what you know about dividing whole numbers to help.

B.
The total area is 84.8. The pieces of the model represent the areas for the partial quotients.

Convince Me! Reasoning How could Amy use estimation to make sure the decimal point is in the correct place in the quotient?

Guided Practice

Do You Understand?
In 1 and 2, use the example on the previous page.

Question 1.
Where is 5.3 shown in the diagram?

Question 2.
How can you check that the quotient 5.3 is reasonable? Explain.

Do You Know How?

In 3 and 4, complete the division problem.

Question 3.

306.25 ÷ 49 = 6.25
Explanation:
In the above image we can observe the division operation  306.25 ÷ 49 = 6.25. The nearest possible multiple value of 49 is 294(49 x 6). Subtract 294 from 306.25 so as to get the remainder 12.25. 49 cannot be a multiple of 12.25 and so we have kept a decimal in the quotient and now 49 has the least possible multiple value of 9.8(49 x 0.2) near to 12.25. After subtracting 9.8 from 12.25, we are now having 6.2 as quotient and 2.45 as remainder. Now the possible multiple value of 49 near to 2.45 is 2.45 itself(49 x 0.05 = 2.45). Now the remainder is zero and the final quotient is 6.25.

Question 4.

28.5 ÷ 15 = 1.9
Explanation:
In the above image we can observe the division operation  28.5 ÷ 15 = 1.9. The nearest possible multiple value of 15 is 15(15 x 1). Subtract 15 from 28.5 so as to get the remainder 13.5. 15 cannot be a multiple of 13.5 and so we have kept a decimal in the quotient and now the possible multiple value of 15 near to 13.5 is 13.5 itself(15 x 0.9 = 13.5). Now the remainder is zero and the final quotient is 1.9.

Independent Practice

Leveled Practice In 5-6, find each quotient and label the model.

Question 5.

78.2 ÷ 17 = 4.6
Explanation:
In the above image we can observe the division operation  78.2 ÷ 17 = 4.6. The nearest possible multiple value of 17 is 68(17 x 4). Subtract 68 from 78.2 so as to get the remainder 10.2. 17 cannot be a multiple of 10.2 and so we have kept a decimal in the quotient and now the possible multiple value of 17 near to 10.2 is 10.2 itself(17 x 0.6 = 10.2). Now the remainder is zero and the final quotient is 4.6.

Question 6.

304.75 ÷ 53 = 5.75
Explanation:
In the above image we can observe the division operation  304.75÷ 53 = 5.75. The nearest possible multiple value of 53 is 265(53 x 5). Subtract 265 from 304.75 so as to get he remainder 39.75. 53 cannot be a multiple of 39.75 and so we have kept a decimal in the quotient and now 53 has the least possible multiple value of 37.10(53 x 0.7) near to 39.75. After subtracting 37.10 from 39.75, we are now having 5.7 as quotient and 2.65 as remainder. Now the possible multiple value of 53 near to 2.65 is 2.65 itself(53 x 0.05 = 2.65). Now the remainder is zero and the final quotient is 5.75.

In 7-10, find each quotient.

Question 7.

91.8 ÷ 27 = 3.4
The quotient is 3.4
Explanation:
In the above image we can observe the division operation  91.8 ÷ 27 = 3.4. The nearest possible multiple value of 27 is 81(27 x 3). Subtract 81 from 91.8 so as to get the remainder as 10.8. 27 cannot be a multiple of 10.8 and so we have kept a decimal in the quotient and now the possible multiple value of 27 near to 10.8 is 10.8 itself(27 x 0.4 = 10.8). Now the remainder is zero and the final quotient is 3.4.

Question 8.

3.9 ÷ 15 = 0.26
The quotient is 0.26
Explanation:
In the above image we can observe the division operation  3.9 ÷ 15 = 0.26. 15 cannot be a multiple of 3.9 so we have kept a decimal in the quotient and now 15 has the least possible multiple value of 3.0(15 x 0.2) near to 3.9. Subtract 3.0 from 3.9 then the remainder is 0.9.  Now the possible multiple value of 15 near to 0.9 is 0.9 itself(15 x 0.06 = 0.9). Now the remainder is zero and the final quotient is 0.26.

Question 9.

39.6 ÷ 12 = 3.3
The quotient is 3.3
Explanation:
In the above image we can observe the division operation  39.6 ÷ 12 = 3.3. The nearest possible multiple value of 12 is 36(12 x 3). Subtract 36 from 39.6 so as to get the remainder as 3.6. 12 cannot be a multiple of 3.6 and so we have kept a decimal in the quotient and now the possible multiple value of 12 near to 3.6 is 3.6 itself(12 x 0.3 = 3.6). Now the remainder is zero and the final quotient is 3.3.

Question 10.

247.5 ÷ 50 = 4.95
Explanation:
In the above image we can observe the division operation  247.5 ÷ 50 = 4.95. The nearest possible multiple value of 50 is 200(50 x 4). Subtract 200 from 247.5 so as to get the remainder 47.5. 50 cannot be a multiple of 47.5 and so we have kept a decimal in the quotient and now 50 has the least possible multiple value of 45.0(50 x 0.9) near to 47.5. After subtracting 45.0 from 47.5, we are now having 4.9 as quotient and 2.5 as remainder. Now the possible multiple value of 50 near to 2.5 is 2.5 itself(50 x 0.05 = 2.5). Now the remainder is zero and the final quotient is 4.95.

