Name Class Date 5-6
Dividing Fractions and Mixed Numbers
Modeling
Essential question: How do you divide fractions and mixed numbers?
CC.6.NS.1
1
EXPLORE
video tutor
Modeling Fraction Division
You have _
​ 3 ​ cup of salsa for making burritos. Each burrito requires
4
_​ 1 ​ cup of salsa.
How many burritos can you make?
8
To find the number of burritos that can
be made, you need to determine how
many __​ 81 ​s are in ​ __34 ​.
How many ​ __18 ​s are there in __​ 34 ​?
3
4
1
8
You have enough salsa to make
burritos.
REFLECT
1a. Division can be checked by using multiplication. What would you
multiply to check your answer above?
TRY THIS!
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1b. How many burritos could you make with __​ 12 ​cup of salsa?
1
2
1
8
1c. Five people share __​ 12 ​pound of chocolate equally. How much chocolate
does each person receive?
Chapter 5
pound
221
Lesson 6
Another way to divide fractions is to use reciprocals.
___________
​  numerator
   
  
 ​· ___________
​ denominator
 ​ 
=1
numerator
denominator
Two numbers whose product is 1 are reciprocals. To find the
reciprocal of a fraction, switch the numerator and denominator.
CC.6.NS.1
2
EXAMPLE
Reciprocals
Find the reciprocal of each fraction or mixed number.
A
Check:
__​ 58 ​
Switch the numerator and denominator: ____
​    ​
The reciprocal of __​ 58 ​is ​ ____  ​.
B
8
__​ 16 ​
Switch the numerator and denominator: ____
​    ​
Simplify:
The reciprocal of __​ 16 ​is
C
____
​    ​× _
​ 5 ​  = ____
​    ​= 1
.
1​ __27 ​
​    ​
Change to an improper fraction: 1​ __27 ​= ____
Switch the numerator and denominator: ____
​    ​
The reciprocal of 1​ __27 ​is ​ ____  ​.
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TRY THIS!
Find the reciprocal of each fraction or mixed number.
2a. __​ 78 ​ 9
2b. __
​ 15
  ​ 1  ​ 2c. __
​ 11
2d. 2​ __45 ​ REFLECT
2e. Is any number its own reciprocal? If so, what number(s)?
2f. Does every number have a reciprocal? Explain.
2g. The reciprocal of a whole number is a fraction with
Chapter 5
222
in the numerator.
Lesson 6
Notice that dividing by a whole number is equivalent to
multiplying by its reciprocal. This is also true when dividing
by fractions. To divide by a fraction, multiply by its reciprocal.
24 ÷ 3 = 8
24 × __​ 13 ​= 8
CC.6.NS.1
3
EXAMPLE
Using Reciprocals to Divide Fractions
Divide.
__
​ 5 ​  ÷ __
​ 5 ​  
8 6
Step 1 Rewrite the problem as multiplication using the reciprocal of the
second fraction.
A
5  ​ ÷ _
​ _
​ 5 ​  = _
​ 5 ​  × ____
​    ​
8 6 8
Step 2 Multiply and simplify.
_​ 5 ​  × ____
​ 30 ​  
​    ​= __
8
40
30 ​  = ____
​    ​
​ __
40
5  ​ ÷ _
​ _
​ 5 ​  =
8 6
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B
3 ​  ÷ __
1​ __
​ 2 ​  
7 5
Step 1 Convert the
mixed number to a
fraction.
3  ​ = ____
​    ​
1​ _
7
Step 2 Rewrite the problem
as multiplication using
the improper fraction
and the reciprocal of the
second fraction.
Step 3 Multiply and
simplify.
​    ​= ____
​    ​
​ ____  ​× ____
3 ​   ÷ _
​ 2 ​  = ____
​    ​ = ____
​    ​× ____
​    ​÷ _
​ 2 ​  = ____
1​ __
​    ​, or 3 ​ ____  ​
7
5
5
3 ​  ÷ _
​ 2 ​  =
1​ _
7 5
TRY THIS!
Divide.
​ 2 ​  = 3a. __
​  9   ​ ÷ _
10 5
Chapter 5
9   ​ ÷ _
3b. 2​ ___
​ 3 ​  = 5
10
223
Lesson 6
CC.6.NS.1
4
EXAMPLE
Solving Problems Involving Area
1
  ​ square feet. The width of the
The area of a rectangular flower bed is 6​ _
2
3
_
flower bed is ​   ​ foot. What is the length? (Hint: area = length × width)
4
To find the length of the flower bed, divide the area by the width.
1  ​ ÷ _
6 ​ _
​ 3 ​  
​    ​÷ _
​ 3 ​  = ____
4
2 4
​    ​= ​ ____  ​=
= ____
​    ​× ____
​ ____  ​=
The length of the flower bed is
A = 6 1 ft2
2
​ ____  ​
w = 3 ft
4
=?
feet.
pra c t i c e
Find the reciprocal of each fraction or mixed number.
