Lesson 7.

5
Name
Compare Fraction Factors and Products Learning Objective You will use an area
model to compare the size of a product to the
Essential Question How does the size of the product compare to the size of one factor and a number line to show
size of one factor when multiplying fractions? the relationship between products when a
fraction is multiplied or resized by a number.

Unlock
Unlock the
the Problem
Problem
Multiplication can be thought of as resizing one
number by another number. For example, 2 × 3
will result in a product that is 2 times as great as 3.

What happens to the size of a product when a number is

multiplied by a fraction rather than a whole number?

One Way Use a model.

A During the week, the Delgado family ate ​3__4​of a box of cereal.

• Shade the model to show ​3

__​ of a box of cereal.
4

• Write an expression for ​3

__​ of 1 box of cereal. ​3
__​ ×
_
4 4

• Will the product be equal to, greater than, or less than 1?

B  The Ling family has 4 boxes of cereal. They ate ​ 3_4 ​

of all the cereal during the week.
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• Shade the model to show ​ 3_4 ​ of 4 boxes of cereal.

• Write an expression for ​ 3_4 ​ of 4 boxes of cereal. ​ 3_4 ​ × _

• Will the product be equal to, greater than, or less than 4?

C  The Carter family has only ​ 1_2 ​of a box of cereal at the beginning
of the week. They ate ​ 3_4 ​of the ​ 1_2 ​box of cereal.

• Shade the model to show ​ 3_4 ​ of ​ 1_2 ​ box of cereal.

• Write an expression to show ​ 3_4 ​ of ​ 1_2 ​ box of cereal. ​ 3_4 ​ × _

• Will the product be equal to, greater than, or less than ​ 1_2 ​? than ​ 3_4 ​?

Chapter 7   445

 Another Way Use a diagram.
You can use a diagram to show the relationship between the products when a
fraction is multiplied or scaled (resized) by a number.
Graph a point to show ​ 3_4 ​scaled by 1, ​ 1_2 ,​ and 4.

A 1 × ​3_​
4

Think: Locate ​ 3_4 ​ on the diagram and

shade that distance from 0. Then
0 1 2 3 4 graph a point to show 1 of ​ 3_4 ​.

B 1_​ ​× ​3_​
2 4

Think: Locate ​ 3_4 ​ on the diagram and

shade that distance from 0. Then
0 1 2 3 4 graph a point to show ​ 1_2 ​ of ​ 3_4 ​.

C 4 × ​3_​
4

Think: Locate ​ 3_4 ​ on the diagram and

shade that distance from 0. Then
0 1 2 3 4 graph a point to show 4 times ​ 3_4 ​.

Math
Math Processes
6
Talk Math Processes and Practices 2
Complete each statement with equal to, greater than,
and Practices Reason Abstractly What if ​__3​
5
or less than. Explain your choices. was multiplied by ​1 __​ or by the
6
whole number 7? Would the
• The product of 1 and ​ 3_4 ​ will be __ ​ 3_4 ​. products be equal to, greater
than, or less than ​3
__​? Explain.
5

• The product of a number less than 1 and ​ 3_4 ​ will be

__ ​ 3_4 ​ and __ the other factor.

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• The product of a number greater than 1 and ​ 3_4 ​ will

be __ ​ 3_4 ​ and __ the other factor.

446
 Name

MATH
Share
Share and
and Show
Show BOARD

Complete the statement with equal to, greater than, or less than.
1. 4 × ​7_8​will be __ ​7_8​.

0 1 2 3 4

2. _3​5​ × ​2_7​will be __ ​3_5​. 3. ​5_​ × 6 will be __ ​5_8​.

8

On
On Your
Your Own
Own
Complete the statement with equal to, greater than, or less than.
9
4. _7​ ​ × ​3_​ will be __ ​3_​. 9 ​ will be __ ​ __
5. 7 × ​ __ 10
​.
8 5 5 10

6
6. 5 × ​1_​ will be __ ​1_3​. 6 ​ × 1 will be __ ​ __
7. ​ __
11 11
​.
3

Problem
Problem Solving
Solving •• Applications
Applications
8. Peter is planning on spending ​ 2_3 ​as many hours watching television this week as
he did last week. Is Peter going to spend more hours or fewer hours watching television this week?

9. DEEPER Mrs. Rodriguez has 18 packages of pens in stock at her store on Monday.
On Tuesday, she has ​ 5_6 ​ the number of pens she had on Monday. On Wednesday,
she has ​ 2_5 ​ of the number of pens she had on Tuesday. How many packages of pens
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does she have on Wednesday?

