Fraction
Operations
FLORIDA
CHAPTER
5
Name
Class
Date
Lesson
Worktext
Copyright © by Holt McDougal. All rights reserved.
Remember It?
Student
Textbook
183 –184
Rev. MA.5.A.6.1
5-1 Least Common Multiple
194 –197
Rev. MA.5.A.2.2
5-2 Adding and Subtracting with
Unlike Denominators
198 –201
Rev. MA.5.A.2.2
5-3 Adding and Subtracting Mixed
Numbers
202–205
Rev. MA.5.A.2.2
5-4 Regrouping to Subtract Mixed
Numbers
206 –209
MA.6.A.1.1,
MA.6.A.1.2,
MA.6.A.1.3
5-5 Multiplying Fractions by
Whole Numbers
185 –192
212–215
MA.6.A.1.1,
MA.6.A.1.2,
MA.6.A.1.3
5-6 Multiplying Fractions
193 –200
216 –219
MA.6.A.1.1,
MA.6.A.1.2,
MA.6.A.1.3
5-7 Multiplying Mixed Numbers
201–208
220 –223
MA.6.A.1.1,
MA.6.A.1.2,
MA.6.A.1.3
5-8 Dividing Fractions and
Mixed Numbers
209 –216
224 –227
Study It!
219 –221
Write About It!
222
Chapter 5 Fraction Operations 181
CHAPTER
Benchmark
5
Chapter at a Glance
Vocabulary Connection
LA.6.1.6.5 The student will relate new vocabulary to familiar words.
Key Vocabulary
Vocabulario
Vokabilè
reciprocal
recíproco
resipwòk
multiplicative inverse
inverso multiplicativo
envès (pou miltiplikasyon)
To become familiar with some of the vocabulary terms in the chapter, consider the
following. You may refer to the chapter, the glossary, or a dictionary if you like.
1. The word reciprocal means “inversely related.” What do you think the reciprocal of
a fraction will look like?
CHAPTER
Notes
Copyright © by Holt McDougal. All rights reserved.
5
182 Chapter 5 Fraction Operations
5-1
Name
Class
Date
THROUGH
5-4
Remember It?
Review skills and prepare for future lesson
lessons.
Lesson
5-1
Least Common Multiple (Student Textbook pp. 194–197)
Rev. of MA.5.A.6.1
Find
Fi
d th
the L
Leastt Common Multiple (LCM).
4, 6, and 8
4: 4, 8, 12, 16, 20, 24, 28, . . . List multiples of
6: 6, 12, 18, 24, 30, . . .
8: 8, 16, 24, 32, . . .
LCM: 24
4, 6, and 8.
Find the smallest
number that is in
all the lists.
Find the least common multiple ( LCM ).
1. 3, 5, and 10
2. 6, 8, and 16
3. 3, 9, and 27
4. 4, 12, and 30
5. 25 and 45
6. 12, 22, and 30
Copyright © by Holt McDougal. All rights reserved.
Lesson
5-2
Adding and Subtracting with Unlike Denominators
(Student Textbook pp. 198–201)
Rev. of MA.5.A.2.2
Add or subtract.
b
Write each answer in simplest form.
7
__
+ __23
9
7 + __
6
__
9
9
13 = 1__
4
__
9
9
1
__
- __14
3
3
4 - ___
___
12
12
1
__
12
Write equivalent fractions.
Add.
Write equivalent fractions.
Subtract.
Add or subtract. Write each answer in simplest form.
7. __15 + __58
7
8. __16 + __
12
13
9. __
- __45
15
10. __78 - __23
3
11. __
+ __58
10
12. __38 + __13
13. About _13 of the animals at the zoo are birds. The mammals make
up _25 of the zoo’s population. What fraction of the zoo’s animals are
mammals or birds?
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Chapter 5 Fraction Operations 183
5-3
Lesson
Adding and Subtracting Mixed Numbers (Student Textbook pp. 202–205)
Rev. of MA.5.A.2.2
Find
Fi
d each
h sum or difference. Write the answer in simplest form.
5 __58 - 3 __16
Write equivalent fractions.
7
12__35 + 16 __
10
15
4
5 ___
- 3 __
24
24
Subtract.
6
7
12___
+ 16 __
10
10
13 = 29 __
3
28 __
10
10
11
2__
24
Write equivalent
fractions.
Add. Then simplify.
Find each sum or difference. Write the answer in simplest form.
3 + 3 __
2
14. 1__
5
10
15. 4__59 - 1__12
16. 3__34 + 2__14
5
17. 20__59 + 14__
12
4 - 16__
1
18. 48__
11
6
19. 18__58 - 4__14
20. 18__13 + 16__16
5
21. 5__34 + 3__
12
22. 12__12 - 8__25
7
23. Angela had __
gallon of paint. She used _13 gallon for a project.
10
How much paint did she have left?
Lesson
5-4
Regrouping to Subtract Mixed Numbers (Student Textbook pp. 206–209)
Rev. of MA.5.A.2.2
Subtract.
S
bt t Write
it each answer in simplest form.
7
9
4___
- 2__
10
10
17
9
3___
- 2__
10
10
8
1__
10
7 as 3__
17
Regroup 4___
.
10
10
Subtract.
Simplify.
10__77 - 5__57
7.
Regroup 11 as 10 + __
7
Subtract.
5__27
Subtract. Write each answer in simplest form.
24. 7__29 - 3 __59
7
25. 3__15 - 1__
10
7 - 2 __
11
26. 8__
12
12
27. 5__38 - 2__34
13
28. 11__67 - 4 __
14
29. 10 - 8__78
9
30. 4__25 - 2__
10
31. 9__16 - 5__56
32. Georgette needs 8 feet of ribbon to decorate gifts. She has 3_14 feet of
ribbon. How many more feet of ribbon does Georgette need?
184 Chapter 5 Fraction Operations
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1__45
11 - 5__57
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Class
5-5
Date
Explore It!
MA.6.A.1.1 Explain and justify
procedures for multiplying…fractions.