1.​ __25 ​ 2.​ __91 ​ 3.​ __53 ​ 4  ​ 4.​ __
11
5. 4​ __51 ​ 6. 3​ __18 ​ 7.​ __43 ​÷ __​ 53 ​=
3  ​÷ __
8.​ __
​ 45 ​=
10
9.​ __12 ​÷ __​ 25 ​=
10.​ __89 ​÷ __​ 12 ​=
11. 4​ __41 ​÷ __​ 34 ​=
12. 4 ÷ 1​ __18 ​=
Divide.
loaves
14. Ayita made 5​ __12 ​cups of trail mix. She wants to divide the trail
mix into __​ 34 ​cup servings. How many servings will she have?
serving(s)
15. Dao has 2​ __83 ​pounds of hamburger meat. He is making __​ 14 ​ -pound
burgers. How many hamburgers can he make?
hamburger(s)
16. A rectangular piece of land has an area of ​ __34 ​square mile and
is __​ 12 ​mile wide. What is the length?
mile(s)
17. Write a real-world problem whose solution requires dividing the fractions
__
​ 13 ​and ​ __34 ​. Then solve your problem.
Chapter 5
224
Lesson 6
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13. A recipe for one loaf of banana bread requires ​ __23 ​cup of oil.
You have 2 cups of oil. How many loaves of banana bread
can you make?
Name Class 5-6
Date Name ________________________________________ Date __________________ Class__________________
Fraction Operations
Additional
Practice
LESSON
6
Practice B: Dividing Fractions and Mixed Numbers
Find the reciprocal.
1. 5
7
________________________
4. 1
10
________________________
7. 1 1
3
________________________
2. 9
8
3. 3
5
_______________________
________________________
6. 13
14
5. 4
9
_______________________
8. 2 4
5
________________________
9. 3 1
6
_______________________
________________________
Divide. Write each answer in simplest form.
________________________
13. 3 1  2 3
4
4
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________________________
16. 2 6  6
9
7
________________________
11. 2 3  1 4
7
4
_______________________
14. 9  3
10
________________________
15. 3  9
4
_______________________
17. 5  2 3
6
10
_______________________
19. The rope in the school gymnasium is 10 1 feet long. To
2
make it easier to climb, the gym teacher tied a knot in
the rope every 3 foot. How many knots are in the rope?
4
20. Mr. Fulton bought 12 1 pounds of ground beef for the
2
cookout. He plans on using 1 pound of beef for each
4
hamburger. How many hamburgers can he make?
21. Mrs. Marks has 9 1 ounces of fertilizer for her plants.
4
She plans on using 3 ounce of fertilizer for each plant.
4
How many plants can she fertilize?
Chapter 5
12. 7  2
8
3
________________________
18. 2 1  3 1
8
4
________________________
__________________________
__________________________
__________________________
225
Practice and Problem Solving
34
Holt McDougal Mathematics
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10. 5  5
6
Name ________________________________________ Date __________________ Class __________________
Fraction
Operations
Problem
Solving
LESSON
6
Problem Solving: Dividing Fractions and Mixed Numbers
Write the correct answer in simplest form.
1. Horses are measured in units called
hands. One inch equals 1 hand. The
4
average Clydesdale horse is 17 1
5
hands high. What is the horse’s height
in inches? in feet?
2. Cloth manufacturers use a unit of
measurement called a finger. One
finger is equal to 4 1 inches. If 25
2
inches are cut off a bolt of cloth, how
many fingers of cloth were cut?
________________________________________
________________________________________
3. People in England measure weights in
units called stones. One pound equals
1 of a stone. If a cat weighs 3 stone,
4
14
how many pounds does it weigh?
4. The hiking trail is 9 mile long. There
10
are 6 markers evenly posted along
the trail to direct hikers. How far apart
are the markers placed?
________________________________________
________________________________________
A 24 tablespoons
B 8 tablespoons
C
3 tablespoon
32
6. Printed letters are measured in units
called points. One point equals 1 inch.
72
If you want the title of a paper you are
typing on a computer to be 1 inch tall,
2
what type point size should you use?
F 144 point
H
1 point
36
G 36 point
J
1 point
144
D 9 tablespoons
7. Phyllis bought 14 yards of material to
make chair cushions. She cut the
material into pieces 1 3 yards long to
4
make each cushion. How many
cushions did Phyllis make?
8. Dry goods are sold in units called
pecks and bushels. One peck equals
1 bushel. If Peter picks 5 1 bushels
4
2
of peppers, how many pecks of
peppers did Peter pick?
A 4 cushions
C 8 cushions
F 1 3 pecks
8
H 20 pecks
B 6 cushions
D 24 1 cushions
2
G 11 pecks
J 22 pecks
Chapter 5
226
34
Holt McDougal Mathematics
Practice and Problem Solving
© Houghton Mifflin Harcourt Publishing Company
5. A cake recipe calls for 1 1 cups of
2
butter. One tablespoon equals 1
16
cup. How many tablespoons of butter
do you need to make the cake?
© Houghton Mifflin Harcourt Publishing Company
Choose the letter for the best answer.