Go through the **enVision Math Common Core Grade 4 Answer Key Topic 6 Use Operations with Whole Numbers to Solve Problems** regularly and improve your accuracy in solving questions.

## enVision Math Common Core 4th Grade Answers Key Topic 6 Use Operations with Whole Numbers to Solve Problems

**Essential Questions:**

How is comparing with multiplication different from comparing with addition? How can you use equations to solve multi-step problems?

Answer:

In an additive comparison, we find the relation between two amounts by asking or telling how much more is one compared to the other. In multiplicative comparison, there are two different sets being compared. The first set contains a certain number of items. The second set contains multiple copies of the first set

Example:

Additive Comparison:

2 + 2 + 5 + 3 + … n

Multiplicative Comparison:

3 × 4 × 5 × … n

**enVision STEM Project: Energy and Multiplication**

**Do Research**

Use the Internet or other sources to find and describe 3 examples of renewable energy?

Answer:

The examples of renewable energy are:

a. Solar energy

b. Wind energy

c. Tidal energy

d. Geothermal energy

**Journal**: Write a Report that Includes what you found. Also in your report:

A)

Solar panels are made up of smaller modules or sections called cells. Find a picture of a solar panel. How many cells are in 6 solar panels? How many cells are in 9 solar panels? How many more cells are in the 9 solar panels than the 6 solar panels?

Answer:

It is given that

Solar panels are made up of smaller modules or sections called cells

Now,

We know that,

A solar power panel is made of photovoltaic cells arranged in a configuration that can contain 32, 36, 48, 60, 72, and 96 cells

Now,

Let a solar panel contains 32 cells

So,

The number of cells that present in 6 solar panels = (The number of solar panels) × (The number of cells present in a solar panel)

= 6 × 32

= 6 × (30 + 2)

= (6 × 30) + (6 × 2)

= 180 + 12

= 192 cells

So,

The number of cells that present in 9 solar panels = (The number of solar panels) × (The number of cells present in a solar panel)

= 9 × 32

= 9 × (30 + 2)

= (9 × 30) + (9 × 2)

= 270 + 18

= 288 cells

So,

The difference between the two solar panels = 288 – 192

= 96 cells

Now,

The picture of a solar panel is:

Hence, from the above,

We can conclude that

The difference of the cells between 9 solar panels and 6 solar panels is: 96 cells

B)

Find examples of other items that use solar power?

Answer:

Some of the items that use solar power are:

A) Solar charger.

B) Solar cell phone charger. Strawberry Tree.

C) Solar chimney.

D) Solar-powered waste compacting bin.

E) Solar cooker.

F) Solar dryer.

G) Solar-powered fan.

H) Solar furnace.

**Review What You Know**

**Vocabulary**

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

- Associative Property of Multiplication
- Commutative Property of Multiplication
- Compatible numbers
- Distributive Property

Question 1.

The __________ says that factors can be multiplied in any order and the product remains the same.

Answer:

We know that,

The “Commutative Property of Multiplication” says that factors can be multiplied in any order and the product remains the same.

Hence, from the above,

We can conclude that the best term that is suitable for the given definition is: Cumulative Property of Multiplication

Question 2.

Breaking apart a multiplication problem into the sum or difference of two simpler multiplication problems is an example of using the __________

Answer:

We know that,

Breaking apart a multiplication problem into the sum or difference of two simpler multiplication problems is an example of using the “Distributive Property”

Hence, from the above,

We can conclude that the best term that is suitable for the given definition is: Distributive Property

Question 3.

According to the ________, factors can be regrouped and the product remains the same.

Answer:

We know that,

According to the “Associative property of Multiplication”, factors can be regrouped and the product remains the same.

Hence, from the above,

We can conclude that the best term that is suitable for the given definition is: Associative Property of Multiplication

**Dividing by 1-Digit Numbers**

Estimate each quotient.

Question 4.

16 ÷ 3

Answer:

The given expression is: 16 ÷ 3

Now,

By using the Long Division method,

Hence, from the above,

We can conclude that the quotient value for the given expression is: 5

Question 5.

25 ÷ 4

Answer:

The given expression is: 25 ÷ 4

Now,

By using the Long Division method,

Hence, from the above,

We can conclude that the quotient value for the given expression is: 6

Question 6.

155 ÷ 4

Answer:

The given expression is: 155 ÷ 4

Now,

By using the Long Division method,

Hence, from the above,

We can conclude that the quotient value for the given expression is: 38

Question 7.

304 ÷ 3

Answer:

The given expression is: 304 ÷ 3

Now,

By using the Long Division method,

Hence, from the above,

We can conclude that the quotient value for the given expression is: 101

Question 8.

1,283 ÷ 6

Answer:

The given expression is: 1,283 ÷ 6

Now,

By using the Long Division method,

Hence, from the above,

We can conclude that the quotient value for the given expression is: 213

Question 9.

1,999 ÷ 4

Answer:

The given expression is: 1,999 ÷ 4

Now,

By using the Long Division method,

Hence, from the above,

We can conclude that the quotient value for the given expression is: 499

**Multiplying by 1-Digit Numbers**

Find each product.

Question 10.

53 × 9

Answer:

The given expression is: 53 × 9

Now,

By using the partial Products method,

53 × 9

= (50 + 3) × 9

= (50 × 9) + (3 × 9)

= 450 + 27

= 477

Hence, from the above,

We can conclude that the product value for the given expression is: 477

Question 11.

1,127 × 7

Answer:

The given expression is: 1,127 × 7

Now,

By using the partial Products method,

1,127 × 7

= (1,000 + 100 + 27) × 7

= (1,000 × 7) + (100 × 7) + (27 × 7)

= 7,000 + 700 + 189

= 7,889

Hence, from the above,

We can conclude that the product value for the given expression is: 7,889

Question 12.

2,769 × 5

Answer:

The given expression is: 2,769 × 5

Now,

By using the partial Products method,

2,769 × 5

= (2,000 + 700 + 60 + 9) × 5

= (2,000 × 5) + (700 × 5) + (60 × 5) + (9 × 5)

= 10,000 + 3,500 + 300 + 45

= 13,845

Hence, from the above,

We can conclude that the product value for the given expression is: 13,845

Question 13.

3 × 215

Answer:

The given expression is: 3 × 215

Now,

By using the partial Products method,

3 × 215

= 3 × (200 + 15)

= (3 × 200) + (3 × 15)

= 600 + 45

= 645

Hence, from the above,

We can conclude that the product value for the given expression is: 645

Question 14.

914 × 5

Answer:

The given expression is: 914 × 5

Now,

By using the partial Products method,

914 × 5

= (900 + 14) × 5

= (900 × 5) + (14 × 5)

= 4,500 + 70

= 4,570

Hence, from the above,

We can conclude that the product value for the given expression is: 4,570

Question 15.

1,238 × 5

Answer:

The given expression is: 1,238 × 5

Now,

By using the partial Products method,

1,238 × 5

= (1,000 + 200 + 30 + 8) × 5

= (1,000 × 5) + (200 × 5) + (30 × 5) + (8 × 5)

= 5,000 + 1,000 + 150 + 40

= 6,190

Hence, from the above,

We can conclude that the product value for the given expression is: 6,190

**Problem Solving**

Question 16.

**Construct Arguments** Explain why the array represents 3 × 21

Answer:

The given array is:

Now,

We know that,

An “Array” is a collection of rows and columns

Now,

From the given figure,

We can observe that

The number of rows is: 21

The number of columns is: 3

So,

To find the number of cells in a given array,

We have to multiply the rows and columns

Hence, from the above,

We can conclude that the given array represents 21 × 3 since it represents the number of rows and number of columns

Question 17.

James multiplies 38 by 55. He finds three of the four partial products: 40, 150, and 400. Which partial product is James missing? What is the solution?

Answer:

It is given that

James multiplies 38 by 55. He finds three of the four partial products: 40, 150, and 400.

Now,

The given expression is: 38 × 55

Now,

By using the partial Products method,

38 × 55

= (30 + 8) × (50 + 5)

= (30 × 50) + (30 × 5) + (8 × 50) + (8 × 5)

= 1,500 + 150 +400 + 40

= 2,090

Hence, from the above,

We can conclude that

The Partial product James is missing is: 1,500

The solution for the given expression is: 2,090

**Pick a Project**

PROJECT 6A

How tall is tall?

Project: Model the Height of a Redwood Tree

PROJECT 6B

What are some interesting number facts about a manatee?

Project: Make a Presentation with Manatee Number Facts

PROJECT 6C

How does temperature affect an alligator egg?

Project: Create a Bar Diagram

PROJECT 6D

How many eggs does the loggerhead turtle lay?

Project: Model a Turtle’s Egg

### Lesson 6.1 Solve Comparison Problems

**Solve & Share**

Sarah is making a square pillow with edges that each measure 18 inches long. She needs a strip of fabric 4 times as long as one edge of the pillow to make a border around the pillow. How long does the strip of fabric need to be? Solve this problem any way you choose.

I can … use multiplication or addition to compare one quantity to another.

Answer:

It is given that

Sarah is making a square pillow with edges that each measure 18 inches long. She needs a strip of fabric 4 times as long as one edge of the pillow to make a border around the pillow

Now,

Let the length of the strip of fabric be x

So,

According to the given data,

The length of the strip (x) = 4 × (The length of the pillow at the edges)

= 4 × 18

= 72 inches

Hence, from the above,

We can conclude that

The length of the fabric needed is: 72 inches

**Look Back!**

**Model with Math**

How could a bar diagram help you write and solve an equation for the problem?

Answer:

It is given that

Sarah is making a square pillow with edges that each measure 18 inches long. She needs a strip of fabric 4 times as long as one edge of the pillow to make a border around the pillow

Now,

The representation of the given situation in the form of a bar diagram is:

So,

From the above bar diagram,

We can observe that

The length of the string = 4 × (The length of the edge of the pillow)

= 4 × 18

= 72 inches

Hence, from the above,

We can conclude that

By using the bar diagram, the length of the string is: 72 inches

**Essential Question**

How Is Comparing with Multiplication Different from Comparing with Addition?

Answer:

In an additive comparison, we find the relation between two amounts by asking or telling how much more is one compared to the other. In multiplicative comparison, there are two different sets being compared. The first set contains a certain number of items. The second set contains multiple copies of the first set

Example:

Additive Comparison:

2 + 2 + 5 + 3 + … n

Multiplicative Comparison:

3 × 4 × 5 × … n

Visual Learning Bridge

Max said the Rangers scored 3 times as many runs as the Stars. Jody said the Rangers scored 8 more runs than the Stars. Could both Max and Jody be correct?

Compare with Multiplication

Find 3 times as many as 4 runs.

m = 3 × 4

m = 12 runs

According to Max, the Rangers scored 12 runs.

Compare with Addition

Find 8 more than 4 runs.

j = 4 + 8

j = 12 runs

According to Jody, the Rangers scored 12 runs.

Max and Jody are both correct.

**Convince Me!
**

**Construct Arguments**Describe when you would use multiplication and when you would use addition to make a comparison?

Answer:

When distinguishing multiplicative comparison from additive comparison,

We should note that “Additive comparisons” focus on the difference between two quantities whereas “Multiplicative comparisons” focus on the difference between two qualities

**Guided Practice**

**Do You Understand?**

Question 1.

Use the information from the previous page. If the Rangers scored 5 times as many runs as the Stars, how many runs would the Rangers have scored?

a. Compare using multiplication. Write and solve an equation.

Answer:

It is given that

Rangers scored 5 times as many runs as the Stars

Now,

The given figure is:

Now,

The representation of the given situation in the form of a bar diagram is:

So,

The total number of runs rangers scored = 5 × (The number of runs Stars scored)

= 5 × 4

= 20 runs

Hence, from the above,

We can conclude that the number of runs scored by Rangers is: 20

b. How could you use addition to compare the runs for the two teams?