Also MA.6.A.1.2
Multiplying Fractions by Whole Numbers
Tools
Model Multiplying Fractions by
Whole Numbers
Go to thinkcentral.com
You can use models to model products of fractions
and whole numbers.
Activity 1
Find _23 × 6.
1 Think of _23 × 6 as _23 of 6.
Draw 6 bars to model 6.
2 Because the denominator of _23 is 3, divide your model into
3 equal-sized groups.
Copyright © by Holt McDougal. All rights reserved.
3 Because the numerator of _23 is 2, identify how many bars
are in 2 of the 3 equal-sized groups.
There are
bars in 2 groups, so _23 × 6 =
.
Try This
Use a model to find each product.
1. __34 × 16
2. __13 × 12
3. __16 × 18
Find each product.
4. __25 × 10
5. __13 × 6
8. __34 × 8
9. __38 × 16
6. __36 × 12
10. __13 × 9
7. __23 × 9
11. __34 × 12
5-5 Multiplying Fractions by Whole Numbers 185
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Activity 2
Find 3 × _34.
1 Shade a square or rectangle to model _34 .
2 Shade rectangles to model 3 groups of _34 .
3 Count the number of shaded fourths in your model.
fourths. So, 3 × _34 = ______
, or 2_14.
4
There are
Try This
Write the multiplication problem each model represents. Then write the product.
12.
13.
×
=
14.
×
=
×
=
15.
=
Find each product.
16. 2 × __34
17. 4 × __13
18. 5 × __16
Draw Conclusions
×2
12
20. Why is 6 × _23 not equal to 6____
= __
?
6×3
18
21. Write a rule for multiplying a fraction by a whole number.
186 5-5 Multiplying Fractions by Whole Numbers
19. 4 × __12
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×
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Name
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Class
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5-5
Date
MA.6.A.1.2 Multiply…fractions…
efficiently. Also MA.6.A.1.3
Multiplying Fractions by Whole Numbers (Student Textbook pp. 212–215)
Lesson Objective
Multiply fractions by whole numbers
Example 1
Multiply.
M
lti l W
Write each answer in simplest form.
A. 7 · __19
7·1
7 · __19 = ____
·
1 9
Write 7 as a
. Multiply
and
.
=
B. 6 · __18
6·1
6 · __18 = ____
·
1 8
Write 6 as a
. Multiply
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and
.
=
=
Write your answer in simplest form.
C. 8 · __23
8·2
8 · __23 = ____
·
1 3
Write 8 as a
. Multiply
and
.
=
=
Lesson Tutorial Videos
5-5 Multiplying Fractions by Whole Numbers 187
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M
Multiply. Write each answer in simplest form.
1a. 9 · __34
1c. 5 · __56
1b. 4 · __52
Example 2
Evaluate 4x for each value of x. Write each answer in simplest form.
1
A. x = __
10
4x
Write the expression.
4·
Substitute
1 =
· ___
for x.
Multiply.
10
Write your answer in simplest form.
=
B. x = __38
Write the expression.
4·
Substitute
12
· __3 = __
8
Check It Out!
2a. x = __19
Multiply.
8
=
for x.
or
Write your answer in simplest form.
Evaluate 3x for each value of x. Write each answer in simplest form.
2b. x = __25
188 5-5 Multiplying Fractions by Whole Numbers
2c. x = __56
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4x
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Example 3
Th
There
are 25 students in the music club. Of those students, _35 are also in the band.
How many music club students are in the band?
To find _35 of 25, multiply.
3 · 25 = __
3·
__
5
5
= ______
5
Divide
by 5 and write your answer in
simplest form.
=
There are
music club students in the band.
Check It Out!
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3a. There are 50 people on the football team. Of those people, _15 are also on
the basketball team. How many people on the football team are also on the
basketball team?
3b. There are 16 players on the baseball team. Of these players, _14 are girls.
How many girls play on the baseball team?
Lesson Tutorial Videos
5-5 Multiplying Fractions by Whole Numbers 189
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5-5
Learn It!
Summarize It!
Name
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Class
Apply It!
Date
LA.6.4.2.2 The student will record
information…related to a topic,
including visual aids to organize and
record information…
Summarize It!
Multiplying Fractions by Whole Numbers
Think and Discuss
1. Describe a model you could use to show the product of 4 · __15 .
2. Choose the expression that is correctly multiplied.
2 · __37 = __67
6
2 · __37 = __
14
3. Explain how you know without actually multiplying that __58 · 16 is greater than 8.
3 19
Method 1
Use repeated
addition.
190 5-5 Multiplying Fractions by Whole Numbers
Method 2
Write the whole number
with 1 in the denominator
and multiply.
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4. Get Organized Complete the graphic organizer. List the steps you take to use
each method to find the given product.
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5-5
Date
MA.6.A.1.2 Multiply…fractions…
efficiently. Also MA.6.A.1.3
Multiplying Fractions by Whole Numbers
Multiply. Write your answers in simplest form.
1
1. 5 × __
25
2
2. 7 × __
21
3. 3 × __19
4. 3 × __15
5. 6 × __38
6. 2 × __23
3
7. 7 × __
10
8. 8 × __47
9. 13 × __12
13
10. 4 × __
16
11. 5 × __49
3
12. 10 × __
11
13. 6 × __89
14. 8 × __78
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Evaluate 4x for each value of x. Write your answers in simplest form.
15. x = __14
16. x = __38
17. x = __49
18. x = __67
3
19. x = __
11
20. x = __12
21. x = __45
5
22. x = __
16
23. You have 20 buckets of sand and you want to make a sandcastle. If the bottom
part of the castle takes _47 of the sand, how many buckets do you have left to
make the top?
24. There are 39 children in an after-school program. About _14 of the children are
under the age of 5. Estimate the number of children in the program who are
5 years of age or older.
25. A muffin recipe calls for _34 cup of flour. Josie has 2 cups of flour. Does Josie
have enough flour to double the recipe? Justify your answer.
