Multi-Part Lesson
2-2
Multiply Fractions
PART
A
B
C
D
Multiply Fractions
Main Idea
Multiply fractions using
models.
NGSSS
MA.6.A.1.1 Explain
and justify
procedures for
multiplying and dividing
fractions and decimals.
glencoe.com
In Lesson 1-1D, you used
decimal models to multiply
decimals. You can use a
similar model to multiply
fractions.
BACKYARDS One half of Mr. Lytle’s backyard is a garden. One
third of his garden is covered with flowers. What fraction of
Mr. Lytle’s backyard is covered with flowers?
What do you need to find? what fraction of Mr. Lytle’s backyard
is covered with flowers
1 _
To find _
× 1 , find the area of a _- by _-unit rectangle.
3
1
3
2
1
2
Divide a square in half.
Then divide it in thirds.
1
2
1
3
Shade a rectangle that
is _-unit wide and
1
2
1
_
-unit long.
3
1
2
One sixth of the square is shaded.
1
3
So, _ of Mr. Lytle’s backyard is covered with flowers.
1
6
and Apply
Find each product using a model.
1 _
1. _
×1
4
102
2
Chapter 2 Multiply and Divide Fractions
1 _
2. _
×1
3
4
1 _
3. _
×1
2
5
BAKING Mia is baking brownies. She covers
3
_
of the brownies with icing.
5
2
Of the brownies with icing, she covers _ of them with sprinkles. What
3
fraction of Mia’s brownies are covered with icing and sprinkles?
What do you need to find? the fraction of Mia’s brownies that are covered
with icing and sprinkles
3 _
3
2
To find _
× 2 , find _
of _
.
5
3
5
3
Divide a square into thirds.
Then divide it in fifths.
2
3
3
5
Shade a rectangle that
is _-unit wide and
2
3
3
_
-unit long.
2
3
5
3
5
6
2 _
=_
, 2 of the brownies are
Six out of 15 parts are shaded. Since _
5 5
15
covered with both icing and sprinkles.
and Apply
Find each product using a model. Then write in simplest form.
3 _
4. _
×2
4
2 _
5. _
×5
3
5
6
4 _
6. _
×3
5
8
the Results
10
2 _
7. Draw a model to show that _
× 5 =_
. Then explain how the model
3
10
5
shows that _
simplifies to _
.
18
6
18
9
8. Explain the relationship between the numerators of the problem and the
numerator of the product. What do you notice about the denominators
of the problem and the denominator of the product?
9. MAKE A CONJECTURE Write a rule you can use to multiply fractions.
Lesson 2-2 Multiply Fractions
103
Multi-Part Lesson
2-2
Multiply Fractions
PART
Main Idea
Multiply fractions.
NGSSS
MA.6.A.1.2 Multiply
and divide fractions
and decimals efficiently.
MA.6.A.1.3 Solve realworld problems involving
multiplication and division
of fractions and decimals.
Also addresses MA.6.A.1.1.
A
B
C
D
Multiply Fractions
REPTILES A chameleon’s body is about
_1 the length of its tongue. A certain
2
chameleon has a tongue that is
_2 foot long. The shaded rectangle
3
1
2
1 _
of _
or _
× 2.
represents _
2
3
3
1. Refer to the model. What fraction
1 _
represents _
× 2?
2
glencoe.com
2
3
2. What is the relationship between
the numerators and denominators
of the factors and the numerator
and denominator of the
product?
2
Tongue is 3 foot.
Body is
1
2
2 of 3 .
Multiply Fractions
Words
Multiply the numerators and multiply the denominator.
Examples
Numbers
Algebra
2×1
_2 × _1 = _
5
2
a×c
_a × _c = _
, where b and d are not 0.
5×2
d
b
b×d
Multiply Fractions
_ _
Find 1 × 1 .
1
3
3
4
1×1
_1 × _1 = _
3
4
3×4
1
=_
12
Multiply the numerators.
Multiply the denominators.
1
4
Simplify.
Multiply. Write in simplest form.