Answer:

It is given that

Rangers scored 5 times as many runs as the Stars

Now,

The given figure is:

Now,

The representation of the given situation in the form of a bar diagram is:

So,

The total number of runs rangers scored = (The number of runs Stars scored) + (The number of runs Stars scored) + (The number of runs Stars scored) + (The number of runs Stars scored) + (The number of runs Stars scored)

= 5 + 5 + 5 + 5 +5

= 20 runs

Hence, from the above,

We can conclude that the number of runs scored by Rangers is: 20

**Do You Know How?**

For 2-3, complete the comparison sentence. Find the value of the variable that makes the sentence true.

Question 2.

Sam’s baseball team has 4 times as many helmets as Ed’s team. Ed’s team has 21 helmets. Let h = the number of helmets Sam’s team has.

h is ___________ times as many as ________.

h = _________

Answer:

It is given that

Sam’s baseball team has 4 times as many helmets as Ed’s team. Ed’s team has 21 helmets.

So,

The number of helmets Sam’s team has (h) = 4 × (The number of helmets Ed’s team has)

= 4 × 21

= 4 × (20 + 1)

= (4 × 20) + (4 × 1)

= 80 + 4

= 84 helmets

Hence, from the above,

We can conclude that

h is 4 times as many as the number of helmets Ed’s team has

The number of helmets Sam’s team has is: 84 helmets

Question 3.

There are 128 more trees in the park than at Ty’s house. There are 3 trees at Ty’s house. Let t = the number of trees in the park.

_______ more than _____ is t.

t = __________

Answer:

It is given that

There are 128 more trees in the park than at Ty’s house. There are 3 trees at Ty’s house.

So,

The number of trees in the park (t) = (The number of trees at Ty’s house) + (The number of more trees in the park)

= 3 + 128

= 131 trees

Hence, from the above,

We can conclude that

128 trees more than 3 trees is t

The total number of trees in the park is: 131 trees

**Independent Practice**

Use times as many as or more than to compare the amounts.

For 4-9, write a comparison sentence. For 4-7, find the value of the variable that makes the sentence true.

Question 4.

Katy has 6 times as many nickels as Shaun. Shaun has 18 nickels. Let n = the number of nickels Katy has.

n is _____________

n = ___________.

Answer:

It is given that

Katy has 6 times as many nickels as Shaun. Shaun has 18 nickels.

So,

The number of nickels Katy has (n) = 6 × (The number of nickels Shaun has)

= 6 × 18

= 6 × (10 + 8)

= (6 × 10) + (6 × 8)

= 60 + 48

= 108 nickels

Hence, from the above,

We can conclude that

n is 6 times as many as the number of nickels Shaun has

The number of nickels Katy has: 108 nickels

Question 5.

Kyle has watched 238 movies. Jason has watched 49 more movies than Kyle. Let m = the number of movies Jason has watched.

m is _____________

m = _____________

Answer:

It is given that

Kyle has watched 238 movies. Jason has watched 49 more movies than Kyle

So,

The number of movies Jason has watched (m) = The number of movies Kyle watched + 49

= 238 + 49

= 287 movies

Hence, from the above,

We can conclude that

m is 49 mere than the number of movies Kyle has watched

The number of movies Jason has watched is: 287 movies

Question 6.

Amber tied 89 knots to make a macrame wall hanging. Hunter tied 3 times as many knots as Amber. Let k= the number of knots Hunter tied. k is _____________

k = _____________

Answer:

It is given that

Amber tied 89 knots to make a macrame wall hanging. Hunter tied 3 times as many knots as Amber

So,

The number of knots Hunter tied (k) = 3 × (The number of knots Amber tied)

= 3 × 89

= 3 × (80 + 9)

= (3 × 80) + (3 × 9)

= 240 + 27

= 267

Hence, from the above,

We can conclude that

k is 3 times as many knots as Amber

The number of knots Hunter tied is: 267 knots

Question 7.

Tia sells 292 newspapers. Tess sells 117 more newspapers than Tia. Let n = the number of newspapers Tess sells.

n is _____________

n = _____________

Answer:

It is given that

Tia sells 292 newspapers. Tess sells 117 more newspapers than Tia

So,

The number of newspapers Tess sells (n) = 117 + (The number of newspapers tia sells)

= 117 + 292

= 409 newspapers

Hence, from the above,

We can conclude that

n is 117 more than Tia’s newspapers

The number of newspapers Tess sells is: 409 newspapers

Question 8.

Trent has 48 markers. Sharon has 8 markers.

Answer:

Question 9.

Lucy has 317 bottles. Craig has 82 bottles.

Answer:

**Problem Solving**

Question 10.

**Model with Math**

Roger swam 19 laps in the pool. Anna Maria swam 4 times as many laps as Roger. How many laps did Anna Maria swim? Draw a bar diagram and write an equation to solve the problem.

Answer:

It is given that

Roger swam 19 laps in the pool. Anna Maria swam 4 times as many laps as Roger

So,

The representation of the given situation in the form of a bar diagram is:

So,

The number of laps did Anna maria swam = 4 × (The number of laps did Roger swam)

= 4 × 19

= 4 × (10 + 9)

= (4 × 10) + (4 × 9)

= 40 + 36

= 76 laps

Hence, from the above,

We can conclude that the number of laps did Anna swam is: 76 laps

Question 11.

**Critique Reasoning** Nina says the equation 600 = 12 × 50 means 600 is 12 times as many as 50. Julio says the equation means 600 is 50 times as many as 12. Who is correct? Explain.

Answer:

It is given that

Nina says the equation 600 = 12 × 50 means 600 is 12 times as many as 50. Julio says the equation means 600 is 50 times as many as 12

Now,

We know that,

According to Commutative Property of Multiplication,

a × b = b × a

So,

12 × 50 is the same as 50 × 12

“12 times as many as 50” is the same as “50 times as many as 12”

Hence, from the above,

We can conclude that

Both Nina and Julio are correct

Question 12.

**Vocabulary** The amount that is left after dividing a number into equal parts is called the _________.

13 ÷ 4 = ________

Answer:

We know that,

The amount that is left after dividing a number into equal parts is called the “Remainder”

Now,

The given expression is: 13 ÷ 4

Now,

By using the Long division method,

Hence, from the above,

We can conclude that the remainder for the given expression is: 1

Question 13.

**Higher-Order Thinking** A shirt is on sale for d dollars. The regular price is 4 times as much. Todd has enough money to buy 2 shirts at the regular price. How many shirts can Todd buy at the sale price? Explain.

Answer:

It is given that

A shirt is on sale for d dollars. The regular price is 4 times as much. Todd has enough money to buy 2 shirts at the regular price

So,

The regular price of a shirt = 4 × (The sale price of a shirt)

= 4 × d

= 4d

Now,

According to the given information,

The sale price of the shirts that Todd can buy = (Regular price of a shirt) × (The number of shirts Todd can buy)

= 4d × 2

= 8d

So,

The number of shirts Todd can buy = \(\frac{The sale price of the shirts Todd can buy}{The sale price of a shirt}\)

= \(\frac{8d}{d}\)

= 8

Hence, from the above,

We can conclude that Todd can buy 8 shirts at the sale price

**Assessment Practice**

Question 14.

Select all the sentences that involve a comparison using multiplication.

☐ k is 26 times as many as 7.

☐ u is 18 more than 314.

☐ Tom ran 4 miles. Cindy ran 2 more miles than Tom. How many miles did Cindy run?

☐ Yuan has 2 dogs and Jon has 3 times as many dogs. How many dogs does Jon have?

☐ Kris has 4 times as many pairs of shoes as her brother. Her brother has 8 pairs of shoes.

Answer:

The sentences that involve a comparison using multiplication are:

Question 15.

Select all the statements that can be represented by the equation 5 × 9 = w.

☐ w is 9 more than 5.

☐ w is 5 times as much as 9.

☐ Henry sang 9 songs during practice. He sang 5 times as many as Joe, who sang w songs.

☐ Greg carried 9 buckets of water to his sister’s baby pool. His mother carried w buckets, which were 5 times as many.

☐ Tom has 9 pens. Joan has w pens, which is 5 fewer.

Answer:

The sentences that can be represented by the equation 5 × 9 = w are:

### Lesson 6.2 Continue to solve Comparison Problems

**Solve & Share**

The students in Ms. Chang’s fourth-grade class plant a tree every year. One tree they planted is now 288 inches tall. This height is 6 times as great as when the tree was first planted. How tall was the tree when it was first planted? Solve this problem any way you choose. Show your work.

I can … use multiplication or division to compare one quantity to another.

Answer:

It is given that

The students in Ms. Chang’s fourth-grade class plant a tree every year. One tree they planted is now 288 inches tall. This height is 6 times as great as when the tree was first planted

Now,

Let the height of the tree when first planted be x

Now,

According to the given information,

6 × x = 288

x = \(\frac{288}{6}\)

x = 48 inches

Hence, from the above,

We can conclude that

The height of the tree when it was first planted is: 48 inches

**Look Back!** Compare the heights of the 2 trees using addition.

Answer:

From part (a),

We can observe that

The height of the tree when it was first planted is: 48 inches

The current height of the tree is: 288 inches

So,

The difference between the heights of the two trees = (The current height of the tree) – (The height of the tree when it was first planted)

= 288 – 48

= 240 inches

Hence, from the above,

We can conclude that the current height of the tree is 240 inches more than the height of the tree when it was first planted

**Essential Question**

How Can You Solve a Problem Involving Multiplication as Comparison?

Answer:

Interpret a multiplication equation as a comparison

Example:

Interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5.

Visual Learning Bridge

Carla and Calvin are twins that attend different colleges. Carla’s college is four times as far from home as Calvin’s college. How far does Calvin travel to college?

The number of miles Carla travels, or 192 miles, is 4 times as far as Calvin travels.

Write a multiplication equation to find the number of miles Calvin travels to college.

192 = 4 × m

m = 48 miles

What number times 4 equals 192?

If 192 = 4 × m, them m = 192 ÷ 4.

Calvin travels 48 miles to college.

**Convince Me!**

**Use Structure** When do you use division to make a comparison?

Answer:

A ratio is a comparison of two numbers by division. The value of a ratio is the quotient that results from dividing the two numbers.

Example:

The value of the ratio 35: 7 is 5, which you find by computing 35 ÷ 7 = 5.

**Guided Practice**

**Do You Understand?**

Question 1.

Carla travels 3 times as far to college as her friend Mandy travels to college. Write and solve a related multiplication and division equation to find how far Mandy travels to college.

Answer:

It is given that

Carla travels 3 times as far to college as her friend Mandy travels to college

Now,

From the example on the previous page,

We can observe that

Carla travels 192 miles to college

Now,

Let x be how far Mandy travels to college

So,

From the given information,

192 = 3 × x

x = \(\frac{192}{3}\)

x = 64 miles

Hence, from the above,

We can conclude that Mandy travels 64 miles to the college

**Do You Know How?**

Question 2.

Complete the comparison sentence. Find the value of the variable that makes the sentence true.

If 3 × m = 48, then m = 48 ÷ 3.

3 times as many as m is 48.

m = 16

Answer:

The given expression is:

3 × m = 48

m = \(\frac{48}{3}\)

m = 16

Hence, from the above,

We can conclude that

3 times as many as m is 48

The value of m is: 16

**Independent Practice**

For 3-8, write a comparison sentence and an equation. Find the value of the variable that makes the sentence true.

Question 3.

Connor has 77 magazines. That is 7 times as many magazines Kristen has. How many magazines, n, does Kristen have?

Answer:

It is given that

Connor has 77 magazines. That is 7 times as many magazines Kristen has

Now,

Let n be the number of magazines Kristen has

So,

According to the given information,

7 × n = 77

n = \(\frac{77}{7}\)

n = 11

Hence, from the above,

We can conclude that

The number of magazines that Kristen has: 11

Question 4.

Eric completed 75 math problems. That is 5 times as many math problems as Katie completed. How many math problems, m, did Katie complete?

Answer:

It is given that

Eric completed 75 math problems. That is 5 times as many math problems as Katie completed

Now,

Let m be the math problems Katie completed

So,

According to the given information,

5 × m = 75

m = \(\frac{75}{5}\)

m = 15

Hence, from the above,

We can conclude that

The number of math problems Katie completed is: 15

Question 5.