Math Processes
10. 2 Represent a Problem Ariel goes running for ​ _5 ​ of an hour.
and Practices 6
The next day, she runs for ​ _3 ​ as much time. Does she spend more or
4
less time running the second day? Draw a diagram or make a model to represent the problem.

Chapter 7 • Lesson 5   447

Math Processes and Practices COMMUNICATE • PERSEVERE • CONSTRUCT ARGUMENTS

A scale model is a representation of an object

with the same shape as the real object. Models
can be larger or smaller than the actual object but are
often smaller.

Architects often make scale models of the buildings or

structures they plan to build. Models can give them an
idea of how the structure will look when finished. Each
measurement of the building is scaled up or down by
the same factor.

Bob is building a scale model of his bike. He wants

his model to be 1_5​  ​as long as his bike.

11. If Bob’s bike is 60 inches long, how long will his model be? __

12. SMARTER
If one wheel on Bob’s model is 4 inches across, how
many inches across is the actual wheel on his bike? Explain.

WRITE Math • Show Your Work

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Personal Math Trainer
13. SMARTER Write each multiplication expression in the correct box.
5 5 ​5_​× 4_​ ​ ​5_​× 7_​ ​ __​× 5 ​5_​× 5_​ ​
_6​ ​× 2_3​ ​ 2 × ​_6​ 6 4 6 3
​10
10
_
6​ ​ 6 6

Product is equal to 5_​ ​. Product is greater than ​5_​. Product is less than ​5_​.
6 6 6

448
 Practice and Homework
Name
Lesson 7.5
Compare Fraction Factors and Products
Learning Objective You will use an area
model to compare the size of a product to
the size of one factor and a number line to
show the relationship between products
when a fraction is multiplied or resized by a
Complete the statement with equal to, greater than, or less than. number.

less than
1. 3_5​  ​× ​ 4_7 ​will be ____ ​  4_
2. 5 × ​ 7_8 ​will be ____ ​ 7_8 ​.
7 ​.

Think: ​ _​is multiplied by a

4
7
number less than 1;
so, ​ _​∙ ​ _​will be less than ​ _​.
3 4 4
5 7 7

3. 6 × ​ 2_5 ​will be ____ ​ 2_5 ​. 4. 1_9​  ​× 1 will be ____ ​ 1_9 ​.

5. ​ __49 ​× ​ __38 ​will be ____ ​ __38 ​. 6. 4_5​  ​× ​ 7_7 ​will be ____ ​ 4_5 ​.

Problem
Problem Solving
Solving
7. Starla is making hot cocoa. She plans to multiply 8. Miles is planning to spend ​ 2_3 ​as many hours
the recipe by 4 to make enough hot cocoa for bicycling this week as he did last week. Is Miles
the whole class. If the recipe calls for ​ 1_2 ​teaspoon going to spend more hours or fewer hours
vanilla extract, will she need more than bicycling this week than last week?
1_ 1_
2​  ​teaspoon or less than ​ 2 ​teaspoon of vanilla
extract to make all the hot cocoa?
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9. WRITE Math Explain how you can compare the size of a product to
the size of a factor when multiplying fractions without actually doing the
multiplication. Include a model.

Chapter 7 449
Lesson Check
1. Trevor saves ​ 2_3 ​of the money he earns at his 2. Suppose you multiply a whole number greater
after-school job. Suppose Trevor starts saving ​ 1_4 ​ than 1 by the fraction ​ 3_5 ​. Will the product be
as much as he is saving now. Will he be saving greater than, less than, or equal to ​ 3_5 ​?
less, more, or the same amount?

Spiral Review
3. In the next 10 months, Colin wants to save $900 4. What is the total cost of 0.5 pound of peaches
for his vacation. He plans to save $75 each of selling for $0.80 per pound and 0.7 pound of
the first 8 months. How much must he save oranges selling for $0.90 per pound?
each of the last 2 months in order to meet his
goal if he saves the same amount each month?

5. Megan hiked 15.12 miles in 6.3 hours. If Megan 6. It is 42​ 1_2 ​miles from Eaton to Baxter, and
hiked the same number of miles each hour, how 37​ 4_5 ​miles from Baxter to Wellington. How far
many miles did she hike each hour? is it from Eaton to Wellington, if you go by way
of Baxter?
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