5-5 Multiplying Fractions by Whole Numbers 191
Explore It!
5-5
Learn It!
Summarize It!
Name
Practice It!
Class
Apply It!
Apply It!
Date
MA.6.A.1.2 Multiply…fractions…
efficiently. Also MA.6.A.1.3
Multiplying Fractions by Whole Numbers
1
1. About __
of all people have an extra rib
20
bone. In a crowd of 60 people, about how
many people are likely to have an extra
rib bone?
6. Maricel is putting fences around
rectangular garden plots. What is the total
length of the fencing needed for 5 plots?
1 yd
3
1 yd
2
1
2. Human fingernails can grow up to __
of
10
a millimeter each day. How much can
fingernails grow in one week?
3. Sarah walks _78 miles each day in each of the
4 months that have 30 days. How would you
find the total number of miles Sarah walks
in all the months that have 30 days?
5. The Statue of Liberty is about 305 feet tall
from the ground to the tip of her torch. The
statue’s pedestal makes up about _12 of its
height. A story of a building is about 10 feet.
Estimate how many stories tall the statue’s
pedestal is.
8. Gridded Response
Today, the United
States flag has 50
stars—one for each
state. The first official
U.S. flag, approved in
3
1795, had __
as many
10
stars as today’s flag.
How many stars were
on the first official U.S.
flag?
192 5-5 Multiplying Fractions by Whole Numbers
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4. The Appalachian National Scenic Trail
extends through 14 states for about 2,000
miles. Last year, Carla hiked _15 of the trail.
Carla wants to hike the entire trail. How
many miles does she have left to hike?
7. A serving of oatmeal has 4_12 g of fiber. Katie
eats oatmeal for breakfast 5 days a week.
Explain whether Katie gets more than
50 g of fiber from oatmeal after two weeks.
(Hint: Write the mixed number as an
improper fraction.)
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5-6
Date
MA.6.A.1.1 Explain and justify
procedures for multiplying…fractions.
Also MA.6.A.1.2
Multiplying Fractions
Model Fraction Multiplication
You can use shading of squares or rectangles to model
products of fractions.
Tools
Go to thinkcentral.com
Activity
A. Find _12 × _13.
1 Think of _12 × _13 as _12 of _13 . Shade _13 of a square or rectangle vertically.
2 So that you can find _12 of _13 , divide the square or rectangle into
halves horizontally.
3 Look at _12 of the part you shaded. What fraction
of the whole that you started with is this?
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So, _12 × _13 =
.
B. Find _34 × _12.
4 Think of _34 × _12 as _34 of _12 . Shade _12 of a square or rectangle vertically.
5 So that you can find _34 of _12 , divide the rectangle into
fourths horizontally.
6 Shade _34 of the 4 horizontal sections.
7 How many sections did you shade in both Step 4 and Step 6?
What is the total number of sections?
8 __3 × __1 =
4
2
number of sections shaded twice
total number of sections
5-6 Multiplying Fractions 193
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Try This
Write the multiplication problem each model represents. Then write the product.
1.
2.
×
=
3.
×
=
×
=
4.
×
=
Use a model to find each product.
5. __34 × __23
6. __13 × __14
7. __12 × __16
8. __23 × __14
9. __56 × __12
10. __12 × __12
11. __23 × __13
12. __14 × __34
13. Make up a problem you can solve using the grid at the right.
Solve the problem by shading the grid. Explain the steps
you took.
14. Write a rule for multiplying fractions.
194 5-6 Multiplying Fractions
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Draw Conclusions
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Class
5-6
Date
Learn It!
MA.6.A.1.2 Multiply…fractions…
efficiently. Also MA.6.A.1.3
Multiplying Fractions (Student Textbook pp. 216–219)
Lesson Objective
Multiply fractions
Example 1
Multiply.
M
lti l W
Write each answer in simplest form.
1
2
__
__
A. ·
4
5
1 · __
2 = _____
1·2
__
4 5
4·5
Multiply
. Multiply
=
The GCF of 2 and 20 is
=
The answer is in simplest form.
.
.
4
B. __57 · __
15
1
5 · __
4 = __
1 · __
4
__
7 153 7 3
Use the GCF to simplify the fractions before
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.
The greatest common factor of 5 and 15 is
1·4
= ____
7·3
Multiply
=
The answer is in simplest form.
. Multiply
.
.
6
C. __49 · __
10
4·6
6
4 · __
__
= _____
9 10
9 · 10
Multiply
. Multiply
=
The GCF of
and
=
The answer is in simplest form.
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is
.
.
5-6 Multiplying Fractions 195
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Multiply. Write each answer in simplest form.
M
1a. __73 · __59
1c. __58 · __25
1b. __12 · __67
Example 2
E
l
h expression b · _25 for each value of b. Write the answer in simplest
Evaluate
the
form.
A. b = __13
2
· __25
·2
1____
3·5
Substitute
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b · __5
for b.
Multiply.
The answer is in simplest form.
B. b = __38
2
b · __5
· __25
Substitute
3 · __
21
__
8 5
4
Use the
·1
3____
4·5
Multiply.
for b.
to simplify.
The answer is in simplest form.
196 5-6 Multiplying Fractions
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C. b = __57
2
b · __5
· __25
Substitute
for b.
5·2
____
7·5
Multiply
10
__
35
Use the GCF to simplify. The GCF of 10 and 35 is
. Multiply denominators.
.
The answer is in simplest form.
Check It Out!
Evaluate
the expression c · __14 for each value of c. Write each
E
answer in simplest form.
a
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2a. c = __17
2b. c = __45
2c. c = __23
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5-6 Multiplying Fractions 197
Explore It!
5-6
Learn It!
Summarize It!
Name
Practice It!
Class
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Date
LA.6.4.2.2 The student will record
information…related to a topic,
including visual aids to organize and
record information…
Summarize It!
Multiplying Fractions
Think and Discuss
1. Determine whether the product of two proper fractions is greater than or less
than each factor.
1
2
· _23 = __
2. Record the missing denominator in the equation _____
.