Clare counted 117 different colors at the paint store. That is 9 times as many as the number James counted. How many different colors, c, did James count?

Answer:

It is given that

Clare counted 117 different colors at the paint store. That is 9 times as many as the number James counted.

Now,

Let c be the number of colors James counted

So,

According to the given information,

9 × c = 117

c = \(\frac{117}{9}\)

c = 13

Hence, from the above,

We can conclude that

The number of different colors James counted is: 13

Question 6.

Alisa has 153 dominoes. That is 3 times as many dominoes as Stan has. How many dominoes, d, does Stan have?

Answer:

It is given that

Alisa has 153 dominoes. That is 3 times as many dominoes as Stan has

Now,

Let d be the number of dominoes does Stan have

So,

According to the given information,

3 × d = 153

d = \(\frac{153}{3}\)

d = 51

Hence, from the above,

We can conclude that

The number of dominoes does Stan have is: 51

Question 7.

Justin practiced piano for h hours. His sister practiced for 12 hours, which was 3 times as many hours as Justin practiced. Write and solve an equation to find how many hours Justin practiced piano.

Answer:

It is given that

Justin practiced piano for h hours. His sister practiced for 12 hours, which was 3 times as many hours as Justin practiced

Now,

Let h be the number of hours Justin practiced piano

So,

According to the given information,

h × 3 = 12

h = \(\frac{12}{3}\)

h = 4

Hence, from the above,

We can conclude that

The number of hours that Justin practiced piano is: 4 hours

Question 8.

Mary practiced violin for 2 hours and her brother practiced trombone t times as long or 8 hours. Write and solve an equation to find how many times as long Mary’s brother practiced trombone.

Answer:

It is given that

Mary practiced violin for 2 hours and her brother practiced trombone t times as long or 8 hours

Now,

Let t be the number of hours Mary’s brother practiced trombone

So,

According to the given information,

2 × t = 8

t = \(\frac{8}{2}\)

t = 4

Hence, from the above,

We can conclude that

The number of hours mary’s brother practiced trombone is: 4 hours

**Problem Solving**

Question 9.

**Model with Math**

Dave is making soup that includes 12 cups of water and 3 cups of broth. How many times as much water as the broth will be in the soup? Draw a bar diagram and write and solve an equation.

Answer:

It is given that

Dave is making soup that includes 12 cups of water and 3 cups of broth.

Now,

Let x be the number of cups of water present in the broth

So,

According to the given information,

3 × x = 12

x = \(\frac{12}{3}\)

x = 4

Hence, from the above,

We can conclude that

The number of cups of water present in the broth is: 4

Question 10.

Trevor wants to buy three light fixtures that cost $168 each. He has $500. Does he have enough money to buy the three light fixtures? Use a comparison sentence to explain your reasoning.

Answer:

It is given that

Trevor wants to buy three light fixtures that cost $168 each. He has $500.

So,

The cost of three light fixtures that Trevor wants to buy = 3 × (The cost of each light fixture)

= 3 × 168

= 3 × (100 + 68)

= (3 × 100) + (3 × 68)

= 300 + 204

= $504

So,

From the given information,

$500 < $504

Hence, from the above,

We can conclude that

The cost of the three light fixtures that Trevor wants to buy is $4 more than the amount of money Trevor has

He does not have enough money to buy the three light fixtures

Question 11.

Miranda has 4 times as many leaves in her collection as Joy Joy has 13 more leaves than Armani. Armani has 10 leaves in his collection. How many leaves does Miranda have in her collection? Explain.

Answer:

It is given that

Miranda has 4 times as many leaves in her collection as Joy Joy has 13 more leaves than Armani. Armani has 10 leaves in his collection

Now,

Let the number of leaves Miranda has in her collection be x

Let the number of leaves Joy has be j

Let the number of leaves Armani has be a

So,

According to the given information,

a = 10

j = a + 13

4 × j = x

Now,

j = 10 + 13

j = 23

So,

4 × 23 = x

x = 4 × (20 + 3)

x = (4 × 20) + (4 × 3)

x = 80 + 12

x = 92 leaves

Hence, from the above,

We can conclude that the number of leaves Miranda has in her collection is: 92

Question 12.

**Higher-Order Thinking** Jordan needs $9,240 for her first year of college tuition. Each of her two grandfathers said they would match what she saves. She has 8 years before she goes to college. How much does Jordan need to save on her own each year to have enough for her first year with the help from her two grandfathers?

Answer:

It is given that

Jordan needs $9,240 for her first year of college tuition. Each of her two grandfathers said they would match what she saves. She has 8 years before she goes to college

So,

The amount of money that will be saved by Jordan and her 2 grandfathers = \(\frac{$9,240}{3}\)

= $3,080

So,

The amount of money Jordan has to save each year so that she would have enough money for her first year = \(\frac{$3,080}{8}\)

= $385

Hence, from the above,

We can conclude that

Jordan needs to save on her own each year to have enough for her first year with the help from her two grandfathers will be: $385

**Assessment Practice**

Question 13.

Tina walked 20 miles for a fundraiser. Lia walked m miles. Tina walked 4 times as far as Lia. Which equation can be used to find m, the number of miles Lia walked?

A. m = 4 × 20

B. 20 = 4 × m

C. 20 = m ÷ 4

D. m = 20 + 4

Answer:

It is given that

Tina walked 20 miles for a fundraiser. Lia walked m miles. Tina walked 4 times as far as Lia

So,

According to the given information,

4 × m = 20

Hence, from the above,

We can conclude that the number of miles Lia walked is:

Question 14.

Jason and Raul kept a reading log for the year. Jason read 7 books and Raul read 105. How many times as many books as Jason did Raul read?

A. 13 times

B. 14 times

C. 15 times

D. 16 times

Answer:

It is given that

Jason and Raul kept a reading log for the year. Jason read 7 books and Raul read 105

Now,

Let the number of times Jason read as many as Raul did be x

So,

(The number of books Jason read) × (The number of times Jason read as many as Raul did) = The number of books Raul did

7 × x = 105

x = \(\frac{105}{7}\)

x = 15

Hence,from the above,

We can conclude that the number of times Raul read as many books as Jason is:

### Lesson 6.3 Model Multi-Step Problems

**Solve & Share**

Last year, 18 people went on a family camping trip. This year, three times as many people went. How many more people went this year than last year? Complete the bar diagram and show how you solve it?

I can … find hidden questions and use bar diagrams and equations to model and solve multi-step problems.

Answer:

It is given that

Last year, 18 people went on a family camping trip. This year, three times as many people went

So,

The number of people that went on a family camping trip this year = 3 × (The number of people that went on a family camping trip last year)

= 3 × 18

= 3 × (10 + 8)

= (3 × 10) + (3 × 8)

= 30 + 24

= 54 people

So,

The representation of the number of people that went on a camping trip in the form of a bar diagram is:

Hence, from the above,

We can conclude that the number of people that went on a camping trip this year is: 54

**Look Back!**

How can you use estimation to decide if your answer is reasonable?

Answer:

From the above problem,

We can observe that

The expression we used to find the number of people that went on a camping trip this year is:

3 × 18

Now,

Estimate:

3 × 18

= 3 × 20

= 3 × 2 ×10

= (3 × 2) × 10

= 6 × 10

= 60

So,

The difference between the number of people we obtained in the previous problem and in this problem is not large

Hence, from the above,

We can conclude that your estimation for the answer is reasonable

**Essential Question**

How Can You Use Diagrams and Equations to Solve Multi-Step Problems?

Answer:

The process we can use to solve Multi-step problems is:

Step 1:

Circle and underline. Circle only the necessary information and underline what ultimately needs to be figured out.

Step 2:

Figure out the first step/problem in the paragraph and solve it.

Step 3:

Find the answer by using the information from Steps 1 and 2

Visual Learning Bridge

A school uses 8 buses to transport the fourth-grade students and 6 vans to transport the third-grade students on a field trip. How many fewer third-grade students than fourth-grade students are on the field trip?

**Step 1**

Find and answer the hidden questions.

Hidden Question: How many fourth-grade students are on the field trip?

F = 8 × 24

= 192 fourth-grade students

Hidden Question: How many third-grade students are on the field trip?

T= 6 × 15

= 90 third-grade students

**Step 2**

Use the answers to the hidden questions to answer the original question.

Original Question: How many fewer third-grade students than fourth-grade students are on the field trip?

S = the number fewer third-grade students than fourth-grade students

S = 192 – 90

= 102

102 fewer third-grade students than fourth-grade students are on the field trip.

**Convince Me!**

**Construct Arguments** Amanda says she can solve the problem using the equation below. Do you agree? Explain.

S = (8 × 24) – (6 × 15)

Answer:

It is given that

Amanda says she can solve the problem using the equation below.

S = (8 × 24) – (6 × 15)

Now,

From the example on the previous page,

We have to find out the number of fewer third-grade students than the fourth-grade students

So,

We can find the difference between the third-grade and the fourth-grade students by finding the total number of third-grade and the fourth-grade students

So,

The total number of fourth-grade students = (The number of buses) × (The number of students in each bus)

= 8 × 24

The total number of third-grade students = (The number of vans) × (The number of students in each van)

= 6 × 15

So,

The difference between the fourth-grade and the third-grade students = (8 × 24) – (6 × 15)

Hence, from the above,

We can conclude that we can agree with Amanda

**Another Example!**

The Lopez family gets 2 pizzas and 4 hot dogs at Eat-It-Up. What is the total cost?

**Guided Practice**

**Do You Understand?**

Question 1.

Doug says he can use the equation T = 18 + 18 + 6 + 6 + 6 + 6 to solve the problem above. Is Doug correct? Explain.

Answer:

It is given that

Doug says he can use the equation T = 18 + 18 + 6 + 6 + 6 + 6 to solve the problem above.

Now,

From the example on the previous page,

We can observe that

The cost of each pizza is $18 and there are 2 pizzas

The cost of each hot-dog is $6 and there are 4 hot-dogs

So,

The total cost = (The total cost of pizzas) + (The total cost of hot-dogs)

= [(The cost of each pizza) × (The total number of pizzas)] + [(The cost of each hot-dog) × (The total number of hot-dogs)]

= [18 × 2] + [6 × 4]

= [18 + 18] + [6 + 6 + 6 + 6]

Hence, from the above,

We can conclude that Doug is correct

Question 2.

Write one or more equations you could use to find the total cost C, of 2 turkey wraps, 2 hamburgers, and 4 drinks.

Answer:

It is given that

There are 2 turkey wraps, 2 hamburgers, and 4 drinks

Now,

The given table is:

So,

The total cost of the three items (C) = [(The cost of each turkey wrap) × (The number of turkey wraps)] + [(The cost of each hamburger) × (The number of hamburgers)] + [(The cost of each drink) × (The number of drinks)]

= (2 × $4) + (2 × $8) + (4 × $2) ———-> 1 way of writing the equation to find the total cost

= ($4 + $4) + ($8 + $8) + ($2 + $2 + $2 + $2) —– > Other way of writing the equation to find the total cost

= $8 + $16 + $8

= $32

Hence, from the above,

We can conclude that

The total cost of all the given items is: $32

**Do You Know How?**

Question 3.

In the problem on the previous page, suppose the school fills 11 buses and lets the remaining students ride in vans. How many students ride in vans? Use one or more bar diagrams and equations to show how you solve.

Answer:

It is given that

In the problem on the previous page, the school fills 11 buses and the remaining students ride in vans.

Now,

From the example on the previous page,

The total number of students = (The number of fourth-grade students that travel in buses) + (The number of third-grade students that travel in vans)

= 192 + 90

= 282 students

Now,

We know that,

The number of students each bus carries is: 24

So,

The total number of students present in 11 buses = (The number of students present in each bus) × (The number of buses)

= 24 × 11

= 24 × (10 + 1)

= (24 × 10) + (24 × 1)

= 240 + 24

= 264 students

So,

The number of remaining students who travel in vans = (The total number of students) – (The number of students who travel in 11 buses)

= 282 – 264

= 18 students

Now,

The representation of the given problem in the form of bar diagrams is:

Hence, from the above,

We can conclude that the number of students that ride in vans is: 18

**Independent Practice**

Draw a bar diagram, and write an equation to solve the problem. Use a variable to represent an unknown quantity and tell what the variable represents.