Problem Solving

Question 11.
Sharon pays $98.75 for twenty-five 14-ounce boxes of Yummy Flakes cereal. How much does one box of cereal cost? Answer: Question 12. Javier bought a new TV for$479.76. He will make equal payments each month for 2 years. How can Javier use compatible numbers to estimate each payment?
Estimate $479.76 ÷ 24. Use compatible numbers. Look for compatible numbers.$479.76 ÷ 24 is closes to $480 ÷ 24 =$20.
The Estimated quotient is $20. Javier pays each month$20.
Explanation:
Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. Javier bought a new TV for $479.76. He will make equal payments each month for 2 years. 2years means 24 months. In the above division problem$479.76 ÷ 24 is closes to 480 ÷ 24. The actual dividend $479.76 is compatible to 480. Perform division operation 480 ÷ 24 = 20. The Estimated quotient is 20. Javier pays each month$20.

Question 13.
Higher Order Thinking The area of the rectangular flower bed shown is 20.4 square meters. How many meters of edging are needed to go around the flower bed? Explain.

Question 14.

### Topic 6 Fluency Practice

Activity

Solve each problem. Follow products that are multiples of 20 to shade a path from START to FINISH. You can only move up, down, right, or left.

In the above image we can observe some multiplication problems. Products that are multiples of 20 to shade a path from START to FINISH. The products that are shaded with green color are multiples of 20.

Topic 6 Vocabulary Review

Glossary

Understand Vocabulary

Write always, sometimes, or never.

Word List
• estimate
• exponent
• hundredths
• power
• quotient
• rounding
• tenths
• thousandths

Question 1.
A digit in the hundredths place has $$\frac{1}{10}$$ the value of the same digit in the tenths place. ______
A digit in the hundredths place has $$\frac{1}{10}$$ the value of the same digit in the tenths place. Sometimes.

Question 2.
The answer to a division problem is less than the divisor. ______
The answer to a division problem is less than the divisor. Sometimes

Question 3.
A whole number divided by a decimal number is a whole number. ____
A whole number divided by a decimal number is a whole number. Sometimes

Question 4.
Dividing by 103 moves the decimal point in the dividend three places to the left. ____
Dividing by 103 moves the decimal point in the dividend three places to the left. Always

Question 5.
Multiplying the dividend and the divisor by the same power of 10 changes the quotient. ____
Multiplying the dividend and the divisor by the same power of 10 changes the quotient. Never

Question 6.
The answer to a division problem is greater than the divisor. ______
The answer to a division problem is greater than the divisor. Sometimes

Write T for true or F for false.

_____ Question 7.
3.65 ÷ 5.2 < 1
3.65 ÷ 5.2 < 1
0.70 < 1
True
Explanation:
Perform division operation on 3.65 ÷ 5.2 = 0.70. The quotient is 0.70. The quotient 0.70 is less than 1. So the above expression is True.

Question 8.
48 ÷ 0.6 = 0.8
48 ÷ 0.6 = 0.8
48 ÷ 0.6 = 80
The answer for above division operation is not correct.  So the answer is False.
Explanation:
Perform division operation on 48 ÷ 0.6 = 80. The quotient is 80.  In the above division operation the quotient is 0.8. So the above expression is False.

_____ Question 9.
2.42 ÷ 2.1 > 1.
2.42 ÷ 2.1 > 1.
1.15 > 1
The answer for above division operation is 1.15 is greater than 1. So the answer is True.
Explanation:
Perform division operation on 2.42 ÷ 2.1 = 1.15. The quotient is 1.15. The quotient 1.15 is greater than 1. So the above expression is True.

_____ Question 10.
4.9 ÷ 0.8 < 4.9
4.9 ÷ 0.8 < 4.9
6.125 < 4.9
The answer for above division operation is 6.125 is not less than 4.9. So the answer is False.
Explanation:
Perform division operation on 4.9 ÷ 0.8 = 6.125. The quotient is 6.125. The quotient 6.125 is  not less than 4.9. So the above expression is False.

Use Vocabulary in Writing

Question 11.
Mary says the digits in the quotient of 381.109 0.86 are 4 4 315, but she doesn’t know where to place the decimal point. How can Mary use number sense to place the decimal point? Use at least three terms from the Word List in your answer.

### Topic 6 Reteaching

Set A
pages 229-232
Find 340.5 ÷ 100.
Dividing by 10, or 101, means moving the decimal point one place to the left.
Dividing by 100, or 102, means moving the decimal point two places to the left.
Dividing by 1,000, or 103, means moving the decimal point three places to the left.

Remember that when dividing decimals by a power of 10, you may need to use one or more zeros as placeholders.

Use mental math to find each quotient.

Question 1.
34.6 ÷ 101
34.6 ÷ 101
34.6 ÷ 10 = 3.46
The quotient is 3.46.
Explanation:
Dividing by 10, or 101, means moving the decimal point one place to the left. Move the dividend decimal point 34.6 one place to the left then the result is 3.46. The quotient is 3.46.

Question 2.
6,483 ÷ 102
6,483 ÷ 102
6,483 ÷ 100 = 64.83
The quotient is 64.83.
Explanation:
Dividing by 100, or 102, means moving the decimal point two places to the left. Move the dividend decimal point 6,483 two places to the left then the result is 64.83. The quotient is 64.83.

Question 3.
148.3 ÷ 100
148.3 ÷ 100 = 1.483
The quotient is 1.483.
Explanation:
Dividing by 100, or 102, means moving the decimal point two places to the left. Move the dividend decimal point 148.3 two places to the left then the result is 1.483. The quotient is 1.483.