1 _
a. _
×3
2
104
5
Chapter 2 Multiply and Divide Fractions
1 _
b. _
×3
3
4
2 _
c. _
×5
3
6
If the numerators and the denominators have a common factor, you
can simplify before you multiply.
Factor two or more
numbers that are
multiplied together to form
a product; Example: 1, 2, 3,
and 6 are all factors of 6
Simplify Before Multiplying
_ _
Find 3 × 5 .
1
1
Estimate _
×1=_
2
2
6
4
1
3×5
_3 × _5 = _
4×6
6
4
Divide both the numerator and the denominator by 3.
2
5
=_
8
Simplify. Compare to the estimate.
3
4
d. _
×_
9
5
e. _
×_
9
4
LAWN MOWING Frank had
_
3
f. _
× 10
10
6
5
_1 of the lawn left to mow. On
2
Saturday, he mowed 2 of what was left. What fraction of the
3
entire lawn did Frank mow on Saturday?
1
1×2
_1 × _2 = _
2
3
2×3
Divide both the numerator and denominator by 2.
1
1
=_
3
Simplify.
1
So, Frank mowed _
of the lawn on Saturday.
3
1
g. SUB SANDWICHES Rick has _
of a footlong sub left from
2
1
yesterday. He ate _
of the leftover sandwich as a snack. What
3
fraction of the entire sandwich did he eat as a snack?
Examples 1 and 2
(pp. 104–105)
Multiply. Write in simplest form.
1 _
1. _
×1
4
3. _
× 10
3
5
5
3
5
3 _
5. _
×_
6. _
×5
6
5
6
10
3
MONEY Juanita spent _ of her allowance at the mall. Of the money
4
1
spent at the mall, _
was spent on new earphones. What part of her
2
8
2
3
4. _ × 12
4
Example 3
(p. 105)
7
2 _
2. _
×4
allowance did Juanita spend on earphones?
Lesson 2-2 Multiply Fractions
105
= Step-by-Step Solutions begin on page R1.
Extra Practice begins on page EP2.
Multiply. Write in simplest form.
Examples 1 and 2
(pp. 104–105)
1 _
8. _
×2
1 _
9. _
×3
3 _
10. _
×5
2 _
11. _
×3
5
7
3
12. _
×2
4
_2 × 4
5
14. _
× 15
6
3 _
17. _
×5
5
7
3
15. _
× 11
8
4 _
18. _
×3
9
8
3
5
8
4
4
13
8
3
2 _
16. _
×1
3
4
2 _
_
19. × 5
5
6
7
1
20. SPORTS The bleachers at a football game are _
full, and _
of the fans
Example 3
8
(p. 105)
2
in the bleachers are rooting for the home team. What fraction of the
bleachers are filled with home-team fans? Justify your procedure.
21. ELECTIONS The table shows the fraction of
the votes that each candidate received. If
230 students voted, how many students
voted for each candidate?
B
22. HORSES The height of a horse is often
measured in hands, which is the distance
across an adult’s palm. A pony measures
Candidate
Fraction
of Votes
Nyemi
_3
Luke
3
_
Natalie
1
_
5
10
10
1
13 hands. If a hand is equal to _
foot, about how tall is the pony?
3
2
23. INVENTORY A paint store has 35 gallons of paint in storage, _
of which
5
Real-World Link
Clydesdales can
stand as tall as
20 hands and
weigh as much
as 2,000 pounds.
are for outdoor use. The others are for indoor use. If each gallon costs
$22, what is the total cost of the indoor paint in storage?
24. PIZZA Tico ate _ of a pizza. If there were 16 slices of pizza, how
5
8
many slices did Tico eat?
Multiply.
1 _
1
25. _
× 1 ×_
2
3
4
15
1 _
27. _
× 2 ×_
2
5
16
2 _
2
26. _
× 3 ×_
3
3
4
5
2 _
28. _
× 9 ×_
3
9
10
29. PHYSICAL EDUCATION Mr. Williams’ P.E. class lasts
7
for _
hour.