Question 4.

Four boys and five girls went to the movies together. Between them, they had $120 to spend. Tickets cost $8 each. How much money did they have to buy refreshments?

Answer:

It is given that

Four boys and five girls went to the movies together. Between them, they had $120 to spend. Tickets cost $8 each

So,

The total number of people that went to the movies = 4 + 5

= 9

Now,

We know that,

The total money spent by 9 people = (The amount of money spent on Tickets) + (The amount of money spent to buy refreshments)

$120 = [(The total number of people) × (The cost of each ticket)] + (The amount of money spent to buy refreshments)

$120 = (9 × $8) + (The amount of money spent to buy refreshments)

The amount of money spent to buy refreshments = $120 – $72

= $48

Now,

The representation of the given information in the form of a bar diagram is:

Hence, from the above,

We can conclude that the amount of money spent on refreshments is: $48

**Problem Solving**

Question 5.

**Model with Math**

A power plant has 4 tons of coal. A ton of coal produces 2,460 kilowatt-hours of electricity. The plant reserves enough electricity to power 9 light bulbs for a year. How many additional kilowatt-hours of electricity is produced? Draw the bar diagrams and write one or more equations to show how you solve them. Tell what your variables represent.

Answer:

It is given that

A power plant has 4 tons of coal. A ton of coal produces 2,460 kilowatt-hours of electricity. The plant reserves enough electricity to power 9 light bulbs for a year

Now,

The energy produced by 4 tons of coal = (The energy produced by a ton of coal) × (The number of tons of coal)

= 2,460 × 4

(2,000 + 400 + 60) × 4

= (2,000 × 4) + (400 × 4) + (60 × 4)

= 8,000 + 1,600 + 240

= 9,280 Kilowatt-hours

Now,

The energy produced by 9 light bulbs for a year = (The energy produced by a light bulb for a year) × (The number of light bulbs)

= 876 × 9

= (800 + 70 + 6) × 9

= (800 × 9) + (70 × 9) + (6 × 9)

= 7,200 + 630 + 54

= 7,884 Kilowatt-hours

So,

The number of additional Kilowatt-hours produced = (The energy produced by 4 tons of coal) – (The energy produced by 9 light bulbs for a year)

= 9,280 – 7,884

= 1,396 Kilowatt-hours

Now,

The representation of the given information in the form of a bar diagram is:

Hence, from the above,

We can conclude that the amount of additional Kilowatt-hours produced is: 1,39 Kilowatt-hours

Question 6.

**Higher-Order Thinking** You have $360 to buy 26 tickets for a baseball game. You need to buy some of each kind of seat. You want to spend most of the money. How many of each type of ticket can you buy? Find two different solutions to the problem. Use an equation to show each solution.

Answer:

It is given that

You have $360 to buy 26 tickets for a baseball game. You need to buy some of each kind of seat. You want to spend most of the money.

Now,

The given cost for each type of seat is:

Now,

Let the number of tickets for Box seats be b

Let the number of tickets for Upper deck be u

So,

In terms of total number of tickets,

b + u = 26 —- (1)

So,

In terms of cost of the tickets,

Total cost =[ (The cost of each ticket of box seat) × (The number of tickets of box seats)] + [(The cost of each ticket of Upper deck) × (The number of tickets of Upper deck)]

18b + 12u = 360 —- (2)

So,

b = 26 – u —- (3)

Now,

By using the method of substitution,

18 (26 – u) + 12u = 350

(18 × 26) – 18u + 12u = 350

[18 × (20 + 6)] – 6u = 350

[(18 × 20) + (18 × 6)] – 6u = 350

[360 + 108] – 6u = 350

468 – 6u = 350

6u = 468 – 360

6u = 108

u = \(\frac{108}{6}\)

u = 18 tickets

So,

b = 26 -u

= 26 -18

= 8 tickets

Hence, from the above,

We can conclude that

The number of tickets of box seat is: 8

The number of tickets of Upper desk is: 18

**Assessment Practice**

Question 7.

The gym teacher has $250 to spend on volleyball equipment. She buys 4 volleyball nets for $28 each volleyball costs $7 each. How many volleyballs can she buy? Explain how you solve. Use one or more equations and bar diagrams in your explanation. Tell what your variables represent.

Answer:

It is given that

The gym teacher has $250 to spend on volleyball equipment. She buys 4 volleyball nets for $28 each volleyball costs $7 each.

Now,

Let the number of volleyballs be x

So,

The total cost of volleyball equipment = (The total cost of volleyball nets) + (The total cost of volleyballs)

$250 = [(The cost of each volleyball net) × (The number of volleyballs)] + [(The number of volleyballs) × (The cost of each volleyball)]

$250 = (4 × $28) + (x × $7)

$250 = 4 × (20 + 8) + x × 7

$250 = (4 × 20) + (4 × 8) + x× 7

$250 = 80 + 32 + 7x

$250 = 112 + 7x

7x = 250 – 112

7x = 138

x = \(\frac{138}{7}\)

x = 19.7

x ≈ 20

Now,

The representation of the given information in the form of a bar diagram is:

Hence, from the above,

We can conclude that the number of volleyballs is: 20

### Lesson 6.4 More Model MultiStep Problems

**Solve & Share**

Three eight-year-old children, one adult, and one senior had $125 saved to go to the Happy Days amusement park. After buying tickets, how much money did they have left to buy refreshments? Complete the bar diagram, and use it to write one or more equations to solve the problem. Tell what your variables represent.

I can … Solve multi-step problems by drawing bar diagrams and writing expressions and equations.

Answer:

It is given that

Three eight-year-old children, one adult, and one senior had $125 saved to go to the Happy Days amusement park

Now,

The given table is:

Now,

From the given table,

The ticket cost of 1 8-year old child is: $12

The ticket cost of 1 adult is: $25

The ticket cost of 1 senior is: $18

So,

The total amount of money saved by the given people = (The total ticket cost of the people) + (The total cost of buying refreshments for the people)

= $125 = [(3 × The ticket cost of each 8-year old child) + (The ticket cost of 1 adult) + (The ticket cost of 1 senior)] + (The total cost of buying refreshments for the people)

$125 = [(3 × $12) + $25 + $18] + (The total cost of buying refreshments for the people)

$125 = ($36 + $25 + $18) + (The total cost of buying refreshments for the people)

The total cost of buying refreshments for the people = $125 – $79

The total cost of buying refreshments for the people = $46

Now,

The representation of the given information in the form of a bar diagram is:

Hence, from the above,

We can conclude that

The total cost spent on refreshments by the given people is: $46

**Look Back!** What expressions and equations did you use to solve this problem? What hidden questions did you need to answer?

Answer:

The equation used to solve the above problem is:

The total amount of money saved by the given people = (The total ticket cost of the people) + (The total cost of buying refreshments for the people)

Now,

The hidden question we have to answer for the above problem is:

What is the total cost of the tickets for 3 8-year old children, 1 adult, and 1 senior?

**Essential Question**

How Can You Model and Solve Multi-Step Problems?

Answer:

The steps to solve Multi-steps Problems are:

Step 1:

Circle and underline. Circle only the necessary information and underline what ultimately needs to be figured out.

Step 2:

Figure out the first step/problem in the paragraph and solve it.

Step 3:

Find the answer by using the information from Steps 1 and 2

Visual Learning Bridge

Chef Angela needs 8 cartons of eggs to make the cakes that are ordered. She has 2 cartons of eggs and 4 single eggs in the refrigerator. How many more eggs does she need to make all of the cakes?

**One Way**

How many eggs does the chef have?

(2 × 12) + 4 = 24 + 4

= 28 eggs

How many more eggs does the chef need?

N = number of additional eggs needed

N = (8 × 12) – 28

= 96 – 28 = 68

The expression 8 × 12 is how many eggs the chef needs in all.

The chef needs 68 more eggs.

Since 68 is close to 72, the answer is reasonable.

**Another Way**

How many full cartons should the chef buy?

8 – 2 = 6 full cartons

How many more eggs does the chef need?

N = number of additional eggs needed

N =(6 × 12) – 4

= 72 – 4 = 68

The expression 6 × 12 is the number of eggs in 6 cartons.

The chef needs 68 more eggs.

Since 68 is close to 72, the answer is reasonable.

**Convince Me!**

**A model with Math**

Draw the bar diagrams to represent a solution to the problem above.

Answer:

The representations of the bar diagrams for the above problem are:

**Guided Practice**

**Do You Understand?**

Question 1.

Explain the meaning of the expression (2 × 12) + 4 in the problem on the previous page.

Answer:

In the problem on the previous page,

The meaning of the expression (2 × 12) + 4 is:

2 × 12:

The number of single eggs present in the 2 cartons

4:

The number of single eggs that are present separately

Hence, from the above,

We can conclude that

(2 × 12) + 4:

The total number of single eggs present for chef Angela to make a cake

Question 2.

In the problem on the previous page, how many eggs will the chef have leftover? Explain.

Answer:

In the problem on the previous page,

It is given that

The total number of cartons the chef needed is: 8 cartons

The number of cartons the chef have: 2 cartons

So,

The remaining number of cartons the chef needed = (The total number of cartons the chef needed) – (The number of cartons the chef have)

= 8 – 2

= 6 cartons

Now,

We know that,

A single carton has 12 eggs

So,

The number of eggs present in 6 cartons = 12 × 6

= 72 eggs

But,

It is given that the 4 single eggs are present

So,

From 72 eggs, only 8 single eggs are needed along with 2 cartons

Hence, from the above,

We can conclude that

The number of eggs will the chef have leftover is: 4 eggs

**Do You Know How?**

Question 3.

Carrie’s stamp book has 20 pages and each page can hold 15 stamps. She has 45 international stamps. She has 4 times as many U.S. stamps. Can all her stamps fit in her book? Use the bar diagram below to help solve this problem. Draw other bar diagrams as needed. Show the equations used to solve this problem.

Answer:

It is given that

Carrie’s stamp book has 20 pages and each page can hold 15 stamps. She has 45 international stamps. She has 4 times as many U.S. stamps

So,

The total number of stamps present in Carrie’s stamp book = (The number of pages present in Carrie’s stamp book) × (The number of stamps that each stamp can hold)

= 20 × 15

= 20 × (10 + 5)

= (20 × 10) + (20 × 5)

= 200 + 100

= 300 stamps

Now,

According to the given information,

The number of U.S stamps Carrie have = 4 × (The number of International stamps)

= 4 × 45

= 4 × (40 + 5)

= (4 × 40) + (4 × 5)

= 160 +20

= 180 U.S stamps

So,

300 > 180

Now,

The representation of the given information in the form of a bar diagram is:

Hence, from the above,

We can conclude that all of Carrie’s stamps fit in her book since the number of stamps Carrie’s book can accommodate is greater than the number of stamps Carrie have

**Independent Practice**

For 4-5, draw the bar diagrams, and write equations to solve each problem. Use variables to represent unknown quantities and tell what each variable represents.

Question 4.

Five toymakers each carved 28 blocks and 17 airplanes. Three other toymakers each carved the same number of airplanes and twice as many blocks. How many toys did the eight carve in all?

Answer:

It is given that

Five toymakers each carved 28 blocks and 17 airplanes. Three other toymakers each carved the same number of airplanes and twice as many blocks.

So,

The total number of toys the eight people carved = (The number of toys 5 toymakers carved) + (The number of toys the remaining 3 toymakers carved)

= ([The number of blocks] + [The number of airplanes]) + ([The number of airplanes as the same as 5 toymakers made] + [The number of blocks as the same as 5 toymakers made] × 2)

= (28 + 17) + (17 + [28 × 2])

= 45 + (17 + 56)

= 45 + 73

= 118 toys

Now,

The representation of the given information in the form of a bar diagram is:

Hence, from the above,

We can conclude that

The total number of toys that the eight toymakers carved are: 118 toys

Question 5.