21
6
4 · __
.
3. List two different ways to find the product of __
12 10
5 3
6 20
Method 1
Method 2
Simplify then
multiply.
Multiply then
simplify.
198 5-6 Multiplying Fractions
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4. Get Organized Complete the graphic organizer. Record how to use each method
to find the given product.
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Class
5-6
Date
Practice It!
MA.6.A.1.2 Multiply…fractions…
efficiently. Also MA.6.A.1.3
Multiplying Fractions
Multiply. Write each answer in simplest form.
1. __23 · __38
1
2. __15 · __
16
3. __37 · __14
4. __16 · __29
11 · __
1
5. __
13 8
6. __15 · __56
7. __12 · __49
8. __37 · __34
10. __67 · __13
3
9. __29 · __
11
Evaluate the expression x · _23 for each value of x. Write each answer in
simplest form.
Copyright © by Holt McDougal. All rights reserved.
1
11. x = __2
9
12. x = __
11
13. x = __35
14. x = __4
8
15. x = __
13
1
16. x = __
12
3
Compare. Write <, >, or =.
17. __37 · __29
9
2 · __
__
3 11
4
18. __23 · __
10
8 · __
1
__
15 2
19. __56 · __14
3
4 · __
__
5 5
20. Latoya has _34 of a bag of popcorn. If she eats half of the popcorn at a movie,
how much does she have left?
3
21. If Rupert flew __
of the way around the world, but George only flew _49 as
16
far Rupert, how far around the world did George fly? Simplify your answer.
22. Explain why you might want to simplify _12 · _23 · _34 · _45 before multiplying.
5-6 Multiplying Fractions 199
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Date
MA.6.A.1.2 Multiply… …efficiently.
Also MA.6.A.1.3
Multiplying Fractions
Use the circle graph for 1–3. Write each
answer in simplest form.
Brass, 18
4. On a snack table, _13 of the snacks involve
a dip of some kind. Of the dips, _12 include
cheese. The table has a total of 12 kinds of
snacks. How many of the snacks are dips
that include cheese?
Woodwinds, 14
5. Write three different pairs of fractions
whose product is the same.
Percussion, 18
Strings, 12
1. Of the students playing stringed
instruments, _34 play the violin. What fraction
of the whole orchestra are violin players?
3. There are 40 students in the orchestra. How
many students play either percussion or
brass instruments? Justify your answer.
200 5-6 Multiplying Fractions
7. Short Response
Consider the following fractions: _12 , _23 , and _34.
Find three different products by multiplying
pairs of these fractions. Which product
is equal to one of the original fractions?
Explain how to determine which product
has the least value.
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2. Two-thirds of the students who play a
percussion instrument are boys. What
fraction of the musicians in the orchestra
are boys who play percussion?
6. Jemma’s square handkerchief measures _56 ft
by _56 ft. What is the area of the handkerchief?
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Class
5-7
Date
Explore It!
MA.6.A.1.1 Explain and justify
procedures for multiplying…fractions.
Also MA.6.A.1.2
Multiplying Mixed Numbers
Model Mixed Number Multiplication
Tools
Go to thinkcentral.com
You can use shading of squares or rectangles to
model the product of a fraction and a mixed number,
or of two mixed numbers.
Activity 1
Find _12 · 2 _12 .
1 Think of _12 · 2 _12 as _12 of 2 _12 . Shade 2 _12 squares.
2 Divide the squares into halves.
3 Look at _12 of the part you shaded.
Copyright © by Holt McDougal. All rights reserved.
What does each small square represent?
How many small squares are there?
So, _12 of 2 _12 is _54, or 1 _14.
Try This
Write the multiplication expression modeled by each group of rectangles.
Then write the product.
1.
2.
×
=
3.
×
=
×
=
4.
×
=
5-7 Multiplying Mixed Numbers 201
Explore It!
Learn It!
Summarize It!
Practice It!
Apply It!
Use a model to find each product.
5. __13 · 1__13
6. __23 · 2__13
7. __14 · 2__23
Activity 2
Find 2_12 · 1_13.
1 Shade 1 _13 squares.
2 Think: I need 2_12 groups of 1_13 squares.
3 What does each small square represent?
How many small squares are there?
20
So, 2_12 · 1_13 = __
, or 3_13 .
6
Try This
Use a model to find each product.
9. 1__14 · 1__12
10. 1__12 · 1__13
Draw Conclusions
12. What product is shown on the model? Explain your reasoning.
202 5-7 Multiplying Mixed Numbers
11. 2__34 · 1__12
Copyright © by Holt McDougal. All rights reserved.
8. 2__12 · 1__12
Explore It!
Learn It!
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Name
Practice It!
Class
Apply It!
5-7
Date
Learn It!
MA.6.A.1.2 Multiply…fractions…
efficiently. Also MA.6.A.1.3
Multiplying Mixed Numbers (Student Textbook pp. 220–223)
Lesson Objective
Multiply mixed numbers
Example 1
Multiply.
M
lti l W
Write
i each answer in simplest form.
A. _14 · 1_13
_1 ·
Write 1__13 as an improper fraction. 1__13 =
·4
1____
4·3
Multiply
4
.
. Multiply
.
Write the answer in simplest form.
B. 3__12 · __45
Copyright © by Holt McDougal. All rights reserved.
· __45
·4
7____
2·5
Write 3__12 as an improper fraction. 3__12 =
Multiply
.
. Multiply
.
Write the answer in simplest form.
=
You can write the answer as a mixed number.
12 · 2__
3
C. __
13
8
12 ·
__
13
Write 2__38 as an improper fraction. 2__38 =
3
12 · ___
19
___
13 8 2
Use the GCF to simplify before
· 19
3_____
13 · 2
.
.
=
You can write the answer as a mixed number.
Lesson Tutorial Videos @ thinkcentral.com
5-7 Multiplying Mixed Numbers 203
Explore It!
Check It Out!
1a. 2__23 · __56
Learn It!
Summarize It!
