Lesson 7
Divide with Fractions
Name: Prerequisite: Divide with Unit Fractions
Study the example problem showing how to solve a word
problem that involves dividing with unit fractions. Then
solve problems 1–6.
Example
The students in Mrs. Marco’s art class use 5 jars of paint
altogether. Each student uses ​ 1 ​jar of paint. How many
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students are in the class?
To answer this question, you need to find how many ​ 1 s​ are
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in 5. You can draw a model to understand the problem.
5 4 ​ 1 ​ 5 15
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There are 15 students in Mrs. Marco’s art class.
1 Explain how the model represents 5 4 ​  1 ​ 5 15.
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2 Complete the equation. Explain how the model also
shows this equation.
3 5 15
3 Andy divided his ​ 1 ​jar of paint equally between
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2 projects. What fraction of a jar of paint did Andy use
for each project? Explain how to draw a model to find
the answer.
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Lesson 7 Divide with Fractions
67
Solve.
4 Judi is making a rope ladder. She uses ​ 1 ​yard of rope
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for each step. How many steps can she make with
6 yards of rope?
Show your work.
Solution: 5 Harry has ​ 1 ​of an apple pie that he wants to cut into
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3 equal slices. What fraction of the whole pie is each
slice?
Show your work.
Solution:
6 Ryan wants to plant ​ 1 ​packet of seeds in each row of
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his garden. He has 4 packets of seeds. Ryan used the
expression ​ 1 ​ 4 4 to find the number of rows he can
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plant. Explain what is wrong with his expression. Then
write an equation to show the correct number of rows.
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Lesson 7 Divide with Fractions
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Lesson 7
Name: Divide a Whole Number by a Fraction
Study the example problem showing how to divide a
whole number by a fraction. Then solve problems 1–6.
Example
On a field trip, students ate ​  3  ​ of a box of oranges. Altogether they ate
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6 pounds of oranges. How many pounds of oranges were in the full box?
You can draw a model to
represent the problem.
You can also use an equation
6 lb
2 lb 2 lb 2 lb
to represent that ​  3  ​ of the full
10
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box is 6 pounds: ​  3  ​  3 ? 5 6. To solve a
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missing factor problem, divide: 6 4 ​  3  ​  5 ?.
10
··
?
To divide by a fraction, multiply by its reciproal.
6 4 ​  3  ​  5 6 3 ​ 10  ​ 5 20
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3
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There were 20 pounds of oranges in the full box.
Vocabulary
1 Look at the model. Explain why each tenth of the
model is 2 pounds.
2 How can you use the model in the example to find
multiplicative
inverse a number is the
multiplicative inverse of
another number if their
product is 1.
how many pounds of oranges were in the box?
9 3 ​ 91 ​ 5 1
T he fraction ​ 1 ​is the
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multiplicative inverse of 9.
3 Suppose ​ 4 ​of a different box of oranges weighs 8 pounds.
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How many pounds of oranges are in the full box?
··
reciprocal the
multiplicative inverse of
a number; with fractions,
the numerator and
denominator are
switched.
8
5
​ 5 ​is the reciprocal of ​ 8 ​.
··
··
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Lesson 7 Divide with Fractions
69
Solve.
4 Ling walks ​ 3 ​of the distance home from school in
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9 minutes. She wants to know how long it will take her
to walk the entire distance at the same speed. Ling
uses the expressions 9 4 ​ 3 ​and 9 3 ​  3 ​to find the
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8
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answer. Explain what is wrong with Ling’s expressions
and then write the correct solution.
5 Daniel has 20 quarts of water. How many 2​ 1 ​ -quart
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containers can he fill?
Show your work.
Solution: 6 Write a word problem that you can represent with the
expression 8 4 ​  2 ​. Draw a model and use equations to
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show the solution.
70
Lesson 7 Divide with Fractions
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Lesson 7
Name: Divide a Fraction by a Fraction
Study the example problem showing how to divide a
fraction by a fraction. Then solve problems 1–7.
Example
A construction company is building a fence that is ​ 2 ​ mile
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1
long. They can build ​   ​mile of the fence every hour. How
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many hours will it take them to complete the fence?
You can draw a picture to represent the
problem.
0
1
3
2
3
1
You can also use a division equation:
​ 2 ​ 4 ​ 1 ​. Think: How many ​ 1 ​s are in ​ 2 ?​
3
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6
··
6
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0
1
3
··
​ 2 ​ 4 ​ 1 ​ 5 ​ 2 ​ 3 ​ 6 ​ 5 4
3
··
6
··
3
··
1
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It will take 4 hours for the company to build the fence.
1 How can you use the number lines to find how many
hours it will take the company to build the fence?
2 Suppose you were told that the company built the
fence in 4 hours and that they completed ​ 1 ​mile of the
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fence each hour. How would you use the double
number line to help you find the length of the fence?
3 Suppose the length of the fence was 1​ 1 ​mile. How
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would you change the number lines to solve the
problem?
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Lesson 7 Divide with Fractions
71
Solve.
4 A chef cooks ​ 5 ​of a pound of pasta. She plans to
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1
serve ​    ​ of a pound to each customer. How many
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customers can she serve? Explain.