Kendra is using 27 blue patches and some white patches to make a quilt. The quilt has a total area of 540 square inches. Each patch has an area of 9 square inches. How much of the area of the quilt is white?

Answer:

It is given that

Kendra is using 27 blue patches and some white patches to make a quilt. The quilt has a total area of 540 square inches. Each patch has an area of 9 square inches.

Now,

Let the area of the white patch be x

Now,

The total number of patches = (The number of blue patches) + (The number of white patches)

= (27 + x)

Now,

The total area of the patches (or) The total area of the quilt = (The total number of patches) × (The area of each patch)

= (27 + x) × 9 square inches

Now,

According to the given information,

(27 + x) × 9 = 540

27 + x = \(\frac{540}{9}\)

27 + x = 60

x = 60 – 27

x = 33 square inches

Now,

The representation of the given information in the form of a bar diagram is:

Hence, from the above,

We can conclude that

The area of the white patch is: 33 square inches

**Problem Solving**

Question 6.

**Make Sense and Persevere** A ticket to a movie for a student is $7. The cost for an adult is $2 more than for a student. How much would it cost 5 adults and 29 students for tickets to the movie?

Answer:

It is given that

A ticket to a movie for a student is $7. The cost for an adult is $2 more than for a student

Now,

The cost of a ticket for an adult = $2 + (The cost of a ticket for a student)

= $2 + $7

= $9

So,

The cost of the tickets for 5 adults = 5 × (The cost of a ticket for an adult)

= 5 × $9

= $45

So,

The cost of the tickets for 29 students = 29 × (The cost of a ticket for a student)

= 29 × $7

= (20 + 9) × $7

= (20 × $7) + (9 × $7)

= 140 + 63

= $203

So,

The total cost of the tickets for 5 adults and 29 students = $45 + $203

= $248

Hence, from the above,

We can conclude that the total cost of the tickets for 5 adults and 29 students to a movie is: $248

Question 7.

**Vocabulary** Give an example of an expression. Then give an example of an equation.

Answer:

An example of an expression is:

25 × 3

An example of an equation is:

20 × 3 + a × 3 = 250

Where,

‘a’ is an unknown variable

Question 8.

**Higher-Order Thinking** Cody and Max both solve the problem below correctly. Explain how each solves.

Emma has $79 to spend at the toy store. She wants to buy a building set, a board game, and 2 action figures from her favorite movie. What else can she buy?

Answer:

It is given that

Emma has $79 to spend at the toy store. She wants to buy a building set, a board game, and 2 action figures from her favorite movie

Now,

The given table is:

Now,

According to Max:

Let the money Emma has after she bought the mentioned items be x

So,

The amount of money Emma has left = (The total amount of money Emma has) – (The total amount of money spent on buying the items)

= $79 – ($32 + $19 + 2 ($8))

= $79 – ($32 + $19 + $16)

= $79 – 67

= $12

So,

From the given table,

We can observe that

For $12, we can buy a doll or an action figure

Now,

According to Cody:

The amount of money Emma have after she bought a building set = (The total amount of money Emma have) – (The amount of money spent by Emma on buying a building set)

= $79 – $32

= $47

The amount of money Emma have after she bought a board game = (The remaining amount of money Emma have after she bought a building set) – (The amount of money spent by Emma on buying a board game)

= $47 – $19

= $28

The amount of money Emma have after she bought 2 action figures = (The remaining amount of money Emma have after she bought a board game) – (The amount of money spent by Emma on buying 2 action figures)

= $28 – 2 ($8)

= $28 – $16

= $12

Hence, from the above,

We can conclude that Emma can buy a doll or an action figure

**Assessment Practice**

Question 9.

A company has 2 geothermal plants which can power a total of 2,034 homes. After they build 3 additional, more powerful, geothermal plants, they can power a total of 5,799 homes. How many homes does each of the new plants’ power? Explain how you solve. Use one or more equations and bar diagrams in your explanation. Tell what your variables represent.

Answer:

It is given that

A company has 2 geothermal plants which can power a total of 2,034 homes. After they build 3 additional, more powerful, geothermal plants, they can power a total of 5,799 homes

So,

The total number of geothermal plants = 2 + 3

= 5 geothermal plants

The number of homes that can be powered by all the geothermal plants =(The number of homes powered by 2 geothermal plants) + (The number of homes powered by 3 geothermal plants)

= 2,034 + 5,799

= 7,833 homes

So,

The number of homes powered by each geothermal plant = \(\frac{The number of homes powered by all the geothermal plants}{5}\)

= \(\frac{7,833}{5}\)

= 1,766.5

≅ 1,777 homes

Now,

The representation of the given information in the form of a bar diagram is:

Hence, from the above,

We can conclude that the number of homes provided by each geothermal plant is: 1,777 homes

### Lesson 6.5 Solve Multi-Step Problems

**Solve & Share**

A farmer needs $3,500 to buy a previously owned truck. If she sells 45 maple trees and 27 pine trees, will she earn enough to buy the truck? If not, how much more money does she need? Solve this problem any way you choose.

I can … solve multi-step problems by finding and solving hidden questions and by writing expressions and equations.

Answer:

It is given that

A farmer needs $3,500 to buy a previously owned truck and she sells 45 maple trees and 27 pine trees

Now,

The given table is:

Now,

The amount earned by selling 45 maple trees = (The number of maple trees) × (The cost of each maple tree)

= 45 × $56

= (40 + 5) × (50 + 6)

= (40 × 50) + (40 × 6) + (5 × 50) + (5 × 6)

= 2,000 + 240 + 250 + 30

= $2,520

Now,

The amount earned by selling 27 pine trees = (The number of pine trees) × (The cost of each pine tree)

= 27 × $33

= (20 + 7) × (30 + 3)

= (20 × 30) + (20 × 3) + (7 × 30) + (7 × 3)

= 600 + 60 + 210 + 21

= $891

So,

The total amount of money earned by selling 45 maple trees and 27 pine trees = $2,520 + $891

= $3,411

But,

A farmer needs $3,500

Hence, from the above,

We can conclude that she will not earn enough money to buy the truck

**Look Back!** In the question above, how many more of each tree could the farmer sell to get enough money to buy the truck? Explain.

Answer:

From the above problem,

We can observe that

The total amount of money earned by selling 45 maple trees and 27 pine trees is: $3,411

She will need $3,500

Now,

The extra money she have = $3,500 – $3,411

= $189

Hence, from the above,

We can conclude that

With the remaining money,

She can buy

The number of maple trees she can buy are: 2

The number of pine trees she can buy are: 2

**Essential Question**

How Can You Use Equations to Solve Multi-Step Problems?

Answer:

The steps to solve Multi-Step problems using equations are:

a. Multiply to clear any fractions or decimals.

b. Simplify each side by clearing parentheses and combining like terms.

c. Add or subtract to isolate the variable term—you may have to move a term with the variable.

d. Multiply or divide to isolate the variable.

e. Check the solution.

Visual Learning Bridge

The students in the fourth and fifth grades are going to a concert. There are 178 students. How many rows are needed for the fourth graders?

**Step 1**

Find and solve the first hidden question.

Hidden Question: How many fifth graders are there?

12 × 8 = s

S = 96

There are 96 fifth graders.

**Step 2**

Find and solve the second hidden question.

Hidden Question: How many fourth graders are there?

178 – 96 = f

f = 82

There are 82 fourth graders.

**Step 3**

Answer the original question.

Original Question: How many rows are needed for the fourth graders?

82 ÷ 8 = r

r = 10 R2

Ten rows will be filled with 2 fourth graders left over. So, 11 rows will be needed.

**Convince Me!**

**Construct Arguments** Does the answer of 11 rows makes sense for the problem above? Explain.

Answer:

From the above problem,

We can observe that

The number of rows needed for the fourth graders is: 10 R 2

Now,

The meaning of 10 R 2 in the given situation is:

10 rows with the remainder 2 seats and we have to arrange that remainder in the next row

So,

The total number of rows is: 11

Hence, from the above,

We can conclude that the answer of 11 rows makes sense for the above problem

**Guided Practice**

**Do You Understand?**

Question 1.

On the previous page, suppose there were only 11 rows of fifth-graders, but the same total number of students. Do you need to solve the whole problem again to find how many rows are needed for the fourth graders? Explain.

Answer:

It is given that

On the previous page, there were only 11 rows of fifth-graders, but the same total number of students

Now,

In the example on the previous page,

It is given that

There are 8 students in each row and the total number of students are 178

So,

The number of rows needed for the fourth-grade students = [(The total number of students) – (The number of rows needed for the fifth-graders) × (The number of students in each row)] ÷ 8

Hence, from the above,

We can conclude that

We need to solve the whole problem again to find the number of rows needed for the fourth-graders

**Do You Know How?**

Question 2.

Show another way to solve the problem on the previous page.

Answer:

The given figure is:

Now,

In the previous example on the previous page,

It is given that

There are a total of 178 students

Now,

The number of students in the fifth-grade = (The number of rows filled by the fifth-graders) × (The number of students in each row)

= 12 × 8

= (10 + 2) × 8

= (10 × 8) + (2 × 8)

= 80 + 16

= 96 students

So,

The number of students in the fourth-grade = (The total number of students) – (The number of students in the fifth grade)

= 178 – 96

= 82 students

So,

The number of rows filled by the fourth-graders = \(\frac{The number of students in the fourth-grade}{8}\)

= \(\frac{82}{8}\)

= 10 R 2

Hence,from the above,

We can conclude that the number of rows filled by the fourth-grade students is: 11

**Independent Practice**

For 3-4, solve each multi-step problem. Write equations to show how you solve. Draw the bar diagrams to help if needed. Use estimation to decide if your answer is reasonable.

Question 3.

Vanya bought 5 medium packages of buttons and 3 small packages of buttons. What was the total number of buttons Vanya bought?

Answer:

The given data is:

So,

The total number of buttons Vanya bought = (The number of buttons present in 5 medium packages of buttons) + (The number of buttons present in 3 small packages of buttons)

= (5 × The number of buttons present in a medium package of buttons) + (3 × The number of buttons present in a small package of buttons)

= (5 × 38) + (3 × 18)

= 5 × (30 + 8) + 3 × (10 + 8)

= (5 × 30) + (5 × 8) + (3 × 10) + (3 × 8)

= 150 + 40 + 30 + 24

= 244 buttons

Hence, from the above,

We can conclude that

The total number of buttons Vanya bought is: 244 buttons

Question 4.

Vance bought 2 packages of large beads, 1 package of medium beads, 2 packages of large buttons, and 2 packages of medium buttons. How many more beads than buttons did Vance buy?

Answer:

The given data is:

So,

The total number of beads Vance bought = (The total number of beads present in 2 packages of large beads) + (The total number of beads present in 1 package of medium beads)

= (2 × 96) + (1 × 64)

= 2 × (90 + 6) + 64

= (2 × 90) + (2 × 6) + 64

= 180 + 12 + 64

= 256 beads

So,

The total number of buttons Vance bought = (The total number of buttons present in 2 packages of large buttons) + (The total numbr of buttons present in 2 packages of medium buttons)

= (2 × 56) + (2 × 38)

= 2 × (50 + 6) + 2 × (30 + 8)

= (2 × 50) + (2 × 6) + (2 × 30) + (2 × 8)

= 100 + 12 + 600 + 16

= 728 buttons

So,

The difference between the number of buttons and beads = 728 – 256

= 472 beads

Hence, from the above,

We can conclude that there are 472 more buttons than beads that Vance bought

**Problem Solving**

Question 5.

en Vision® STEM How much more does it cost to generate 9 megawatt-hours of electricity with conventional coal than with wind energy? Write one or more equations to show how you solve. Tell what your variables represent.