8
_1
a. How many minutes are spent warming
up and cooling down?
playing game
instruction
_1
b. How many minutes are not spent on
instruction? Explain.
warm-up and
cool-down
8
106
_7
Part of -hour Class
Chapter 2 Multiply and Divide Fractions
2
5
3
_
10
30. WATER PARK Alberto rode _ of the water rides at a water park. His
8
sister, Reina, rode half of the rides that Alberto rode. What fraction of
the water rides did Reina not ride? Support your answer with a model.
5
31. RECIPES Lee is making oatmeal chocolate chip cookies and the recipe
3
2
calls for _
cup of chocolate chips. If she wants to make _
of the recipe,
3
4
what fraction of a cup of chocolate chips will she need? Justify your
procedure.
32.
MULTIPLE REPRESENTATIONS Use the
bar diagram.
$100
a. WORDS Write a real-world problem
represented by the bar diagram.
?
b. MODELS Draw an area model to
represent the situation.
c. WORDS Explain how you would solve your problem.
33 BULLETIN BOARD Homeroom 101 and Homeroom 102 share a hallway
3
bulletin board. If Homeroom 101 uses _
of their half to display
5
artwork, what fraction of the bulletin board is used to display
Homeroom 101’s artwork?
2
34. MUSIC About _
of an orchestra is made
3
1
up of string instruments. If _
of the
5
string instruments are cellos, what
fraction of the orchestra is made up
of cellos?
Counterexample
A counterexample is an
example that shows why a
statement is false.
35. REASONING State whether each statement is true or false. If the
C statement is false, provide a counterexample.
a. The product of two fractions that are each between 0 and 1 is also
between 0 and 1.
b. The product of a mixed number between 4 and 5 and a fraction
between 0 and 1 is less than 4.
c. The product of two mixed numbers that are each between 4 and 5
is between 16 and 25.
36. NUMBER SENSE If the product of two positive fractions a and b is _,
56
find three pairs of possible values for a and b.
15
37. CHALLENGE Is the product of two positive fractions that are each less
than 1 also less than 1? Explain.
38.
b _
d
Justify why _a × _
× c ×_
is equal to _a .
b
c
d
e
e
Lesson 2-2 Multiply Fractions
107
NGSSS Practice
MA.4.A.6.3, MA.6.A.1.2, MA.6.A.1.3
39. In a recent survey, _ of pet owners
5
8
40.
stated that they allow their pet to go
1
outside. Of these, _
allow their pet
3
outside without supervision. Which
expression gives the fraction of the pet
owners surveyed that allow their pet
outside without supervision?
5 _
A. _
+1
8
3
5
1
B. _ - _
8
3
EXTENDED RESPONSE Four fifths of
Terrence’s text messages are to his
friends. One half of those messages are
to his friend Bianca.
Part A Draw a model to represent
the situation.
Part B What fraction of Terrence’s
text messages are to Bianca?
5 _
C. _
×1
8
3
5
1
D. _ ÷ _
8
3
Part C Explain how you solved the
problem.
41. Scott is taking a dance class twice a week for 8 weeks.
How many hours altogether will Scott have spent in
dance class at the end of the 8 weeks?
F. 6 hours
H. 12 hours
G. 8 hours
I. 16 hours
Erin’s Dance Studio
3
hour classes
4
-
42. SATELLITES A main satellite sends a signal to each of two smaller
satellites. If each of those two satellites sends a signal to each other
and a signal back to the main satellite, determine the number of
signals sent. (Lesson 2-1E)
Multiply. Write in simplest form.
1
43. 13 × _
3
(Lesson 2-1D)
5
44. 16 × _
3
45. _
× 21
6
7
1
46. MEASUREMENT A game board measures 9_
inches by 11_ inches.
Estimate the area of the game board.
2
3
4
(Lesson 2-1B)
47. GASOLINE Benita spent $38.40 to fill up her gas tank. If she pumped
9 gallons of gasoline, is about $3, $3.50, or $4 a more reasonable estimate
for the cost of each gallon of gasoline? (Lesson 1-3C)
48. DECORATING Selena is considering new
carpeting for her living room. How many
square feet of carpet will Selena need?