Question 4.
29.9 ÷ 101
29.9 ÷ 101
29.9 ÷ 10 = 2.99
The quotient is 2.99
Explanation:
Dividing by 10, or 101, means moving the decimal point one place to the left. Move the dividend decimal point 29.9 one place to the left then the result is 2.99. The quotient is 2.99.

Question 5.
70.7 ÷ 10
70.7 ÷ 10 = 7.07
The quotient is 7.07.
Explanation:
Dividing by 10, or 101, means moving the decimal point one place to the left. Move the dividend decimal point 70.7 one place to the left then the result is 7.07. The quotient is 7.07.

Question 6.
5,913 ÷ 103
5,913 ÷ 103
5,913 ÷ 1000 = 5.913
The quotient is 5.913.
Explanation:
Dividing by 1,000, or 103, means moving the decimal point three places to the left. Move the dividend decimal point 5,913 three place to the left then the result is 5.913. The quotient is 5.913.

Set B
pages 233-236
Estimate 27.3 ÷ 7.1. Use compatible numbers.

So, 27.3 ÷ 7.1 is about 4.
Estimate 42.5 ÷ 11. Use rounding.

So, 42.5 ÷ 11 is about 4.

Remember that compatible numbers are numbers that are easy to compute in your head.

Write a number sentence that shows a way to estimate each quotient.

Question 1.
26.2 ÷ 5
Estimate 26.2 ÷ 5. Use compatible numbers.
Look for compatible numbers.
26.2 ÷ 5 is closes to 25 ÷ 5 = 5.
So, 26.2 ÷ 5 is about 5.
The Estimated quotient is 5.
Explanation:
Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. In the above division problem 26.2 ÷ 5 is closes to 25 ÷ 5. The actual dividend 26.2 is compatible to 25. Perform division operation 25 ÷ 5 = 5. The Estimated quotient is 5.

Question 2.
49.6 ÷ 7.8
Estimate 49.6 ÷ 7.8. Use compatible numbers.
Look for compatible numbers.
49.6 ÷ 7.8 is closes to 49 ÷ 7 = 7.
So, 49.6 ÷ 7.8 is about 7.
The Estimated quotient is 7.
Explanation:
Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. In the above division problem 49.6 ÷ 7.8 is closes to 49÷ 7. The actual dividend 49 is compatible to 25. The actual divisor 7.8 is compatible to 7. Perform division operation 25 ÷ 5 = 5. The Estimated quotient is 7.

Question 3.
121 ÷ 12.75
Estimate 121 ÷ 12.75. Use compatible numbers.
Look for compatible numbers.
121 ÷ 12.75 is closes to 120 ÷ 12 = 10.
So, 121 ÷ 12.75 is about 10.
The Estimated quotient is 10.
Explanation:
Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. In the above division problem 121 ÷ 12.75 is closes to 120 ÷ 12. The actual dividend 121 is compatible to 120. The actual divisor 12.75 is compatible to 12. Perform division operation 120 ÷ 12 = 10. The Estimated quotient is 10.

Question 4.
32.41 ÷ 10.9
Estimate 32.41 ÷ 10.9. Use rounding.
Round to the nearest ten: 32 rounds to 30; 10.9 rounds to 10.
32.41 ÷ 10.9 is about 30 ÷ 10 = 3.
Explanation:
Rounding means replacing a number with an approximate value. In the above division method 32.41 ÷ 10.9. Round the numbers to the nearest ten or hundreds. Here 32.41 is rounded to 30 and 10.9 is rounded to 10. Now perform division operation on 30 ÷ 10 = 3. The estimated quotient is 3.

Question 5.
82.4 ÷ 3.7
Estimate 82.4 ÷ 3.7. Use compatible numbers.
Look for compatible numbers.
82.4 ÷ 3.7 is closes to 80 ÷ 4 = 20.
So, 82.4 ÷ 3.7 is about 20.
The Estimated quotient is 20.
Explanation:
Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. In the above division problem 82.4 ÷ 3.7 is closes to 80 ÷ 4. The actual dividend 82.4 is compatible to 80. The actual divisor 3.7 is compatible to 4. Perform division operation 80 ÷ 4 = 20. The Estimated quotient is 20.

Question 6.
28.5 ÷ 0.94
Estimate 28.5 ÷ 0.94. Use compatible numbers.
Look for compatible numbers.
28.5 ÷ 0.94 is closes to 30 ÷ 1 = 30.
So, 28.5 ÷ 0.94 is about 30.
The Estimated quotient is 30.
Explanation:
Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. In the above division problem 28.5 ÷ 0.94 is closes to 30 ÷ 1. The actual dividend 28.5 is compatible to 30. The actual divisor 0.94 is compatible to 1. Perform division operation 30 ÷ 1 = 30. The Estimated quotient is 30.

Set C
pages 237-240
Find 1.14 ÷ 3.

Remember to use estimation to check the placement of the decimal point in the quotient. Divide. Use models to help.

Question 1.
6.58 ÷ 7

6.58 ÷ 7 = 0.94
Explanation:
In the above image we can observe the division of 6.58 ÷ 7 = 0.94. There are not enough ones to put 1 in each group, so regroup the 6 ones into 60 tenths. Divide 65 tenths into 7 equal groups. Each group gets 9 tenths. Seven groups of 0.9 = 6.3.
Trade the two extra tenth for 20 hundredths to get 28 hundredths. Divide the 28 hundredths into 7 equal groups. Each group gets 4 hundredths. Seven groups of 0.04 = 0.28.