Practice It!
Apply It!
M
Multiply.
Write each answer in simplest form.
1c. 3__35 · __16
1b. 2__14 · __29
Example 2
Find
Fi
d each
h product. Write the answer in simplest form.
A. 1__23 · 2__17
·
Write the mixed number as improper fractions.
· 15
5_____
3·7
2__17 =
Multiply
=
.
. Multiply
.
You can write the improper fraction as a mixed number.
Simplify.
B. 1__38 · 2__25
·
Write the mixed number as improper fractions.
1__38 =
2__25 =
.
11 · ___
123
__
8
5
2
Use the GCF to simplify before
·3
11
_____
2·5
Multiply
=
. Multiply
.
.
Simplify.
204 5-7 Multiplying Mixed Numbers
Lesson Tutorial Videos @ thinkcentral.com
Copyright © by Holt McDougal. All rights reserved.
1__23 =
Explore It!
Learn It!
Summarize It!
Practice It!
Apply It!
C. 2 · 4__25
2 · ( 4 + __25 )
Write 4__25 as a sum.
(2 · 4)
+ ( 2 · __25 )
Use the
Property.
(2 · 4)
+ ( __21 · __25 )
Write 2 as a
with a denominator of 1.
8+
Multiply.
Add.
Check It Out!
Find each product. Write each answer in simplest form.
2a. 1__34 · 2__16
Copyright © by Holt McDougal. All rights reserved.
2b. 3__23 · 2__14
2c. 3__18 · 2__23
Lesson Tutorial Videos @ thinkcentral.com
5-7 Multiplying Mixed Numbers 205
Explore It!
5-7
Learn It!
Name
Summarize It!
Practice It!
Class
Summarize It!
Apply It!
Date
LA.6.4.2.2 The student will record
information…related to a topic,
including visual aids to organize and
record information…
Multiplying Mixed Numbers
Think and Discuss
1. Tell how you multiply a mixed number by a mixed number.
2. Explain two ways you would multiply a mixed number by a whole number.
3. Explain whether the product of a proper fraction and a mixed number is less
than, between, or greater than the two factors.
Multiplying Fractions and Mixed Numbers
Example 1
Example 2
Multiplying a fraction
by a mixed number.
Multiplying a mixed number
by a mixed number.
206 5-7 Multiplying Mixed Numbers
Copyright © by Holt McDougal. All rights reserved.
4. Get Organized Complete the graphic organizer. Record the steps for multiplying
mixed numbers
Explore It!
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Name
Practice It!
Class
Apply It!
5-7
Date
Practice It!
MA.6.A.1.2 Multiply...fractions…
efficiently. Also MA.6.A.1.3
Multiplying Mixed Numbers
Multiply. Write each answer in simplest form.
1.
2__12 · __15
2.
1__14 · __49
6
3. 1__47 · __
11
4. 3__35 · __19
5 · __
4
5. 1__
16 7
6. 5__23 · __12
3
7. 3__13 · 1__
11
8. 1__18 · 2__45
9.
10 · __
3
1__
11 7
10. 6 · 4__13
Copyright © by Holt McDougal. All rights reserved.
Find each product. Write each answer in simplest form.
11. __15 · 2 __23
12. __12 · 1__27
13. 3 __45 · __17
11 ·__1
14. 1__
14 5
15. 2__34 · 3__12
6 · 1__
1
16. 2__
11
3
Compare. Write <, >, or =.
17. __25 · 4__12
7 · 1__
1
__
8
3
18. 1__13 · 1__35
4 · 5__
1
__
9
3
19. __25 · 5__23
2__25 · __23
20. Jamie needs 2_45 yards of ground to make his flower garden. If you only need _58
as much land to make your garden, how much land do you need?
21. Anthony has 2_14 pies. If his friends ate _13 of his pies, how many pies does he
have left?
22. Sydney worked on her scrapbook for 4_23 hours. She spent _14 of that time journaling
in the scrapbook. How much time did Sydney spend on other scrapbook tasks?
5-7 Multiplying Mixed Numbers 207
Explore It!
5-7
Learn It!
Name
Apply It!
Summarize It!
Class
Practice It!
Apply It!
Date
MA.6.A.1.2 Multiply…fractions…
efficiently. Also MA.6.A.1.3
Multiplying Mixed Numbers
Use the recipe to answer 1–3.
Blueberry Muffins
1
_
2
cup butter
4. A bucket can hold 4_34 gallons of water.
Amanda has 7 buckets that are each _13 full.
What is the total amount of water in the
buckets?
1_14 cup sugar
2 eggs
2 cups flour
2 tsp baking powder
1
_
tsp salt
2
1
_
cup milk
4
5. Jerri worked for 2_23 hours on beaded
necklaces. She completed 1_12 necklaces each
hour. How many whole necklaces did Jerri
complete?
2_12 cups blueberries
Makes 1 batch of muffins.
1. If you want to make 2_12 batches of blueberry
muffins, how much flour would you need?
3. Dan multiplied the recipe for blueberry
muffins so that he will use 2_12 cups of sugar.
How many cups of blueberries does Dan
need? Justify your answer.
208 5-7 Multiplying Mixed Numbers
a. What is the total amount of juice Maria
can squeeze from all the oranges?
b. A grapefruit will make 7_12 ounces of
juice. Will the juice of 11 oranges and
1 grapefruit fit in a 24 ounce bottle?
Explain.
c. How many 12 ounce bottles will Maria
need to contain the juice of 11 oranges
and 1 grapefruit? Justify your answer.
Copyright © by Holt McDougal. All rights reserved.
2. If you want to make 1_12 batches of blueberry
muffins, how many cups of milk would you
need?
6. Extended Response Maria has 11
oranges. She can squeeze 1_34 ounces of
orange juice from each orange.
Explore It!
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Summarize It!
Name
Practice It!
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Class
Explore It!
5-8
Date
MA.6.A.1.1 Explain and justify
procedures for…dividing fractions.