16.3 ft
(Lesson 1-1E)
24.6 ft
108
Chapter 2 Multiply and Divide Fractions
Multi-Part Lesson
2-2
Multiply Fractions
PART
Main Idea
Multiply whole
numbers by mixed
numbers. Explain and
justify the process of
multiplying whole
numbers and mixed
numbers.
A
B
C
D
Multiply Whole Numbers
by Mixed Numbers
MUSIC Genevieve is designing
an insert to put into a CD case.
She measured the case and it is
1
5 inches by 5_
inches. What is
2
the area of the case?
NGSSS
MA.6.A.1.1 Explain
and justify
procedures for
multiplying and dividing
fractions and decimals.
What do you need to find? the area of a CD case
Use an area model.
Shade a rectangle
that is 5 units wide
1
5 2 units
_1
and 5 units long.
glencoe.com
2
5 units
Find the total number of shaded units.
25 units
4 half-units 1 half-unit
or 2 whole
units
The area of the case is 27_ square inches.
1
2
and Apply
Find each product using models. Justify your procedure.
2
1. 3 × 2_
5
1
2. 2 × 3_
4
1
3. 5 × 3_
6
Lesson 2-2 Multiply Fractions
109
Multi-Part Lesson
2-2
Multiply Fractions
PART
Main Idea
Multiply mixed
numbers.
NGSSS
MA.6.A.1.2 Multiply
and divide fractions
and decimals efficiently.
MA.6.A.1.3 Solve realworld problems involving
multiplication and division
of fractions and decimals.
Also addresses MA.6.A.1.1
and MA.6.A.5.3.
A
B
D
C
Multiply Mixed Numbers
ANATOMY The Atlantic Giant
Squid has an eyeball that is about
12 times as large as the average
human eyeball. If the average
1
inches across,
human eyeball is 1_
4
how large is the Atlantic Giant
Squid’s eyeball?
1. Write a multiplication expression that shows the size of the
Atlantic Giant Squid’s eyeball.
1
2. Use repeated addition to find 12 × 1_
.
4
glencoe.com
3. Write the multiplication expression from Exercise 1 using
improper fractions.
4. Multiply the improper fractions from Exercise 3. How large is
the Atlantic Giant Squid’s eyeball?
Multiply Mixed Numbers
To multiply mixed numbers, write the mixed numbers as improper
fractions and then multiply as with fractions.
Multiply a Fraction and a Mixed Number
_ _
Find 1 × 1 3 .
3
4
1
Estimate Use compatible numbers. _
×2=1
2
_1 × 1_3 = _1 × _7
3
4
3
4
1×7
=_
3×4
7
=_
12
3 _
Write 1_
as 7 .
4
4
Multiply.
Simplify. Compare to the estimate.
Multiply. Write in simplest form.
1
2
a. _
× 2_
3
110
2
Chapter 2 Multiply and Divide Fractions
3
1
b. _
× 3_
8
3
1 _
c. 3_
×1
2
3
Multiply Mixed Numbers
_ _
Find 1 7 × 3 1 .
8
3
15
10
7
1
1_ × 3_ = _ × _
8
3
8
3
5
5
8
3
15 _
=_
× 10
4
Write 1_ as _. Write 3_ as _.
15
8
7
8
1
3
10
3
Divide the numerator and denominator by 3 and by 2.
1
25
=_
Multiply.
4
1
= 6_
4
Simplify.
_1
DAMS The Hoover Dam contains 4 million cubic yards of
2
concrete. The Grand Coulee Dam, in Washington state, contains
2 2 times as much concrete. How much concrete does it contain?
_
3
Estimate 4 × 3 = 12
8
1
2 _
4_
× 2_
= 9 ×_
2
3
2
3
3
4
2
3
9 _
=_
×8
Real-World Link
The Hoover Dam,
located on the
Arizona-Nevada
border, contains
enough concrete
to pave a highway,
16 feet wide, from
San Francisco to
New York City.