Question 2.
156 ÷ 8

156 ÷ 8 = 19.5
Explanation:
In the above image we can observe the division of 156 ÷ 8 = 19.5. There are enough ones to put 1 in each group, and extra ones is regrouped. Divide the 156 ones into 8 equal groups. Each group gets 19 ones. Eight groups of 19 = 152. Regroup of the 4 ones into 40 tenths.  Divide the 40 tenths into 8 equal groups. Each group gets 5 tenths. Eight groups of 0.5 = 4.

Question 3.
34.2 ÷ 3

34.2 ÷ 3 = 11.4
Explanation:
In the above image we can observe the division of 34.2 ÷ 3 = 11.4. There are enough ones to put 1 in each group, and extra ones is regrouped. Divide the 34 ones into 3 equal groups. Each group gets 11 ones. Three groups of 11 = 33. Regroup of the 1 one into 10 tenths.  Divide the 12 tenths into 3 equal groups. Each group gets 4 tenths. Three groups of 0.4 = 1.2.

Question 4.
5.84 ÷ 4

5.84 ÷ 4 = 1.46
Explanation:
In the above image we can observe the division of 5.84 ÷ 4 = 1.46. There are enough ones to put 1 in each group, and extra ones is regrouped. Divide the 5 ones into 4 equal groups. Each group gets 1 ones. Four groups of 1 = 4. Regroup of the 1 ones into 10 tenths. We can see that there are 18 tenths in 5.84. Divide the 18 tenths into 4 equal groups. Each group gets 4 tenths. Four groups of 0.4 = 1.6.
Trade the one extra tenth for 10 hundredths to get 24 hundredths. Divide the 24 hundredths into 4 equal groups. Each group gets 6 hundredths. Four groups of 0.06 = 0.24.

Question 5.
Michelle pays $66.85 for a costume pattern and 8 yards of fabric. The costume pattern costs$4.85. How much does each yard of the fabric cost?

Set D
pages 241-244

Find 94.5 ÷ 15.
Estimate first
94.5 ÷ 15 is close to 100 ÷ 20 = 5, so start dividing with the ones place.

So, 94.5 ÷ 15 = 6.3

Remember that you can check your calculation by multiplying the quotient by the divisor.

Find each quotient.

Question 1.
91.2 ÷ 16

91.2 ÷ 16 = 5.7
Explanation:
In the above image we can observe the division operation  91.2 ÷ 16 = 5.7. The nearest possible multiple value of 16 is 80(16 x 5). Subtract 80 from 91.2 so as to get the remainder 11.2. 16 cannot be a multiple of 11.2 and so we have kept a decimal in the quotient and now the possible multiple value of 16 near to 11.2 is 11.2 itself(16 x 0.7 = 11.2). Now the remainder is zero and the final quotient is 5.7.

Question 2.
361.5 ÷ 15

361.5 ÷ 15 = 24.1
Explanation:
In the above image we can observe the division operation  361.5 ÷ 15 = 24.1. The nearest possible multiple value of 15 is 360(15 x 24). Subtract 360 from 361.5 so as to get the remainder 1.5. 15 cannot be a multiple of 1.5 and so we have kept a decimal in the quotient and now the possible multiple value of 15 near to 1.5 is 1.5 itself(15 x 0.1 = 1.5). Now the remainder is zero and the final quotient is 24.1.

Question 3.
29.04 ÷ 22

29.04 ÷ 22 = 1.32
Explanation:
In the above image we can observe the division operation  29.04 ÷ 22 = 1.32. The nearest possible multiple value of 22 is 22(22 x 1). Subtract 22 from 29.04 so as to get the remainder 7.04. 22 cannot be a multiple of 7.04 and so we have kept a decimal in the quotient and now 22 has the least possible multiple value of 6.60(22 x 0.3) near to 7.04. After subtracting 6.6 from 7.04, we are now having 1.3 as quotient and 0.44as remainder. Now the possible multiple value of 22 near to 0.44 is 0.44 itself(22 x 0.02 = 0.44). Now the remainder is zero and the final quotient is 1.32.

Question 4.
144 ÷ 45

144 ÷ 45 = 3.2
Explanation:
In the above image we can observe the division operation  144 ÷ 45= 3.2. The nearest possible multiple value of 45 is 135(45 x 3). Subtract 135 from 144 so as to get the remainder 9. 45 cannot be a multiple of 9 and so we have kept a decimal in the quotient and now the possible multiple value of 45 near to 9.0 is 9.0 itself(45 x 0.2 = 9.0). Now the remainder is zero and the final quotient is 3.2.

Question 5.
A 12-ounce bottle of shampoo costs $4.20. A 16-ounce bottle costs$6.88. Which shampoo costs less per ounce? How do you know?
A 12-ounce bottle of shampoo costs $4.20.$4.20 ÷ 12 = $0.35. A 16-ounce bottle costs$6.88.
$6.88 ÷ 16 =$0.43
12- ounce bottle shampoo costs less per ounce is $0.35. Explanation: A 12-ounce bottle of shampoo costs$4.20. Divide $4.20 ÷ 12 =$0.35. A 16-ounce bottle costs $6.88. Divide$6.88 ÷ 16 = $0.43. Compare these 12- ounce and 16- ounce bottle of shampoos. The 12-ounce bottle shampoo costs less per ounce is$0.35.