5 Carla wants to know how many batches of birdseed
she can make with ​ 1 ​cup of sunflower seeds. She
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puts ​ 1 ​cup of sunflower seeds in every batch. Carla
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divides ​ 1 ​ by ​ 1 ​to find the answer. She says this is the
2
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same as multiplying ​ 1 ​by 2. Explain what Carla did
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wrong and show the correct solution.
6 Jared ate ​ 1 ​of a loaf of bread. He cut the rest of the loaf
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1
into ​   ​ -loaf slices. How many slices of bread did he cut?
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Show your work.
Solution: 7 A running track at a school is shaped like oval. The
track is ​ 1 ​mile long. Mr. Perez puts a marker down
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1
every ​   ​mile. How many markers does he need? Show
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how you found your answer.
72
Lesson 7 Divide with Fractions
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Lesson 7
Name: Divide a Mixed Number by a Fraction
Study the example problem showing how to divide a
mixed number by a fraction. Then solve problems 1–6.
Example
Mali has 2​ 1 ​cups of fruit to make smoothies. She puts
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2
​   ​cup of fruit in each smoothie. How many smoothies can
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she make?
1
You can draw a model to
represent the problem.
2 3 cups
You can also use equations.
Think about how many ​ 2 s​
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1
are in 2​   ​.
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2​  1 ​ 4 ​ 2 ​ 5 ​ 7 ​ 4 ​ 2 ​ 
3 ··
3 ··
3 ··
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​  7 ​ 4 ​ 2 ​ 5 ​ 7 ​ 3 ​ 3 ​ 5 3​ 1 ​
3 ··
3 ··
3 ··
2
2
··
··
Mali can make 3​ 1 ​ smoothies.
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2 cup in 1 smoothie
3
1 cup in 1 smoothie
3
2
1 Explain why there is ​ 1 ​cup of fruit in half a smoothie.
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2 Look at the equations. Explain how you know that
2​ 1 ​ is equal to ​ 7 ​.
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3 Explain how the model shows that 2​ 1 ​is equal to ​ 7 ​.
3
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Lesson 7 Divide with Fractions
73
Solve.
4 Otis made 1​ 3 ​cups of oatmeal. He put ​ 2 ​cup of
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oatmeal into each bowl. How many bowls of oatmeal
did Otis make? Use equations to solve the problem.
Show your work.
Solution: 5 Juan wants to know how many ​ 1 ​-cup servings are in
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3
1​   ​cups of juice. She uses the expression ​ 12  ​ 4 ​ 1 ​ to
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4
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find the answer. Explain what is wrong with Juan’s
expression and find the correct solution.
6 Carmela mixes ​ 3 ​kilogram of walnuts, ​ 1 ​kilogram of
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1
almonds, and ​   ​kilogram of pecans together. She
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divides the mixed nuts into ​  3  ​  -kilogram bags. How
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many bags of mixed nuts does she have?
Show your work.
Solution: 74
Lesson 7 Divide with Fractions
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Lesson 7
Name: Divide with Fractions
Solve the problems.
1 Dario divides ​ 4 ​yard of rope equally into ​  1  ​  -yard
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12
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pieces for a craft project. How many pieces of rope
does Dario have? Use the number lines to solve the
problem.
4
6
0
0
What fraction is
each number line
divided into?
1
1
A 4 pieces
B 6 pieces
C 8 pieces
D 12 pieces
2 Check a box in each row to show whether the quotient
of each expression is less than, greater than, or equal
to 1.
quotient is
less than 1
quotient is
equal to than 1
​  3 ​ 4 ​  3  ​ 
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10
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quotient is
greater than 1
Read each
expression as a
question. For
example: How
3
3
many ··
​  10
    ​s are in ​   ​?
5
·
​  3  ​  4 ​ 3 ​
10
··
5
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6 4 ​  2 ​
5
··
2​  3 ​ 4 ​ 3 ​
4
··
4
··
3​ 2 ​ 4 ​ 11  ​
3
··
9
3
··
3
​   ​ 4 ​   ​
8 ··
2
··
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Lesson 7 Divide with Fractions
75
Solve.
3 Omar jogged a total of 3​ 3 ​miles last week. Each
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day that he jogged, he went ​  9  ​ mile. On how many
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days did he jog?
9
Think: How many ​ ··
  ​s 
10
are in 3 ​ ·53 ​?
Show your work.
Solution: 4 Choose all of the expressions that are equivalent to
2​ 1 ​ 4 1​ 2 ​.
2
6
··
··
5
A ​   ​ 3 ​ 6 ​
2 ··
8
··
2
B ​   ​ 3 ​ 6 ​
5 ··
8
··
2
C 1​   ​ 4 2​ 1 ​
6
2
··
··
5
8
D ​   ​ 4 ​   ​
2 ··
6
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5 What is the value of the expression 3 4 ​  3 ​?
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A ​  1 ​
4
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B 2​  1 ​
4
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C 4
Can you draw a
model to help you
2
find 2​ ·21 ​ 4 1​ ··
 ​?
6
How can you use
multiplication to
divide a number
by ​ ··43 ​?
D 12
Jane chose B as the correct answer. How did she get that
answer?
76
Lesson 7 Divide with Fractions
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