1
Write the mixed numbers as improper fractions.
Divide 9 and 3 by their GCF, 3. Then divide 8 and 2 by their GCF, 2.
1
3 _
=_
×4
1
1
12
= _ or 12
1
Multiply the numerators and multiply the denominators.
Simplify.
There are 12 million cubic yards of concrete in the Grand
Coulee Dam.
Check for Reasonableness 12 = 12
d. MEASUREMENT Mr. Wilkins is laying bricks to make a rectangular
3
1
patio. The area he is covering with bricks is 15_
feet by 9_
feet.
2
4
What is the area of the patio?
Examples 1 and 2
(pp. 110–111)
Multiply. Write in simplest form.
3
1
1. _
× 2_
2
1 _
2. 1_
×2
8
2
4
4. 1_
× 2_
3
7
Example 3
(p. 111)
7
3
4
3. 1_
× 2_
2
3
1
1
5. 2_ × 4_
7
5
5
4
9
1
6. 1_ × 1_
3
10
1
COOKING A waffle recipe calls for 2_
cups of flour. If Chun wants to
4
1
make 1_
times the recipe, how much flour does he need?
2
Lesson 2-2 Multiply Fractions
111
= Step-by-Step Solutions begin on page R1.
Extra Practice begins on page EP2.
Examples 1 and 2
(pp. 110–111)
Multiply. Write in simplest form.
1
1
8. _
× 2_
3
5
9. _
× 2_
2
3
4 _
_
11. 1 × 5
5
6
1
1
14. 1_
× 1_
3
4
1
_
_
17. 4 × 2 5
2
6
Example 3
(p. 111)
7 _
10. 1_
×4
6
4
7
_
_
12. × 3 1
8
4
1
1
15. 3_
× 3_
5
6
2
_
_
18. 6 × 3 3
3
10
8
5
3
5
13. _
× 2_
6
10
3
2
16. 3_
× 2_
5
4
3
5
19. 3_
× 5_
5
12
20. MEASUREMENT A reproduction of Claude
Monet’s Water-Lilies has dimensions
1
1
34_
inches by 36_
inches. Find the area
2
2
of the painting.
21 FISH A carp can travel at a speed of
7
3_
miles per hour. At this rate, how
10
1
far can a carp travel in 2_
hours?
2
Multiply. Write in simplest form.
B
3
1 _
22. _
× 2_
×4
2
4
3
1 _
23. 1_
× 2 ×_
5
2
3
5
5
1
1
25 _ × 5_ × 1_
4
7
6
2
1
2
24. 3_
× 4_
× 2_
5
2
3
26. RUNNING Use the formula d = rt to find the distance d a long-distance
3
1
runner can run at a rate r of 9_
miles per hour for time t of 1_
hours.
2
4
27. ASTRONOMY Earth is about 92_ million
9
10
Planet
Approximate
Number of Times
as Far from the
Sun as Earth
b. How far is Mars from the Sun?
Venus
_3
c. How far is Jupiter from the Sun?
Mars
miles from the Sun. Use the table shown.
a. How far is Venus from the Sun?
d. How far is Saturn from the Sun?
4
1
1_
2
1
_
5
4
1
_
9
2
Jupiter
Saturn
28. FIND THE DATA Refer to the Data File on
pages 2–5. Choose some data and write a
real-world problem in which you would
multiply mixed numbers. Then solve your problem.
_
_
_
3
ALGEBRA Evaluate each expression if a = 2 , b = 3 1 , and c = 1 .
3
29. ab
112
1
30. _
c
Chapter 2 Multiply and Divide Fractions
2
31 bc
2
4
1
32. _
a
8
C
33. OPEN ENDED Name two positive mixed numbers, each greater than 1
and less than 2, with a product greater than 1 and less than 2.
34. CHALLENGE Analyze each product in
the table.
First
Factor
Second
Factor
_1
a. Why is the first product less
3
than _
?