Set E
pages 245-248

Find 4.8 ÷ 0.6.

48 tenths ÷ 6 tenths
6 tenths × ? = 48 tenths
? = 8
So, 4.8 ÷ 6 = 8

Remember to use estimation Dates to check the quotient for reasonableness.

Question 1.
6.4 ÷ 3.2
6.4 ÷ 3.2
3.2 x 2 = 6.4
6.4 ÷ 3.2 = 2
The quotient is 2.
Explanation:
Multiply 3.2 with 2 then the product is 6.4. Dividing 6.4 by 3.2 then the quotient is 2.

Question 2.
6.4 ÷ 0.32
0.32 x 20 = 6.4
6.4 ÷ 0.32 = 20
The quotient is 20.
Explanation:
Multiply 0.32 with 20 then the product is 6.4. Dividing 6.4 by 0.32 then the quotient is 20.

Question 3.
9.6 ÷ 0.8
12 x 8 = 96
12 x 0.8 = 9.6
9.6 ÷ 0.8 = 12
The quotient is 12.
Explanation:
Multiply 0.8 with 12 then the product is 9.6. Dividing 9.6 by 0.8 then the quotient is 12.

Question 4.
0.96 ÷ 0.08
12 x 8 = 96
12 x 0.08 = 0.96
0.96 ÷ 0.08 = 12
The quotient is 12.
Explanation:
Multiply 0.08 with 12 then the product is 0.96. Dividing 0.96 by 0.08 then the quotient is 12.

Question 5.
41.8 ÷ 2.2
19 x 2.2 = 41.8
41.8 ÷ 2.2 = 19
The quotient is 19.
Explanation:
Multiply 2.2 with 19 then the product is 41.8. Dividing 41.8 by 2.2 then the quotient is 19.

Question 6.
4.18 ÷ 0.22
0.22 x 19 = 4.18
4.18 ÷ 0.22 = 19
The quotient is 19.
Explanation:
Multiply 0.22 with 19 then the product is 4.18. Dividing 4.18 by 0.22 then the quotient is 19.

Question 7.
81.4 ÷ 7.4
11 x 7.4 = 81.4
81.4 ÷ 7.4 = 11
The quotient is 11.
Explanation:
Multiply 7.4 with 11 then the product is 81.4. Dividing 81.4 by 7.4 then the quotient is 11.

Question 8.
814 ÷ 74
814 ÷ 74 = 11
The quotient is 11.
Explanation:
By Dividing 814 by 74 then the quotient is 11.

Question 9.
9.6 ÷ 0.03
9.6 ÷ 0.03 = 320
The quotient is 320.
Explanation:
By Dividing 9.6 by 0.03 then the quotient is 320.

Question 10.
9.6 ÷ 0.3
9.6 ÷ 0.3 = 32
The quotient is 32.
Explanation:
By Dividing 9.6 by 0.3 then the quotient is 32.

Set F
pages 249-252

Thinking Habits
• What do the numbers and symbols in the problem mean?
• How are the numbers or quantities related?
• How can I represent a word problem using pictures, numbers, or equations?

Zoey has a goal of saving $750 for a vacation. Her vacation will last 6 days. She wants to save the same amount each week for 12 weeks to reach her goal. How much should she save each week? Which quantities do you need to solve the problem? The savings goal is$750; Zoey will save for 12 weeks.

Will Zoey need to save more than or less than $80 each week? Explain your reasoning. Less than; 12 ×$80 = $960, but she only needs to save$750.
How much should she save each week? Write an equation to represent the problem.
$62.50;$750 :12 = $62.50 Remember to check the reasonableness of a solution by making sure your calculations are correct, and that you answered all of the questions that were asked. lan uses 4 feet of ribbon to wrap each package. How many packages can he wrap with 5.6 yards of ribbon? Remember there are 3 feet in a yard. Question 1. Describe one way to solve the problem. Answer: Question 2. What is the solution to the problem? Show your work. Answer: A bushel of apples weighs about 42 pounds. There are 4 pecks in a bushel. It takes 2 pounds of apples to make one pie. How many pies can you make with one peck of apples? Question 3. How are the numbers in the problem related? Answer: Question 4. Describe one way to solve the problem. Answer: Question 5. Solve the problem. Show your work. Answer: ### Topic 6 Assessment Practice Question 1. Mr. Dodd filled the gas tank on his lawn mower with 3.8 gallons of gas. He mowed his yard 10 times on the same tank of gas. He used the same amount of gas each time. How much gas did he use each time? Write an equation to show your work. Explain how the decimal point moves. Answer: Question 2. Kimberly scored a total of 35.08 points in four events for her gymnastic competition. If she scored the same number of points in each event, how many points did she score in each? Write an equation to show your work. Answer: 35.08 points ÷ 4 = 8.77 points. She scored 8.77 points in each event. Explanation: Kimberly scored a total of 35.08 points in four events for her gymnastic competition. If she scored the same number of points in each event. Divide 35.08 by 4 then the quotient is 8.77. She scored 8.77 points in each event. Question 3. Choose the correct quotient for each expression. Use number sense and estimation to help. Answer: Explanation: Perform division operation on 21.6 ÷ 1.8 = 12. The quotient is 12. Put a tick mark below the number 12. Perform division operation on 10.23 ÷ 0.55 = 18.6. The quotient is 18.6. Put a tick mark below the number 18.6. Perform division operation on 78.75 ÷ 3.5 = 22.5. The quotient is 22.5. Put a tick mark below the number 22.5. Perform division operation on 29.67 ÷ 4.6 = 18.6. The quotient is 18.6. Put a tick mark below the number 18.6. Question 4. What is the value of the missing exponent in the equation? A. 1 B. 2 C. 3 D. 4 Answer: 80.5 ÷ 102 = 0.805 The missing exponent in the equation is 2. Question 5. The chef at a restaurant bought 37 pounds of salad for$46.25. How much did she pay for each pound of salad?
A. 0.125
B. $1.25 C.$1.3
D. $12.50 Answer:$46.25 ÷ 37 = $1.25 She pays$1.25 for each pound of salad.
Explanation:
The chef at a restaurant bought 37 pounds of salad for $46.25. Divide$46.25 by 37 pounds then the quotient is $1.25. She pays$1.25 for each pound of salad.