Answer:

The given figure is:

Now,

The cost to generate 9 megawatt hours of electricity with wind energy = 9 × (The cost to generate 1 megawatt hours of electricity with wind energy)

= 9 × $87

= 9 × (80 + 7)

= (9 × 80) + (9 × 7)

= 720 + 63

= $783

Now,

The cost to generate 9 megawatt hours of electricity with conventional coal = 9 × (The cost to generate 1 megawatt hours of electricity with conventional coal)

= 9 × $105

= 9 × (100 + 5)

= (9 × 100) + (9 × 5)

= 900 + 45

= $945

So,

The difference between the energy produced with conventional coal and the wind energy = $945 – $783

= $162

Hence, from the above,

We can conclude that

It takes $162 more to generate 9 megawatt-hours of electricity with conventional coal than the wind energy

Question 6.

**A model with Math** Anna earns $8 an hour babysitting and $6 an hour working in the garden. Last month, she worked 15 hours babysitting and 8 hours in the garden. How much more money does she need to buy a robot which costs $199? Explain how you solve. Use one or more equations in your explanation. Tell what your variables represent.

Answer:

It is given that

Anna earns $8 an hour babysitting and $6 an hour working in the garden. Last month, she worked 15 hours babysitting and 8 hours in the garden.

So,

The amount of money earned by Anna for babysitting = (The number of hours Anna did babysitting) × (The amount of money earned by Anna for 1 hour of babysitting)

= 15 × $8

= (10 + 5) × $8

= (10 × 8) + (5 × 8)

= 80 + 40

= $120

So,

The amount of money earned by Anna for working in the garden = (The number of hours Anna did work in the garden) × (The amount of money earned by Anna for 1 hour working in the garden)

= 8 × $6

= $48

So,

The total amount of money earned by Anna = $120 + $48

= $168

But,

She needed to buy a robot which costs $199

So,

The amount of more money she needed to buy a robot = $199 – $168

= $31

Hence, from the above,

We can conclude that

She needs $31 more to buy a robot which costs $199

Question 7.

**Higher-Order Thinking** Show two different ways to find the answer to the problem below.

Dog and cat food are sold in 20-pound bags. There are 14 bags of dog food and 12 bags of cat food on the store shelves. How many pounds of dog and cat food is on the shelves?

Answer:

It is given that

Dog and cat food are sold in 20-pound bags. There are 14 bags of dog food and 12 bags of cat food on the store shelves

So,

The number of pounds of dog food = (The number of bags of dog food) × (The number of pounds each bag contains)

= 14 × 20

= (10 + 4) × 20

= (10 × 20) + (4 × 20)

= 200 + 80

= 280 pounds

The number of pounds of cat food = (The number of bags of cat food) × (The number of pounds each bag consists)

= 12 × 20

= (10 + 2) × 20

= (10 × 20) + (2 × 20)

= 200 + 40

= 240 pounds

Hence, from the above,

We can conclude that

The number of pounds of dog food present on the shelves is: 280 pounds

The number of pounds of cat food present on the shelves is: 240 pounds

**Assessment Practice**

Question 8.

Chris needs $858 to buy a computer. She has already saved $575. She gets $15 an hour for babysitting and will babysit 12 hours in the next month. She can save $8 a week from her allowance. How many weeks of allowance will it take Chris to save enough to buy the computer? Explain how you solve. Use one or more equations in your explanation. Tell what your variables represent.

Answer:

It is given that

Chris needs $858 to buy a computer. She has already saved $575. She gets $15 an hour for babysitting and will babysit 12 hours in the next month. She can save $8 a week from her allowance

So,

The amount of money she required to buy a computer = (The amount of money Chris needs to buy a computer) – (The amount of money she already saved)

= $858 – $575

= $283

Now,

The amount of money she will earn for babysitting = (The number of hours she did babysitting) × (The amount of money she gets for babysitting for an hour)

= 12 × $15

= (10 + 2) × $15

= (10 × 15) + (2 × 15)

= 150 + 30

= $180

So,

The remaining money she will need to buy a computer = $283 – $180

= $103

Now,

The amount of money she earned from allowance = (The number of weeks) × (The amount of money she will earn for a week as an allowance)

$103 = x × $8

x = \(\frac{103}{8}\)

x = 12.8

x ≈ 13 weeks

Hence, from the above,

We can conclude that

About 13 weeks of allowance will it take Chris to save enough to buy the computer

### Lesson 6.6 Problem Solving

**Make Sense and Persevere**

**Solve & Share**

Ms. Valenzuela had her students design a snake house for the zoo. In the design shown, the anaconda has 538 more square feet than the python. The python has twice as many square feet as the rattlesnake. How much of the house’s area does each snake have? Solve this problem any way you choose.

I can … make sense of problems and keep working if I get stuck.

Thinking Habits

Be a good thinker! These questions can help you.

- What do I need to find?
- What do I know?
- 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?

Answer:

It is given that

Ms. Valenzuela had her students design a snake house for the zoo. In the design shown, the anaconda has 538 more square feet than the python. The python has twice as many square feet as the rattlesnake

Now,

The given figure is:

Now,

Let the area of the rattlesnake be r square feet

Let the area of the python be p square feet

So,

The area of the python = 2 × (The area of the rattlesnake)

= 2r square feet

Now,

According to the given information,

(The area of the anaconda) – (The area of the python) = 538

So,

1,928 – p = 538

p = 1,928 – 538

p = 1,390 square feet

So,

The area of the rattlesnake = \(\frac{The area of the python (p)}{2}\)

= \(\frac{1,390}{2}\)

= 695 square feet

Hence, from the above,

We can conclude that

The area of an anaconda is: 1,928 square feet

The area of python is: 1,390 square feet

The area of the rattlesnake is: 695 square feet

**Look Back!**

**Make Sense and Persevere** How can you check that your solution makes sense?

Answer:

To check whether the given solution makes sense or not,

Estimate the solution

Now,

By the process of Estimation,

The area of the anaconda is: 1,900 square feet

So,

(The area of the anaconda) – (The area of python) = 540

The area of python = 1,900 – 540

= 1,360 square feet

So,

The area of rattlesnake = \(\frac{The area of python}{2}\)

= \(\frac{1,360}{2}\)

= 685 square feet

So,

The area of the anaconda is: 1,900 square feet

The area of python is: 1,360 square feet

The area of the rattlesnake is: 685 square feet

Now,

By observing the answers in the above problem and the answers for this problem,

We can say that your answer is reasonable

Hence, from the above,

We can conclude that your solution makes sense

**Essential Question**

How Do You Make Sense of a Multi-Step Problem and Persevere in Solving It?

Answer:

The steps to make sense of a Multi-Step Problem and Persevere in Solving it are:

a. Identify the Problem. Begin by determining the scenario the problem wants you to solve.

b. Gather Information.

c. Create an Equation.

d. Solve the Problem.

e. Verify the Answer.

Visual Learning Bridge

Bryan and Alex have to buy their own instruments for the band. Alex made $1,025 from a fundraiser. He has a part-time job that pays $8 an hour. How many hours does Alex need to work to have enough money to buy his instrument?

What do you need to do to solve this problem?

I need to determine how much Alex’s tuba costs and how much Alex needs to earn to buy the tuba.

How can I make sense of and solve this 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.

Here’s my thinking

Use the cost of the trumpet to find the cost of the tuba.

$159 x 9 = $1,431 Alex’s tuba costs $1,431.

Find how much more money Alex needs to earn.

$1,431 – $1,025 = $406 Alex needs to earn $406.

Find how many hours Alex needs to work.

$406 ÷ 8 = 50 R6

Alex will not have enough money by working 50 hours, so he needs to work 51 hours.

**Convince Me!**

**Make Sense and Persevere** How can you check to make sure the work and answer given above make sense?

Answer:

To check whether the given answer is reasonable or not,

Estimate the answer by using Estimation methods

Now,

The given figure is:

Now,

From the given figure,

We can observe that

The cost of Bryan’s trumpet is about $160

So,

The cost of Alex’s tuba = 9 × (The cost of Bryan’s trumpet)

= 9 × $160

= 9 × (100 + 60)

= (9 × 100) + (9 × 60)

= 900 + 540

= $1,440

Now,

The more money Alex must have to buy his instrument = (The cost of Alex’s tuba) – (The money raised by fundraiser)

= $1,440 – $1,025

= $415

Now,

The number of hours needed for Alex to earn more money = \(\frac{The more money Alex must have to buy his instrument}{8}\)

= \(\frac{$415}{8}\)

= 51 R 7

Hence, from the above,

We can conclude that Alex needs about 7 hours for Alex to earn more money

Your solution is reasonable since the difference between the number of hours in the above example and this problem is not large

**Guided Practice**

**Make Sense and Persevere** In the problem on the previous page, suppose Alex wanted to know how many weeks it would take him to work

Think about 51 hours. Alex works 3 hours a day and 4 days a week. this question to help you

Question 1.

What are you asked to find and what is the hidden question What’s a good plan for solving question?

Answer:

The hidden question to answer is:

How many weeks it would take him to work?

Now,

A good plan for solving the given question is:

a. The number of hours in a day that Alex works

b. The number of days in a week that Alex works

c. The number of weeks Alex will take to complete the work

Question 2.

Write and solve equations to solve the problem. Be sure to tell what each variable represents.

Answer:

It is given that

Alex works 3 hours a day and 4 days a week

So,

The number of hours in a week that Alex will work = (The number of hours a day) × (The number of days in a week that Alex will work)

= 3 × 4

= 12 hours a week

So,

The number of weeks Alex will need to complete the work = \(\frac{The total number of hore Alex completed the week}{The number of hours in a week that Alex will work}\)

= \(\frac{51}{12}\)

= 4 R 3

Hence, from the above,

We can conclude that Alex will need about 4 weeks to complete the work

Question 3.

Does your answer make sense? Explain.

Answer:

To make whether your answer makes sense or not,

Estimate the solution using the Estimation methods

Now,

The number of hours in a week that Alex will work = (The number of hours a day) × (The number of days in a week that Alex will work)

= 3 × 4

= 12 hours a week

So,

The number of hours in a week that Alex will work is about 10 hours

So,

The number of weeks Alex will need to complete the work = \(\frac{The total number of hore Alex completed the week}{The number of hours in a week that Alex will work}\)

= \(\frac{51}{10}\)

= 5 R 1

So,

We can conclude that Alex will need about 5 weeks to complete the work

Hence, from the above,

We can conclude that your answer is reasonable since the difference between the actual answer and the estimated answer is not large

**Independent Practice**

**Make Sense and Persevere** The high school tennis team is selling tennis balls to raise $500 for new equipment. They sell the balls for $2 each. Will they make enough money if they sell 4 cases?

Question 4.

What are the hidden questions? Write equations to solve each.

Answer:

It is given that

The high school tennis team is selling tennis balls to raise $500 for new equipment. They sell the balls for $2 each

So,

The hidden question for the given problem is:

What is the cost of each case?

Now,

The cost of each case = (The number of packages in each case) × (The number of tennis balls in each package) × (The cost of each tennis ball)

= 24 × 3 × $2

= 24 × $6

= (20 + 4) × 6

= (20 × 6) + (6 × 4)

= 120 + 24

= $144

So,

The cost of 4 cases = 4 × (The cost of each case)

= 4 × $144

= 4 × (100 + 40 + 4)

= (4 × 100) + (4 × 40) + (4 × 4)

= 400 + 160 + 16

= $576

But,

It is given that the high school tennis team is selling tennis balls to raise $500 for new equipment

Hence, from the above,

We can conclude that they will not make enough money if they sell 4 cases

Question 5.

Will the team makes enough money? Explain.

Answer:

From the above problem,

We can observe that

The high school tennis team is selling tennis balls to raise $500 for new equipment

The cost of tennis balls present in 4 cases is: $576

So,

$500 < $576

Hence, from the above,

We can conclude that the team will not make enough money

**Problem Solving**

**Performance Task**

**Designing a Flag** Rainey’s group designed the flag shown for a class project. They used 234 square inches of green fabric. After making one flag, Rainey’s group has 35 square inches of yellow fabric left. How can Rainey’s group determine the total area of the flag?

Question 6.