Also MA.6.A.1.2
Dividing Fractions and Mixed Numbers
Tools
Model Fraction Division
Go to thinkcentral.com
You can use shading of squares or rectangles to help
you understand division of fractions.
Activity 1
Find 4_12 ÷ 3.
1 Shade 4_12 squares. Think of 4_12 ÷ 3 as dividing
4_12 into 3 equal groups.
2 Divide the shaded parts into 3 equal groups.
3 Look at one of the shaded groups.
What number does it represent?
so, 4_12 ÷ 3 =
Copyright © by Holt McDougal. All rights reserved.
Try This
Write the division expression and quotient modeled by each group of rectangles.
1.
÷
=
Use a model to find each quotient.
2. 1__13 ÷ 2 =
3. 2__45 ÷ 2 =
Draw Conclusions
4. How do you know the number of groups into which you must divide the square?
5-8 Dividing Fractions and Mixed Numbers 209
Explore It!
Learn It!
Summarize It!
Practice It!
Apply It!
To divide by a fraction instead of a whole number, use the same method you used in
Activity 1.
Activity 2
A. Find 2_23 ÷ _23 squares.
1 Shade 2_23 squares.
2 Divide the shaded squares and shaded
thirds into equal groups of _23 .
There are 4 groups of _23 in 2_23 , so
2_23 ÷ _23 = 4.
B. Find 3 ÷ _34 .
3 Shade 3 squares. Then divide the squares
into fourths because the denominator of
_3 is 4.
4
4 Divide the shaded squares into equal
groups of _34 .
There are 4 groups of _34 in 3, so
3 ÷ _34 = 4.
Try This
5. 4 ÷ 1__13
6. 3__34 ÷ __34
7. 5__13 ÷ __23
8. 6__23 ÷ 1__23
Draw Conclusions
9. What prediction can you make about the value of 6 ÷ _34 if you know that
3 ÷ _34 is 4?
210 5-8 Dividing Fractions and Mixed Numbers
Copyright © by Holt McDougal. All rights reserved.
Use a model to find each quotient.
Learn It!
Explore It!
Summarize It!
Name
Practice It!
Class
Learn It!
Apply It!
5-8
Date
MA.6.A.1.2 …divide fractions…
efficiently. Also MA.6.A.1.3
Dividing Fractions and Mixed Numbers (Student Textbook pp. 224–227)
Lesson Objective
Divide fractions and mixed numbers
Vocabulary
reciprocal/multiplicative inverse
Example 1
Find
Fi
d th
the reciprocal.
i
A. __19
1·
__
9
1·
__
9
1
Think: __
of what number is 1?
9
=1
_1 of
=1
9
Copyright © by Holt McDougal. All rights reserved.
The reciprocal of _19 is
is 1.
.
B. __23
2·
__
3
2·
__
3
Think: _23 of what number is 1.
=1
_2 of
=1
The reciprocal of _23 is
3
is 1.
.
C. 3__15
16 ·
__
5
16 ·
__
5
=1
80 = 1
= __
80
16
The reciprocal of __
is
5
Lesson Tutorial Videos
16
Write 3_15 as __
.
5
16
__
of
5
is 1.
.
5-8 Dividing Fractions and Mixed Numbers 211
Learn It!
Explore It!
Check It Out!
Summarize It!
Practice It!
Apply It!
Find
each reciprocal.
F
1a. 4__18
1c. 2__14
1b. 5
Example 2
Divide.
Di
id W
Write
it each answer in simplest form.
A. __87 ÷ 7
Rewrite as multiplication using the reciprocal of 7,
.
·1
= 8____
7·7
Multiply by the reciprocal.
=
The answer is in simplest form.
B. __56 ÷ __23
2 = __
5 ÷ __
5·
__
3
6
6
Rewrite as multiplication using the reciprocal of _23 ,
.
1
5 · 3/
= ______
6/ · 2
Simplify before
=
Multiply.
=
You can write the answer as a mixed number.
.
2
212 5-8 Dividing Fractions and Mixed Numbers
Lesson Tutorial Videos
Copyright © by Holt McDougal. All rights reserved.
8
8 ÷ 7 = __
__
·
7
7
Explore It!
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Summarize It!
Practice It!
Apply It!
1
C. 2__34 ÷ 1__
12
1 = __
11 ÷
2__34 ÷ 1__
4
12
11
Write the mixed numbers as improper fractions. 2_34 = __
4
1
and 1__
=
12
11 · __
12
= __
4 13
3
.
Rewrite as multiplication.
11 · ___
12
= __
4 13
Simplify before
=
Multiply.
=
You can write the answer as a mixed number.
.
1
Check It Out!
Divide. Write each answer in simplest form.
Copyright © by Holt McDougal. All rights reserved.
7
2a. __
÷ __15
10
2b. 1__34 ÷ __18
2c. 2__15 ÷ 16__12
Lesson Tutorial Videos
5-8 Dividing Fractions and Mixed Numbers 213
Explore It!
5-8
Learn It!
Summarize It!
Name
Practice It!
Class
Summarize It!
Apply It!
Date
LA.6.4.2.2 The student will record
information…related to a topic,
including visual aids to organize and
record information…
Dividing Fractions and Mixed Numbers
Think and Discuss
1. Explain how you can use mental math to find the value of n in the equation
_5 · n = 1.
8
6
2. Explain how to find the reciprocal of 3__
.
11
3. Get Organized Complete the graphic organizer. Write the definition of
reciprocals and give several examples and nonexamples of reciprocals.
Definition
Examples
214 5-8 Dividing Fractions and Mixed Numbers
Nonexamples
Copyright © by Holt McDougal. All rights reserved.
Reciprocals
Explore It!
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Name
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Class
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5-8
Date
MA.6.A.1.2 …divide fractions…
efficientlly. Also MA.6.A.1.3
Dividing Fractions and Mixed Numbers
Find the reciprocal.
10
1. __
11
2. __12
3. __19
4. __25
6
5. __
15
6. __23
7.
2__37
4
8. 3__
11
Copyright © by Holt McDougal. All rights reserved.