Question 6.
Kathleen spent $231 on concert tickets for herself and 11 friends. Each ticket cost the same amount. A. Estimate the cost of each ticket. Write an equation to show your work. B. Find the exact cost of each ticket. Compare your answer to your estimate to check for reasonableness. Answer: A.$231 ÷ 12 = ?
Estimate $231 ÷ 12 closes to 230 ÷ 12 =$19.16.
The estimated cost of each ticket is $19.16. Explanation: Kathleen spent$231 on concert tickets for herself and 11 friends. Each ticket cost the same amount. Divide 230 by 12 then the quotient is $19.16. The estimated cost of each ticket is$19.16.
B. $231 ÷ 12 =$19.25
The actual cost of each ticket is $19.25. Explanation: The estimated cost of each ticket is$19.16 and the actual cost of each ticket is $19.25. Both estimated answer and actual answer are reasonable. Question 7. Select all of the following equations that are true when 12.5 is used. Use number sense to help. Answer: Explanation: In the above image we can observe the equations that are true when 12.5 is used. Option 1 and option 3, option 5 is correct. 1. When 12.5 is divided by 10 then the quotient is 1.25. So option 1 is correct. 2. When 12.5 is divided by 1 then the quotient is 12.5. So option 2 is not correct. 3. When 12.5 is divided by 1 then the quotient is 12.5. So option 3 is correct. 4. When 12.5 is divided by 100 then the quotient is 0.125. So option 4 is not correct. 5. When 12.5 is divided by 100 then the quotient is 0.125. So option 5 is correct. Question 8. Which division problem does the model Tess made represent? A. 1.35 ÷ 3 = 0.45 B. 1.35 ÷ 3 = 0.54 C. 1.62 ÷ 3 = 0.45 D. 1.62 ÷ 3 = 0.54 Answer: Option D 1.62 ÷ 3 = 0.54 is correct. Explanation: In the above image we can observe 5 tenths and 4 hundredths. By dividing 1.62 by 3 then the quotient is 0.54. The division problem answer option D represents the above image. Question 9. If 8 ounces of canned pumpkin has 82 calories, how many calories are in 1 ounce? Use your answer to find how many calories are in 6 ounces of pumpkin. Answer: 10.25 calories are there in 1 ounce. 10.25 calories = 1 ounce ? calories = 6 ounces 10.25 calories x 6 ounces = 61.5 calories 61.5 calories are in 6 ounces of pumpkin. Explanation: 8 ounces of canned pumpkin has 82 calories. Perform division operation on 82 ÷ 8 = 10.25. There are 10.25 calories in 1 ounce. We have to find out the how many calories are there in 6 ounces. Perform multiplication operation on these two numbers 10.25 calories and 6 ounces then the product is 61.5 calories. There are 61.5 calories in 6 ounces of pumpkin. Question 10. Use the equation 1.6 ÷ n = 0.016. A. What value of n makes the equation true? Write your answer using an exponent. B. Explain how you know your answer is correct. Answer: A. 1.6 ÷ n = 0.016. 1.6 ÷ 0.01 = 0.016. 1.6 ÷ 10-2 = 0.016. The value n = 0.01 makes the equation true. The exponent form of 0.01 is 10-2 . B. We can check the answer is correct or not by performing division operation on 0.016 ÷ 1.6 = 0.01. Question 11. Eileen bought 8 roses for$45.50. Which is the best way to estimate the cost of one rose?
A. $45 ÷ 5 =$9.00
B. $48 ÷ 8 =$6.00
C. $45 ÷ 10 =$0.45
D. $40 ÷ 8 =$0.50
Option B $48 ÷ 8 =$6.00 is the best way to estimate the cost of one rose.

Estimate $45.50÷ 8. Use compatible numbers. Look for compatible numbers.$45.50÷ 8 is closes to $48 ÷ 8 =$6.
The Estimated cost of one rose is $6. Explanation: Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. Eileen bought 8 roses for$45.50. In the above division problem $45.50÷ 8 is closes to 48 ÷ 8. The actual dividend$45.50 is compatible to 48. Perform division operation $48 ÷ 8 = 6. The Estimated cost of one rose is$6.