**Make Sense and Persevere** What hidden question(s) do you need to answer first?

Answer:

It is given that

Rainey’s group designed the flag shown for a class project. They used 234 square inches of green fabric. After making one flag, Rainey’s group has 35 square inches of yellow fabric left

Now,

The hidden questions you need to answer first are:

a. Find the remaining colors present in the flag?

b. What is the area of the remaining colors in the flag?

c. What is the total area of the flag?

Question 7.

**Appropriate Tools** Draw diagrams and write equations to represent the hidden question(s). Be sure to tell what each variable represents.

Answer:

The given figure is:

Now,

It is given that

They used 234 square inches of green fabric

So,

The area of the orange fabric = 2 × (The area of the green fabric)

= 2 × 234

= 2 × (200 + 30 + 4)

= (2 × 200) + (2 × 30) + (2 × 4)

= 400 + 60 + 8

= 468 square inches

So,

The area of the yellow fabric = 3 × (The arae of the green fabric)

= 3 × 234

= (2 × 234) + (1 × 234)

= 468 + 234

= 702 square inches

But,

It is given that Rainey’s group has 35 square inches of yellow fabric left

So,

The area of the yellow fabric = 702 – 35

= 667 square inches

So,

The total area of the flag = (The area of the green flag) + (The area of the orange flag) + (The area of the yellow flag)

= 234 + 702 + 667

= 1,603 square inches

Hence, from the above,

We can conclude that

The total area of the flag is: 1,603 square inches

Question 8.

**Be Precise** Use your drawings and equations to find the total area of the flag. Explain carefully, using the correct units.

Answer:

From the above problem,

We can observe that

The total area of the flag can be found out by adding the areas of the three colors

So,

The total area of the flag = (The area of the green flag) + (The area of the orange flag) + (The area of the yellow flag)

= 234 + 702 + 667

= 1,603 square inches

Hence, from the above,

We can conclude that

The total area of the flag is: 1,603 square inches

Question 9.

**Make Sense and Persevere** What information was not needed to solve the problem?

Answer:

There is not any unnecessary information present that is not used to solve the given problem

### Topic 6 Fluency Practice 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 next to the match. Find a match for every clue.

I can … subtract multi-digit whole numbers.

Clues

A. The difference is exactly 528.

B. The difference is between 550 and 560.

C. The difference is between 800 and 900.

D. The difference is exactly 614.

E. The difference is between 100 and 105.

F. The difference is between 470 and 480.

G. The difference is exactly 392.

H. The difference is between 70 and 80.

Answer:

### Topic 6 Vocabulary Review

Understand Vocabulary

Word List

- addition comparison
- Associative Property of Multiplication
- Commutative Property of Multiplication
- Distributive Property of Multiplication
- equation
- multiplication comparison
- product
- variable

Write T for true and F for false.

Question 1.

__________ Addition comparison is used when you can multiply to find how one quantity is related to another.

Answer:

We know that,

Additive comparisons focus on the difference between two quantities

Hence, from the above,

We can conclude that the given statement is: False

Question 2.

___________ Multiplication comparison is used when one quantity is x times more than another quantity.

Answer:

We know that,

In multiplicative comparison, there are two different sets being compared. The first set contains a certain number of items. The second set contains multiple copies of the first set.

Hence, from the above,

We can conclude that the given statement is: False

Question 3.

__________ A number sentence that uses an equal sign to show that two expressions have the same value is called an equation.

Answer:

We know that,

A number sentence that uses an equal sign to show that two expressions have the same value is called an “Equation”

Hence, from the above,

We can conclude that the given statement is: True

Question 4.

___________ The answer to a subtraction problem is called the product.

Answer:

We know that,

The answer to a multiplication problem is called the “Product”

Hence, from the above,

We can conclude that the given statement is: True

Question 5.

_________ A symbol or letter that stands for a number is called a variable.

Answer:

We know that,

A “Variable” is a symbol that works as a placeholder for expression or quantities that may vary or change

Hence, from the above,

We can conclude that the given statement is: True

Label each example with a term from the Word List.

Question 6.

(3 × 4) × 5 = 3 (4 × 5) _______________

Answer:

We know that,

According to Associative Property of Multiplication,

a × (b × c) = (a × b) × c

So,

we can observe that

The given expression is also in the same form

Hence, from the above,

We can conclude that the term that is suitable for the given problem is: Associative Property of Multiplication

Question 7.

3 × (4 + 5) = (3 × 4) + (3 × 5) _______________

Answer:

We know that,

According to the Distributive Property of Multiplication,

a × (b + c) = (a × b) + (a × c)

So,

we can observe that

The given expression is also in the same form

Hence, from the above,

We can conclude that the term that is suitable for the given problem is: Distributive Property of Multiplication

Question 8.

3 × 4 × 5 = 4 × 3 × 5 _______________

Answer:

We know that,

According to the Commutative Property of Multiplication,

a × b = b × a

So,

We can observe that

The given expression is also in the same form

Hence, from the above,

We can conclude that the term that is suitable for the given problem is: Commutative Property of Multiplication

**Use Vocabulary in Writing**

Question 9.

Seth wrote and solved the following comparison:

Find 6 times as many as 5.

6 × 5 = n

n = 30

Use at least 3 terms from the Word List to describe Seth’s comparison.

Answer:

It is given that

Find 6 times as many as 5 ———-> Multiplication comparison

So,

The given sentence can be written as:

6 × 5 = n ——-> Equation

n = 30 ——-> Product

Where,

n ——> variable

### Topic 6 Reteaching

**Set A pages 225-232**

Write an equation for each comparison. Find the value of the variable that makes the equation true.

k is 9 times as many as 3.

k = 9 × 3

k = 27

m is 6 more than 21.

m = 6 + 21

m = 27

There are 30 apples and 6 bananas in a basket. How many times as many apples as bananas are in the basket?

Let t = the number of times as many apples as bananas.

t times as many as 6 is 30.

t × 6 = 30

Since 5 × 6 = 30, there are 5 times as many apples as bananas in the basket.

**Remember** to use addition or subtraction when you know how much more, and multiplication or division when you know how many times as many.

Write and solve an equation to match each comparison.

Question 1.

x is 21 more than 21.

Answer:

It is given that

x is 21 more than 21

So,

According to the Property of Addition comparison,

x = 21 + 21

x = 42

Hence, from the above,

We can conclude that the value of x is: 42

Question 2.

Macon has 32 rocks in his collection. He has 4 times as many rocks as his brother. How many rocks, r, does Macon’s brother have?

Answer:

It is given that

Macon has 32 rocks in his collection. He has 4 times as many rocks as his brother

Now,

Let the number of rocks Macon’s brother has to be: r

So,

According to the given information,

r × 4 = 32

r = \(\frac{32}{4}\)

r = 8 rocks

Hence from the above,

We can conclude that Macon’s brother has 8 rocks

Question 3.

Pam has 24 pencils and 6 erasers. How many times, t, as many pencils as erasers does Pam have?

Answer:

It is given that

Pam has 24 pencils and 6 erasers

Now,

Let the number of erasers be: x

So,

According to the given information,

x × 6 = 24

x = \(\frac{24}{6}\)

x = 4 erasers

Hence, from the above,

We can conclude that

Pam has the number of pencils 4 times as many as erasers

**Set B pages 233-240**

At a restaurant, a children’s meal costs $5, and an adult meal costs $9. Four children and 2 adults order meals. The family has a $25 gift certificate to use. How much will their total be before tax?

**Remember** to first find and answer any hidden questions.

Question 1.

There are 64 singers in the choir. The tenors and sopranos are in separate rows. There are 8 singers in each row. There are 4 rows of tenors. How many rows of sopranos are there?

Answer:

It is given that

There are 64 singers in the choir. The tenors and sopranos are in separate rows. There are 8 singers in each row. There are 4 rows of tenors.

Now,

Let the number of rows of sopranos be x

So,

The total number of rows = (The number of rows of sopranos) + (The number of rows of tenors)

= (x + 4)

Now,

According to the given information,

The total number of rows = \(\frac{The total number of singers}{The number of singers in each row}\)

= \(\frac{64}{8}\)

= 8

So,

x + 4 = 8

x = 8 – 4

x = 4 rows

Now,

The representation of the given information in the form of a bar diagram is:

Hence, from the above,

We can conclude that

The total number of rows of sopranos is: 4 rows

Question 2.

Samantha has $600 saved for a trip. She buys an airline ticket for $120 and reserves a hotel room for $55 each night for 4 nights. If Samantha’s trip lasts for 5 days and she spends the same amount each day, how much can Samantha spend each day?

Answer:

It is given that

Samantha has $600 saved for a trip. She buys an airline ticket for $120 and reserves a hotel room for $55 each night for 4 nights and Samantha’s trip lasts for 5 days and she spends the same amount each day

Now,

The amount of money Spent by Samantha = (The amount of money spent to buy an airline ticket) + (The amount of money spent to reserve a hotel room for 4 nights)

= $120 + ($55 × 4)

= $120 + (50 + 5) × 4

= 120 + (50 × 4) + (5 × 4)

= 120 + 200 + 20

= $120 + $220

= $340

So,

The amount of money spent by Samantha for 5 days = $340 × 5

= (300 + 40) × 5

= (300 × 5) + (40 × 5)

= 1,500 + 200

= $1,700

But,

Samantha can buy the airline ticket only once

So,

The amount of money Samantha spent for 5 days = $1,700 – $120

= $580

So,

The amount of money spent by Samantha each day = \(\frac{$580}{5}[latex]

= $116

Hence, from the above,

We can conclude that the amount of money spent by Samantha each day is: $116

**Set C pages 241-244**

There are 13 girls and 14 boys signed up to join volleyball. Each team needs 6 players. How many teams can be formed, and how many more players are needed to form one more team?

Find the total number of players signed up.

13 + 14 = p

27 = p

Divide to find the number of teams that can be formed.

27 ÷ 6 = t

4 R 3 = t

There are 3 students left not on a team. Subtract to find the number of players still needed to make a team.

6 – 3 = p

3 = p

4 teams can be formed. 3 more players are needed to make one more team.

**Remember** to draw the bar diagrams to help if needed.

Solve each multi-step problem. Write equations to show how you solve.

Question 1.

Kevin made $96 in necklace sales and half that amount in bracelet sales. How much money did Kevin make in necklace and bracelet sales?

Answer:

It is given that

Kevin made $96 in necklace sales and half that amount in bracelet sales

So,

The amount made by Kevin in bracelet sales = (The amount made by Kevin in necklace sales) ÷ 2

= [latex]\frac{96}{2}\)

= $48

So,

The amount of money made by Kevin in necklace and bracelet sales = (The amount of money made by Kevin in necklace sales) + (The amount of money made by Kevin in bracelet sales)

= $96 + $48

= $144

Hence, from the above,

We can conclude that

The amount made by Kevin in necklace and bracelet sales is: $144

Question 2.

A dog and cat rescue use 40 pounds of dog food and 15 pounds of cat food to feed its animals each day. How many pounds of dog and cat food do they use in seven days?

Answer:

It is given that

A dog and cat rescue use 40 pounds of dog food and 15 pounds of cat food to feed its animals each day

So,

The number of pounds of dog and cat food they use each day = (The number of pounds of dog food) + (The number of pounds of cat food)

= 40 + 15

= 55 pounds

So,

The number of pounds of dog and cat food they use in 7 days = 7 × (The number of pounds of dog and cat food they use each day)

= 7 × 55

= 7 × (50 + 5)

= (7 × 50) + (7 × 5)

= 350 + 35

= 385 pounds

Hence, from the above,

We can conclude that

The number of pounds of dog and cat food they use in 7 days is: 385 pounds

**Set D pages 245-248**

Think about these questions to help you make sense of the problem.

Thinking Habits

- What do I need to find?
- What do I know?
- 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?

**Remember** to make sense of the problem before starting to solve it.

At a local shelter, each large dog can be adopted for $10 and each small dog for $5. There are 17 large dogs up for adoption. If all the dogs are adopted, the shelter will make $215. How many small dogs were at the shelter?

Question 1.

Find the hidden questions. Write equations to solve each.