Divide. Write each answer in simplest form.
9. __12 ÷ 2
10. 2__13 ÷ 2__45
9
10
11. __
÷ __
13 13
6
1 ÷ 2__
12. 4__
7
14
13. __34 ÷ 4
10
14. __
÷9
13
2
15. __89 ÷ __
15
10
16. __
÷ 1__12
11
17. Tina has 7_12 cups of bird food. If her bird feeder holds _12 of a cup of bird food,
how many times can Tina fill the bird feeder?
18. Mrs. Summer’s third grade class is putting on a play. If she has 13_18 yards of
fabric, and each student needs _78 of a yard for a costume, how many students can
make costumes?
1
19. Joseph’s height is 6__
ft. His brothers Thomas and Michael are 5_12 feet tall and
12
4_56 feet tall. What is the average height of the 3 brothers?
20. Pia picked a total of 125 pounds of raspberries this summer. Each bucket holds
15_12 pounds of berries. How many buckets will Pia need to hold all the berries
she picked? Justify your answer.
5-8 Dividing Fractions and Mixed Numbers 215
Explore It!
5-8
Learn It!
Summarize It!
Name
Practice It!
Class
Apply It!
Apply It!
Date
MA.6.A.1.2 …divide fractions…
efficiently. Also MA.6.A.1.3
Dividing Fractions and Mixed Numbers
1. Lynae lives 27 miles from the mall. Her
mother drives there in _12 hour. What was her
mother’s average speed in miles per hour?
2. There are 6 markers evenly posted along a
hiking trail to direct hikers. Use the diagram
to determine how far apart the markers are
placed.
9
10 mi
4. Hugo has a piece of flannel fabric that is
5_34 feet wide. He cuts it into 4 pieces of equal
width. What is the combined width of 3 of
the pieces? Justify your answer.
5. Aaron spent 2_14 hour skateboarding
at different parks on saturday. He
skateboarded for _34 hour at each park.
How many parks did Aaron go to on
Saturday?
7. At Rene’s birthday party, he and his friends
played a game. Each person on two equallysized teams ran across the yard holding a _12
cup measuring cup filled with water. The
person poured the water into a bucket and
ran back to give the measuring cup to the
next person on the team. At the end of the
race, there was a total of 7 cups of water in
the two buckets. Assume that no one spilled
any water during the race. How many
people were on each team?
8. Gridded Response
Dry goods are sold
in units called pecks
and bushels. One peck
equals _14 bushel. If
Peter picks 5_12 bushels
of peppers, how many
pecks of peppers did
Peter pick?
216 5-8 Dividing Fractions and Mixed Numbers
Copyright © by Holt McDougal. All rights reserved.
3. Ariel has a piece of yarn that is 12_16 feet long.
She cuts off pieces that are 2_13 feet long.
What is the maximum number of pieces of
that length that Ariel can cut from the yarn?
6. In printing, fonts are measured in units
1
called points. One point equals __
inch. You
72
are typing a paper on the computer, and
you want the title of the paper to be _12 inch
tall. Explain how to determine how many
points are in _12 inch.
5-5
Name
Class
Date
THROUGH
5-8
Got It?
Ready to Go On?
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Quiz for Lessons 5-5 through 5-8
5-5
Multiplying Fractions by Whole Numbers (Student Textbook pp. 212–215)
Multiply. Write your answer in simplest form.
1. 4 · __23
3. 3 · __56
2. 5 · __14
4. Michelle ordered 5 lb of fruit for a family picnic. Of that fruit, _13 was
watermelon. How much of the fruit was watermelon?
2
5. Philip has 35 comic books. Of those comics, __
take place in space.
10
How many of Philip’s comic books take place in space?
5-6
Multiplying Fractions (Student Textbook pp. 216–219)
Multiply. Write each answer in simplest form.
6. __27 · __34
7. __35 · __23
9. __57 · __45
10 · __
3
10. __
21 5
7 · __
4
8. __
12 5
11. __56 · __23
Evaluate the expression t · __18 for each value of t. Write the answer in
simplest form.
Copyright © by Holt McDougal. All rights reserved.
12. t = __49
5-7
13. t = __45
14. t = __23
Multiplying Mixed Numbers (Student Textbook pp. 220–223)
Multiply. Write each answer in simplest form.
15. __14 · 2__13
16. 1__16 · __23
17. __78 · 2__23
18. 2__14 · 1__16
19. 1__23 · 2__15
20. 3 · 4__27
5-8
Dividing Fractions and Mixed Numbers (Student Textbook pp. 224–227)
Find the reciprocal.
21. __27
5
22. __
12
23. __35
Divide. Write each answer in simplest form.
24. __35 ÷ 4
3 ÷ 3__
1
25. 1__
4
10
26. 1__15 ÷ 2__13
27. 1__57 ÷ 7__15
1
28. 6__25 ÷ 1__
15
29. 36 ÷ 3__35
Chapter 5 Fraction Operations 217
5-5
THROUGH
5-8
Name
Class
Date
Connect It!
MA.6.A.1.1; MA.6.A.1.2;
MA.6.A.5.3
Connect the concepts of Lessons 5-5 through 5-8
It’s in the Cards
Play this game with a partner.
1
4
1
1 21
2
3
3
4
1 31
22
1 43
1
2
3
23
1. Write each of the values shown on an index card.
2. Shuffle the cards and lay them face down. Player A
turns over two cards. Player B tells Player A whether
to multiply or divide the two values. When dividing,
Player B tells the order in which the values are to be
divided. Player A writes down the result as his or her
score, and removes the two cards.
3. Players alternate roles until there are no more cards.
Use estimation to determine which player has a
greater total score.
2
1
1
3
4. Describe any strategies you used for giving your opponent the least
possible score on each turn.
Which planet in the Solar System has the strongest winds?
Circle the letter on every rocket that has an expression whose value is
greater than 1. Arrange the circled letters to form the name of the planet
where wind speeds reach 1,300 mi/h.