1
4
b. Why is the second product equal
4
_3
4
_3
4
×
_3
3
to _
?
_3
×
2
×
2
Product
=
_3
=
_3
=
_9
8
4
8
4
c. Why is the third product greater
3
than _
?
4
35. NUMBER SENSE Without multiplying,
"
1 _
determine whether the product 2_
×2
2
#
0
3
1
$
2
3
is located on the number line at point A,
B, or C. Explain your reasoning.
36.
NGSSS Practice
Summarize how to multiply mixed numbers.
MA.4.A.6.3, MA.6.A.1.2, MA.6.A.1.3
37. Which number when multiplied by _
3
gives a product between _
and 1?
4
3
A. 0
C. _
4
1
1
D. 1_
B. _
4
4
Multiply. Write in simplest form.
5 _
39. _
×3
7
4
2 _
40. _
×1
3
6
3
4
38.
GRIDDED RESPONSE Ben is taking
guitar classes three times a week for
3
8 weeks. Each class will last 1_
hours.
4
How many hours will Ben have spent
in guitar classes in 8 weeks?
(Lesson 2-2B)
3 _
41. _
×2
8
5
1 _
42. _
×4
2
7
43. TREES The Palmers are planting trees along each side of their 52-foot-long
driveway. There is 4 feet between each tree, and the trees are planted
directly across from each other. How many trees do they need? (Lesson 2-1E)
Find each sum or difference. Write in simplest form.
4 _
44. _
+1
5
3
45.
_3 - _1
8
6
46.
(Lesson 0-4)
_6 + _1
7
4
47.
9
1
_
-_
11
2
Lesson 2-2 Multiply Fractions
113
CHAPTE R
2
Mid-Chapter
Check
Estimate each product.
1
1. _
× 15
7
4 _
3. _
×3
7
5
Multiply. Write in simplest form.
(Lesson 2-1B)
1
2. 26 × _
3
3
1
4. 4_
× 2_
8
4
3 _
13. _
×1
1 _
14. _
×4
4
5
7 _
_
15. × 2
3
8
5. CLOTHES A new shirt costs $14.99. If the
1
shirt is on sale for _
off its price, about
5
how much would you save? (Lesson 2-1B)
6
2
6. 8 × _
(Lesson 2-1D)
3
7. _
×9
5
5
_
8. 5 ×
6
4
_
9. 1 × 12
7
10. MEASUREMENT Find the area of the
magnet shown. (Lesson 2-1D)
7
8 in.
5
2 _
16. _
×5
6
7
1
17. NGSSS PRACTICE A recipe calls for 2_
cups
2
of flour. If you triple the recipe, how
much flour is needed? (Lesson 2-2D)
1
B. 7_
c
A. 7 c
Multiply. Write in simplest form.
(Lesson 2-2B)
2
1
D. 8_
c
C. 8 c
2
18. DOG FOOD A cup of dog food weighs
7
2
1_
ounces. A K9 dog eats 5_
cups of dog
8
3
food a day. How many ounces of dog
food does the dog eat each day? (Lesson 2-2D)
19. NGSSS PRACTICE What is the area of the
picture and frame shown? (Lesson 2-2D)
1
7 4 in.
2 in.
2
11 3 in.
11. WEATHER In a recent year, the weather
4
was sunny _
of the days. Assuming there
5
are 365 days in a year, how many days
were not sunny? (Lesson 2-1D)
12. FOOD Rosemaria wants to cut her sub
into thirds. Find the length of each piece.
(Lesson 2-2B)
1
2 ft
7
F. 84_
square inches
12
5
square inches
G. 83_
6
1
square inches
H. 82_
2
1
square inches
I. 77_
6
20. CONSTRUCTION Corey needs 24 boards
1
that are 47_
inches long.
2
a. How many feet of boards should he
buy? Explain.
b. If you can only buy 8-foot boards,
how many should he buy? Explain.
(Lesson 2-2D)
114
Chapter 2 Multiply and Divide Fractions