Answer:

It is given that

At a local shelter, each large dog can be adopted for $10 and each small dog for $5. There are 17 large dogs up for adoption. If all the dogs are adopted, the shelter will make $215.

So,

The hidden questions for the given problem are:

a. What is the cost for the adoption of 17 large dogs?

b. What is the cost for the adoption of the small dogs?

Question 2.

How many small dogs are at the shelter?

Answer:

It is given that

At a local shelter, each large dog can be adopted for $10 and each small dog for $5. There are 17 large dogs up for adoption. If all the dogs are adopted, the shelter will make $215.

Now,

Let the number of small dogs be x

So,

The total cost for the adoption of all dogs = (The total cost for the adoption of large dogs) + (The total cost for the adoption of small dogs)

$215 = [(The number of large dogs) × (The cost for the adoption of each large dog)] + [(The number of small dogs) × (The cost for the adoption of each small dog)]

$215 = (17 × $10) + (x × $5)

$215 = (17 × 1 × 10) + (x × $5)

$215 = $170 + (x × $5)

x × $5 = $215 – $170

x × $5 = $45

x = \(\frac{45}{5}\)

x = 9

Hence, from the above,

We can conclude that the total number of small dogs is: 9 small dogs

### Topic 6 Assessment Practice

Question 1.

Jason and his 3 brothers want to buy a gift for their mother. They have $314 saved. Each of them will save $17 a week until they have at least $515 for her gift. How much money will they save after 3 weeks? Will they have enough money to buy the gift?

A. What are the hidden questions?

Answer:

It is given that

Jason and his 3 brothers want to buy a gift for their mother. They have $314 saved. Each of them will save $17 a week until they have at least $515 for her gift

Hence,

The hidden questions for the given problem are:

a. How much money Jason and his 3 brothers will save each week?

b. How much amount of money do Jason and his brothers need to save?

B. Write an equation that can be used to answer each hidden question. Then solve.

Answer:

It is given that

Jason and his 3 brothers want to buy a gift for their mother. They have $314 saved. Each of them will save $17 a week until they have at least $515 for her gift

Now,

The amount of money jason and his 3 brothers will save each week = 4 × $17

= 4 × (10 + 7)

= (4 × 10) + (4 × 7)

= 40 + 28

= $68

Now,

The amount of enough money Jason and his 3 brothers will have to save = $515 – $314

= $201

Hence, from the above,

We can conclude that

The amount of money Jason and his 3 brothers will save each week is: $68

The amount of enough money Jason and his 3 brothers will have to save is: $201

C. Write and solve an equation to find how much money they will save after 3 weeks. Will they have enough to buy the gift? Explain.

Answer:

From part (a),

We can observe that

The amount of money Jason and his 3 brothers will save each week is: $68

So,

The amount of money Jason and his 3 brothers will save in 3 weeks = 3 × $68

= 3 × (60 + 8)

= (3 × 60) + (3 × 8)

= 180 + 24

= $204

So,

The amount of money Jason and his 3 brothers will save after 3 weeks = $204 – $201

= $3

So,

The amount of money Jason and his 3 brothers will have = $515 + $3

= $518

Hence, from the above,

We can conclude that Jason and his 3 brothers will have enough money to buy the gift

Question 2.

Marco ordered 2 reclining chairs for $230 each and a coffee table for $350. The shipping cost is $100. Marco is going to pay off his bill in 5 equal payments. How much is each of Marco’s payments?

Answer:

It is given that

Marco ordered 2 reclining chairs for $230 each and a coffee table for $350. The shipping cost is $100. Marco is going to pay off his bill in 5 equal payments

So,

The total payment made by Marco = (The payment made by Marco for 2 reclining chairs) + (The payment made by Marco for a coffee table) + (The shipping cost)

= (2 × 230) + 350 + 100

= [(2 × 23) × 10] + 450

= 460 + 450

= $910

Now,

The payment made by Marco in each equal installment = \(\frac{$910}{5}\)

= $182

Hence, from the above,

We can conclude that each of Marco’s payments is: $182

Question 3.

Mitchell wants to beat the record for the most points scored in a season. This season, he has scored 51 points. If he scores 27 points at each of the next 7 games, he will break the record by 1 point. How many total points will break the record by 1 point?

Mitchell will break the record by 1 point if he scores a total of ________ points.

Answer:

It is given that

Mitchell wants to beat the record for the most points scored in a season. This season, he has scored 51 points. and if he scores 27 points at each of the next 7 games, he will break the record by 1 point.

Now,

The number of points scored by Mitchell in the next 7 games = 27 × 7

= (20 + 7) × 7

= (20 × 7) + (7 × 7)

= 140 + 49

= 189 points

So,

The total number of points scored by Mitchell in this season = 189 + 51

= 240 points

So,

The representation of the given situation in the form of a bar diagram is:

Hence, from the above,

We can conclude that

Mitchell will break the record by 1 point if he scores a total of 240 points.

Question 4.

Select all the sentences that describe a comparison using multiplication.

☐ 9 is 3 times as many as p.

☐ 27 more than r is 41.

☐ A bus can travel 3 times as fast as a boat.

☐ It costs d dollars for 9 packages.

☐ There are 4 times as many girls as boys.

Answer:

The sentences that describe a comparison using multiplication are:

Question 5.

Choose the correct word from the box to complete each statement.

Answer:

The completed statements with the suitable word from the given box are:

Question 6.

Dee has $120 to spend. She went to the grocery store and spent $55. She then bought 3 potted flowers for $18 each at the nursery. How much money did Dee have left? Select each correct equation or set of equations that could be used to solve the question.

☐ $120 – $55 = M; M – (3 × $18)

☐ $120 – $55 – (3 × $18) = M

☐ (3 × $18) – $55 – $120 = M

☐ ($120 – $55) – (3 × $18) = M

☐ $120 + $55 – (3 × $18) = M.

Answer:

It is given that

Dee has $120 to spend. She went to the grocery store and spent $55. She then bought 3 potted flowers for $18 each at the nursery.

Now,

The cost of the 3 potted flowers = 3 × $18

= 3 × (10 + 8)

= (3 × 10) + (3 × 8)

= 30 + 24

= $54

So,

The amount of money spent by Dee = $55 + $54

= $109

So,

The amount of money Dee left = (The amount of money Dee has to spend) – (The amount of money Dee has spent)

= $120 – $109

= $11

Hence, from the above,

We can conclude that

Each correct equation or set of equations that could be used to solve the given question is:

Question 7.

Darcy ordered 18 boxes of red balloons and 12 boxes of blue balloons for a party. She ordered a total of 240 balloons. How many balloons are in each box?

A. 6

B. 8

C. 7

D. 9

Answer:

It is given that

Darcy ordered 18 boxes of red balloons and 12 boxes of blue balloons for a party. She ordered a total of 240 balloons

So,

The total number of balloons present in 2 boxes = (The number of red balloons) + (The number of blue balloons)

= 18 + 12

= 30 balloons

So,

The number of balloons present in each box = \(\frac{The total number of balloons}{The total number of balloons present i 2 boxes}\)

= \(\frac{240}{30}\)

= 8 balloons

Hence, from the above,

We can conclude that

The number of balloons present in each box is:

Question 8.

Select all of the sentences that are true for the number 9.

☐ 18 × _____ = 162

☐ 20 more than _____ = 180

☐ _____ times as many as 16 is 145.

☐ 315 is _____ times as many as 35.

☐ 16 × _______ = 128

Answer:

The sentences that are true for the number 9 are:

Question 9.

Select all the expressions that are equal to the product of 14 and 9.

☐ (2 × 7) + 9

☐ 9 times as many as 14

☐ 14 × 9

☐ 14 more than 9

☐ 9 less than 14

Answer:

We know that,

According to the Commutative Property of Multiplication,

a × b = b × a

So,

14 × 9 = 14 × 9

Hence, from the above,

We can conclude that

All the expressions that are equal to the product of 14 and 9 are:

Question 10.

Maggie collected 63 pounds of paper for recycling. Carl collected 9 pounds. How many times as many pounds did Maggie collect as Carl?

A. 3 times

B. 5 times

C. 7 times

D. 8 times

Answer:

It is given that

Maggie collected 63 pounds of paper for recycling. Carl collected 9 pounds

Now,

Let the number of pounds as many times collected be x

So,

According to the given information,

9 × x = 63

x = \(\frac{63}{9}\)

x = 7

Hence, from the above,

We can conclude that

The number of times as many pounds did Maggie collect as Carl are:

### Topic 6 Performance Task

**Ski Jumping**

Jackie and her older brother Robert went skiing. The Ski Jumping table shows their best jumps, including the distance they jumped and the total length of their run.

Question 1.

Jackie wants to find how much farther Robert’s run was than her run.

Part A

What is the total length of Robert’s run? Draw a bar diagram and write and solve an equation to represent the problem. Does this situation use addition or multiplication to compare?

Answer:

The given table is:

Now,

From the given table,

We can observe that

The total length of Robert’s run = 3 × (The jump distance of Robert)

= 3 × 297

= 3 × (200 + 90 + 7)

= (3 × 200) + (3 × 90) + (3 × 7)

= 600 + 270 + 21

= 891 feet

Now,

The representation of the given information about Robert in the form of a bar diagram is:

So,

The given situation uses multiplication to compare

Hence, from the above,

We can conclude that the total length of Robert’s run is: 891 feet

Part B

How much farther was Robert’s run than Jackie’s run? Write equations to represent each step of the problem. Tell what your variables represent.

Answer:

The given table is:

Now,

From the given table,

The total length of Jackie’s run = 3 × (The jump distance of Jackie)

= 3 × 189

= 3 × (100 + 80 + 9)

= (3 × 100) + (3 × 80) + (3 × 9)

= 300 + 240 + 27

= 567 feet

Now,

From part (A),

The total length of Robert’s run is: 891 feet

So,

The difference between Robert and Jackie’s length = 891 – 567

= 324 feet

Hence, from the above,

We can conclude that Robert’s run was 324 feet farther than Jackie’s run

Question 2.

Jackie’s run was the same on two different jumps. In the first, she jumped 145 feet and went an additional 167 feet after the jump. In the second, she jumped 135 feet. How many feet, f, did she go after the jump?

Answer:

It is given that

Jackie’s run was the same on two different jumps. In the first, she jumped 145 feet and went an additional 167 feet after the jump. In the second, she jumped 135 feet

So,

The total run of Jackie in the first jump = (The total run of Jackie before she jumped) + (The additional run of Jackie after she jumped)

= 145 + 167

= 312 feet

Now,

Let the additional run of Jackie after the second jump be x

So,

According to the given information,

312 = 135 + x

x = 312 – 135

x = 177 feet

Hence, from the above,

We can conclude that Jackie had gone 177 feet after her second jump

Question 3.

Use the Beginner Jump information to find how much longer the total length of the ski jump hill for an advanced jump is than a beginner jump. The advanced jump distance is 408 feet.

Part A

What are the hidden questions you need to answer to solve the problem? Name a variable for each question.

Answer:

It is given that

The advanced jump distance is 408 feet.

Now,

The given information is:

Hence, from the above,

We can conclude that

The hidden questions you need to answer to solve the given problem are:

a. What is the jump distance of the beginner jump distance?

b What is the total length of a beginner jump?

Part B

How much longer is d, the total length of an advanced jump hill, then a beginner jump hill? Write equations and explain how to solve the hidden questions and the original question.

Answer:

It is given that

The advanced jump distance is 408 feet.

Now,

The given table is:

Now,

Let the advanced jump distance is x times as the beginner jump distance

So,

According to the given information,

x × 8 = 408

x = \(\frac{408}{8}\)

x = 51 feet

So,

The beginner jump distance is: 51 feet

Now,

According to the given information,

x × 3 = 51

x = \(\frac{51}{3}\)

x = 17 feet

So,

The toatl length of a beginner jump is: 17 feet

So,

The length of an advanced jump hill, greater than a beginner jump hill = 51 – 17

= 34 feet

Hence, from the above,

We can conclude that

The length of an advanced jump hill is 34 feet longer than a beginner jump hill