14 · 57
1 23 ÷ 23
3÷3
5 8
E
T
3÷5
5 3
2 13 · 67
218 Chapter 5 Fraction Operations
2 ·31
5
4
U
S
P
5 ÷ 38
N
2·3
9 4
V
E
1 ÷ 45
N
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Blast Off!
FLORIDA
Name
Class
Study It!
Vocabulary
Date
Multi-Language
Glossary
Go to thinkcentral.com
5
(Student Textbook page references)
reciprocal . . . . . . . . . . . . . . (224)
Lesson 5-5
CHAPTER
multiplicative inverse . . . (224)
Multiplying Fractions by Whole Numbers (Student Textbook pp. 212–215)
M
MA.6.A.1.2,
MA.6.A.1.3
Multiply. Write your answer in simplest form.
3 · __35
3 + __
3 + __
3
__
5
5
5
Write 3
9 or 1__
4
__
5
5
Write your answer in simplest form.
· _35 as addition. Add the numerators.
5 · __34
5 · __
3
__
1 4
·3
5____
1·4
3
15 = 3__
__
4
4
Multiply
Write your answer in simplest form.
Copyright © by Holt McDougal. All rights reserved.
Multiply. Write each answer in simplest form.
1. 5 · __17
2. 2 · __38
3. 3 · __67
4. 4 · __58
5. 6 · __12
6. 2 · __35
Evaluate each expression for the given value of the variable. Write each answer
in simplest form.
11
7. 16y for y = __
16
8. __34b for b = 28
9. __18k for k = 32
10. There are 105 members of the high school band. Of these
members, _15 play percussion instruments. How many members
play percussion?
11. There are 18 players on the baseball team. Of these players, _13 are
girls. How many girls play on the baseball team?
Lesson Tutorial Videos @ thinkcentral.com
Chapter 5 Fraction Operations 219
Lesson 5-6
Multiplying Fractions (Student Textbook pp. 216–219)
M
MA.6.A.1.2,
MA.6.A.1.3
Multiply. Write each answer in simplest form.
1
3 · __
__
4 3
·1
3____
4·3
3 = __
1
__
4
12
3
4 · ___
__
5 16
1
3 = __
3
4 · ___
1 · __
___
5 164
5 4
Multiply
Write your answer in
simplest form.
·3
= 1____
5·4
3
= __
20
Use the GCF to
simplify the fractions
before multiplying.
Multiply numerators.
Multiply denominators.
The answer is in
simplest form.
Multiply. Write each answer in simplest form.
12. __56 · __25
13. __57 · __34
14. __45 · __18
7 · __
2
15. __
10 5
16. __19 · __59
17. __14 · __67
18. Andrew’s hockey team won _45 of their games. Andrew scored in _23 of the games
his team won. In what fraction of his team’s games did Andrew score?
Lesson 5-7
Multiplying Mixed Numbers (Student Textbook pp. 220–223)
M
MA.6.A.1.2,
MA.6.A.1.3
Multiply. Write each answer in simplest form.
1__13 · 3__34
Write 1_23 as _53 .
15
4 · ___
__
3 4
Multiply. Then simplify.
1
155
4 · ___
___
41
3
1
·5
1____
1·1
Write the mixed numbers as
improper fractions.
Use the GCF to simplify the
fractions before multiplying.
Multiply numerators.
Multiply denominators.
5
Multiply. Write each answer in simplest form.
19. __25 · 2__14
20. __34 · 1__23
21. 3__13 · __35
22. 1__13 · __14
23. 2__25 · 1__12
24. 2__34 · 2__13
25. Justin has 18 disc golf discs. Matt has 1__23 times as many discs as
Justin. How many discs does Matt have?
220 Chapter 5 Fraction Operations
Lesson Tutorial Videos @ thinkcentral.com
Copyright © by Holt McDougal. All rights reserved.
2
2 · 1__
__
3
5
5
2 · __
__
5 3
10 = __
2
__
15
3
Lesson 5-8
D
Dividing Fractions and Mixed Numbers (Student Textbook pp. 224–227)
MA.6.A.1.2,
MA.6.A.1.3
Divide. Write each answer in simplest form.
3÷6
__
4
3 · __
1
__
4 6
·1
3____
4·6
3 = __
1
__
24
8
Rewrite as multiplication using the reciprocal of 6.
Multiply numerators. Multiply denominators.
Write the answer in simplest form.
7__12 ÷ 1__12
15 ÷ __
3
___
2
2
15 · __
2
__
2 3
5
15 · 21
______
2 · 31
1
5=5
__
1
Write the mixed numbers as improper fractions.
Rewrite as multiplication using the reciprocal of _32 .
Simplify before multiplying.
Write the answer as a mixed number.
Divide. Write each answer in simplest form.
26. __47 ÷ 3
3
27. __
÷2
10
28. 1__13 ÷ 2__25
29. 9 ÷ __56
30. 1__34 ÷ __18
3
31. 4__15 ÷ 1__
10
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32. A jumbo serving of frozen yogurt is 2_34 cups. A container holds 55
cups of frozen yogurt. How many jumbo servings of frozen yogurt
are in the container?
33. Debbie bought 3_12 lb of ground turkey. She divided the turkey into _14
lb burger patties. How many patties did Debbie make?
34. Beverly needs to measure 2_23 cups of bread crumbs. She has a _13
cup measuring scoop. How many times must she fill the _13 cup
measuring scoop to get 2_23 cups of bread crumbs?
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Chapter 5 Fraction Operations 221
Name
Class
Write About It!
Date
LA.6.4.2.2 The student will record
information (e.g., observations, notes,
lists, charts, legends) related to a
topic…
Think and Discuss
Answer these questions to summarize the important concepts from Chapter 5
in your own words.
1. Explain how to multiply _57 by 6.
3
2 ·_
2. Explain two methods for multiplying __
15 8
3. Explain how to divide 4_34 by 2_12
I need to answer these questions:
222 Chapter 5 Fraction Operations
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Before The Test