Question 12.
Toby’s faucet dripped a total of 1.92 liters of water in 24 hours. The faucet dripped the same amount each hour.
A. Estimate how many liters his faucet dripped each hour. Write an equation to model your work.
B. Find the exact amount of water that dripped each hour.
A. 1.92 liters ÷ 24 = ?
Estimate 1.92 liters ÷ 24 closes to 2 liters ÷ 24 = 0.083 liters.
The estimated amount of water dripped each hour is 0.083 liters.
Explanation:
Toby’s faucet dripped a total of 1.92 liters of water in 24 hours. Divide 2liters by 24 then the result is 0.083 liters. The estimated amount of water dripped each hour is 0.083 liters.
B. 1.92 liters ÷ 24 = 0.08 liters.
The exact amount of water that dripped each hour is 0.08 liters.
Explanation:
Perform division operation on 1.92 liters by 24 hours. The quotient is 0.08 liters. The exact amount of water that dripped each hour is 0.08 liters.
c. The estimated amount of water dripped each hour is 0.083 liters.
The exact amount of water that dripped each hour is 0.08 liters. The answer is reasonable.

### Topic 6 Assessment Practice

Question 13.
Choose the correct quotient for each expression.

Explanation:
Perform division operation on 0.78 ÷ 10 = 0.078. The quotient is 0.078. Put a tick mark below the number 0.078.
Perform division operation on 7,080 ÷ 103 = 7.08. The quotient is 7.08. Put a tick mark below the number 7.08.
Perform division operation on 70.8 ÷ 102 = 0.708. The quotient is 0.708. Put a tick mark below the number 0.708.
Perform division operation on 780 ÷ 103 = 0.78. The quotient is 0.78. Put a tick mark below the number 0.78.

Question 14.
Diego is making a large mural. He draws a hexagon with a perimeter of 10.5 meters. Each side of the hexagon is the same length.

A. How many meters long is each side of Diego’s hexagon? Write an equation to model your work.
B. The total cost of the supplies to paint the mural is $38.70. Diego and 9 friends divide the total cost equally. How much does each person pay? Answer: Question 15. Select all of the following equations that are true when 40.3 is used. Use number sense to help. Answer: Explanation: In the above image we can observe the equations in option 2 and option 3 are true when 40.3 is used. Perform division operation on 40.3 ÷ 102 = 0.403. The quotient is 0.403. So option 2 is correct. Perform division operation on 40.3 ÷ 100 = 40.3. The quotient is 40.3. So option 3 is correct. Question 16. Lou’s Diner spent$12.80 on 8 pounds of potatoes. What was the cost of one pound of potatoes? What would be the total cost if the cost per pound remained the same and the diner bought 7 pounds? Show your work.
$12.80 ÷ 8 pounds =$1.6
The cost of one pound of potatoes is $1.6. ? ÷ 7 pounds =$1.6
$1.6 x 7 pounds =$11.2
The total cost is $11.2. If the cost per pound remains same and the dinner bought 7 pounds. Explanation: Lou’s Diner spent$12.80 on 8 pounds of potatoes. Divide $12.80 ÷ 8 pounds =$1.6. The cost of one pound of potatoes is $1.6. If the cost per pound remains same and the dinner bought 7 pounds. Multiply$1.6 x 7 pounds = $11.2. The total cost is$11.2.

Question 17.
How many quarters are there in $30? Solve the equation 30 ÷ 0.25 to help you. A. 12 quarters B. 20 quarters C. 120 quarters D. 200 quarters Answer: 30 ÷ 0.25 = 120 There are 120 quarters in$30.
Option C is correct.
Explanation:
Perform division operation on 30 ÷ 0.25 = 120 . Here the dividend is 30 and the divisor is 0.25. By dividing the dividend with divisor then the quotient is 120. There are 120 quarters in $30. So option C is correct. Question 18. When solving 6.1 ÷ 102, how is the decimal point moved? A. 1 place to the right B. 1 place to the left C. 2 places to the right D. 2 places to the left Answer: 6.1 ÷ 102 = 0.061 Option D is correct . Explanation: Perform division operation on 6.1 ÷ 102 = 0.061 . Here the divisor is 102 which is 100. So the decimal point is moved two places to the left. The quotient is 0.061. So option D is correct. Question 19. A group of 5 friends bought a bag of grapes to share equally. If the bag of grapes weighs 11.25 pounds, how much is each person’s share? How many friends could share the grapes if each person’s share was 1.25 pounds? Write an equation to model your work. Answer: 11.25 pounds ÷ 5 = 2.25 pounds. Each person share is 2.25 pounds. 11.25 pounds ÷ ? = 1.25 pounds. 11.25 pounds ÷ 9 = 1.25 pounds. 9 friends can share the grapes if each person’s share was 1.25 pounds. Explanation: A group of 5 friends bought a bag of grapes to share equally. The bag of grapes weighs 11.25 pounds. Perform division operation on 11.25 pounds ÷ 5 = 2.25 pounds. Each person share is 2.25 pounds. Divide 11.25 pounds ÷ 1.25 pounds = 9. Nine friends can share the grapes if each person’s share was 1.25 pounds. Question 20. When dividing 560.9 by 100, how should the decimal point be moved? Answer: 560.9 ÷ 100 = 5.609 Explanation: Dividing by 100, or 102, means moving the decimal point two places to the left. Move the dividend decimal point 560.9 two places to the left then the result is 5.609. The quotient is 5.609. Question 21. June says that there should be a decimal point in the quotient below after the 4. Is she correct? Use number sense to explain your answer. 43.94 ÷ 5.2 = 845 Answer: 43.94 ÷ 5.2 = 8.45 She is not correct. The decimal should be placed after 8. Explanation: June says that there should be a decimal point in the quotient after the 4. She is not correct because we have to place the decimal point after 8. Question 22. Three coworkers decided to buy fruit to share at lunchtime. Antonio spent$1.47 on bananas. Laura spent $2.88 on apples. Suzanne spent$2.85 on oranges.
A. Complete the bar diagram to find out how much they spent in all on fruit.

They spent $7.2 in all on fruit. Explanation: Three coworkers decided to buy fruit to share at lunchtime. Antonio spent$1.47 on bananas. Laura spent $2.88 on apples. Suzanne spent$2.85 on oranges. Add all fruit costs. Add $1.47 with$2.88 and $2.85 then the sum is$7.2. They spent $7.2 in all on fruit. B. They evenly divided the cost of the 3 types of fruit. How much did each person pay? Complete the bar diagram to help you. Answer: Each person has to pay$2.4 for fruits.
Explanation:
The three coworkers evenly divided the cost of the 3 types of fruit. The total cost of three fruits is $7.2. Divide$7.2 by 3 then the quotient is $2.4. Each person has to pay$2.4 for fruits.

C. If Laura bought 2.1 pounds of apples, is the price per pound of apples greater than or less than $1? How can you tell? Answer: ### Topic 6 Performance Task Cooking Competition Lydia is organizing a cooking competition at her school. She ordered some basic supplies to share among the teams that are competing. The teams will be bringing other ingredients as well. Use the list at the right to answer the questions. Question 1. If 10 of the teams divide the olive oil equally, how much will each team receive? Write an equation to model your work. Answer: 5.4 liters ÷ 10 = 0.54 liters. Each team receives 0.54 liters. Explanation: 10 of the teams divide the olive oil equally. Here we have total 5.4 liters of olive oil. Perform division operation on 5.4 liters ÷ 10 = 0.54 liters. Each team receives 0.54 liters. Question 2. Eight teams agree to share the flour equally. Part A About how many grams of flour will each team get? Use compatible numbers to estimate. Write an equation to show how you estimated. Answer: 738.4 grams ÷ 8 = ?. Estimate 738.4 grams ÷ 8. Use compatible numbers. Look for compatible numbers. 738.4 ÷ 8 is closes to 738 ÷ 8 = 92.25 grams. The Estimated quotient is 92.25. Each team get 92.25 grams of flour. Explanation: Compatible numbers are pairs of numbers that are easy to add, subtract, multiply, or divide mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers. In the above division problem 738.4 grams ÷ 8 is closes to 738 ÷ 8. The actual dividend738.4 is compatible to 738. Perform division operation 738 ÷ 8 = 92.25. The Estimated quotient is 92.25. Each team get 92.25 grams of flour. Part B Find the actual amount of flour each team will receive. Show your work. Answer: 738.4 grams ÷ 8 = 92.3 grams. Each team will receive 92.3 grams of flour. Explanation: Perform division operation on 738.4 grams ÷ 8 = 92.3 grams. Each team will receive the actual amount of flour is 92.3 grams. Question 3. Several teams agree to share the salt equally. Each team will be given 7.3 grams of salt. How many teams agree to share the salt? Write a division equation to model the problem. Then write an equivalent equation using whole numbers. Answer: 87.6 grams ÷ ? = 7.3 grams. 87.6 grams ÷ 12 = 7.3 grams. 12 teams agree to share the salt. Explanation: Several teams agree to share the salt equally. Each team will be given 7.3 grams of salt. The division equation is 87.6 grams ÷ ? = 7.3 grams. Divide 87.6 with 7.3 then the answer is 12. 12 teams agree to share the salt. Question 4. Malcolm calculated how many liters of milk each team would get if 6 teams shared the milk equally. His work is shown at the right, but he forgot to place the decimal point in the quotient. Where should he place the decimal point? Explain. Answer: 8.25 liters ÷ 6 = 1.375 liters. Malcolm has to place the decimal point before 3 and after 1. Each team get 1.375 liters of milk. Explanation: Malcolm calculated how many liters of milk each team would get if 6 teams shared the milk equally. His work is shown above, but he forgot to place the decimal point in the quotient. He has to place the decimal point before 3 and after 1. Each team get 1.375 liters of milk. Question 5. Lydia decides to provide cheddar cheese for the competition. She buys 4.2 kilograms for$39.90.
Part A
She estimates the cost of 1 kilogram of cheese to be $1. Is her estimate reasonable? Explain. Answer: 4.2 kilograms ÷$39.90 = ?
Estimate 4.2 ÷ $40 =$0.105
The cost of 1 kilogram of cheese to be $0.105. Her Estimation is not reasonable. Explanation: Lydia decides to provide cheddar cheese for the competition. She buys 4.2 kilograms for$39.90. Perform division operation on 4.2 ÷ $40 =$0.105. The cost of 1 kilogram of cheese to be $0.105. Her Estimation is not reasonable. Part B To find the actual cost of 1 kilogram of cheese, Lydia needs to divide$39.90 by 4.2. How can she change the division problem to an equivalent problem using whole numbers? Write and solve the equivalent problem..
Part C
If 7 teams share the cheese equally, how much cheese will each team get?
4.2 kilograms ÷  7 = 0.6 kilograms.
Each team get 0.6 kilograms of cheese equally.
Explanation:
The cheese we have is 4.2 kilograms. Seven teams takes the cheese equally. perform division operation on 4.2 kilograms ÷  7 = 0.6 kilograms. Each team get 0.6 kilograms of cheese equally.