Number, Operations, and Algebraic
Thinking
Focal Point
Use fractions, decimals, and
integers to solve problems
and algebraic equations.
CHAPTER 11
Multiplying and Dividing
Decimals and Fractions
Add, subtract, multiply,
and divide to solve problems and
justify solutions.
CHAPTER 12
Algebra: Integers and
Equations
Add, subtract, multiply, or
divide to solve problems and justify
solutions.
Use equations to solve
problems.
528
Art Wolfe/Getty Images
Math and Science
Cooking Up A Mystery! Have you ever wanted to be a scientist
so you could mix substances together to create chemical reactions?
Come join us on an explosive scientific adventure. Along the
way, you’ll discover the recipe for making a homemade volcano
erupt. You’ll also gather data about real volcanoes. So pack your
math tools and don’t forget a fireproof suit. This adventure is
hot, hot, hot!
Log on to tx.msmath1.com to begin.
Unit 5 Number, Operations, and Algebraic Thinking
529
Multiplying and Dividing
Decimals and Fractions
11
Knowledge
and Skills
•
Add, subtract, multiply, or
divide to solve problems
and justify solutions.
TEKS 7.2
Key Vocabulary
compatible numbers (p. 557)
reciprocal (p. 574)
scientific notation (p. 534)
Real-World Link
Blue Whales Blue whales eat 3.6 metric tons
of krill per day. So in a week, they can eat 3.6 × 7
or 25.2 metric tons of krill.
Multiplying and Dividing Decimals and Fractions Make this Foldable to help you organize your
notes. Begin with eleven sheets of notebook paper.
1 Staple the eleven sheets together to
form a booklet.
2 Cut a tab on the second page
the width of the white space.
On the third page, make the
tab 2 lines longer, and so on.
3 Write the chapter title on the
cover and label each tab with the
lesson number.
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Chapter 11 Multiplying and Dividing Decimals and Fractions
David Tipling/Photographer’s Choice/Getty Images
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GET READY for Chapter 11
Diagnose Readiness You have two options for checking Prerequisite Skills.
Option 2
Take the Online Readiness Quiz at tx.msmath1.com.
Option 1
Take the Quick Quiz below. Refer to the Quick Review for help.
Multiply.
(Used in Lessons 11-1, 11-2, and 11-6
through 11-8)
1.
17 × 28
2.
31 × 6
3.
109 × 14
4.
212 × 62
5.
228 × 19
6.
547 × 31
7.
SLEEP The average adult gets
8 hours of sleep each night. How
many total hours of sleep does
the average adult get in one year
(365 days)?
Divide.
(Used in Lessons 11-3, 11-4, 11-9
and 11-10)
8.
186 ÷ 3
9.
171 ÷ 9
10.
238 ÷ 14
11.
832 ÷ 26
12.
4,356 ÷ 36
13.
1,728 ÷ 6
14.
TRAVEL Four friends drove from
Chicago to Florida and spent
$188 on gasoline. If they split the
cost evenly, how much does each
owe?
Write each number as an improper
fraction. (Used in Lessons 11-8 and 11-10)
3
6
15. 2_
16. 1_
4
7
17. 5_
9
19.
7
1
18. 3_
8
PIANO Trent practiced the
5
piano for 3_
hours last week.
6
Write this time as an improper
fraction.
Example 1
Find 52 × 81.
52
× 81
52
+ 4160
4212
So, 52 × 81 = 4,212.
Example 2
Find 945 ÷ 15.
63
15 945
- 90
45
- 45
0
So, 945 ÷ 15 = 63.
Example 3
Write 4 2 as an improper fraction.
_
3
2
12
2
=_
+_
4_
3
3
14
=_
3
3
4 is equal to _.
12
3
Add _ and _.
12
3
2
3
Chapter 11 Get Ready for Chapter 11
531
Explore
11-1
Main IDEA
Math Lab
Multiplying Decimals by
Whole Numbers
You can use decimal models to multiply a decimal by a whole
number. Recall that a 10-by-10 grid represents the number one.
Use models to multiply
a decimal by a whole
number.
Model 0.5 × 3 using decimal models.
Preparation for
TEKS 7.2 The
student adds,
subtracts, multiplies,
or divides to solve problems
and justify solutions.
(A) Represent multiplication
and division situations
involving fractions and
decimals with models,
including concrete objects,
pictures, words, and numbers.
Draw three 10-by-10
decimal models to
show the factor 3.
3
0.5
Shade five rows of
each decimal model to
represent 0.5.
3
Cut off the shaded rows and
rearrange them to form as
many 10-by-10 grids as
possible.
The product is one and
five tenths.
So, 0.5 × 3 = 1.5.
Use decimal models to show each product.
a.
3 × 0.5
b.
2 × 0.7
c.
0.8 × 4
ANALYZE THE RESULTS
532
1.
MAKE A CONJECTURE Is the product of a whole number and a
decimal greater than the whole number or less than the whole
number? Explain your reasoning.
2.
Test your conjecture on 7 × 0.3. Check your answer by making a
model or by using a calculator.
Chapter 11 Multiplying and Dividing Decimals and Fractions
11-1
Multiplying Decimals
by Whole Numbers
Main IDEA
Estimate and find the
product of decimals and
whole numbers.
Preparation for
TEKS 7.2 The
student adds,
subtracts, multiplies,
or divides to solve problems
and justify solutions. (B) Use
addition, subtraction,
multiplication, and division to
solve problems involving
fractions and decimals.
NEW Vocabulary
SHOPPING CDs are on sale for
$7.99. Tia wants to buy two.
The table shows different ways
to find the total cost.
Cost of Two CDs
Add.
$7.99 + $7.99 = $15.98
$7.99 rounds to $8.
Estimate.
2 × $8 = $16
Multiply. 2 × $7.99 = 1.
Use the addition problem and
the estimate to find 2 × $7.99.
2.
Write an addition problem, an estimate, and a multiplication
problem to find the total cost of 3 CDs, 4 CDs, and 5 CDs.
3.
MAKE A CONJECTURE about how to find $0.35 × 3.
When multiplying a decimal by a whole number, multiply as with
whole numbers. Then use estimation to place the decimal point in the
product. You can also count the number of decimal places.
Multiply Decimals
scientific notation
1 Find 14.2 × 6.
Use estimation.
Round 14.2 to 14.
14.2 × 6 14 × 6 or 84
METHOD 1
21
14.2 Since the estimate is 84,
× 6 place the decimal point
after the 5.
85.2
METHOD 2
Count decimal places.
21
one decimal place
14.2
×6
85.2
Count one decimal
place from the right.
2 Find 9 × 0.83.
Use estimation.
Round 0.83 to 1.
9 × 0.83 9 × 1 or 9
METHOD 1
2
0.83 Since the estimate is 9,
× 9 place the decimal point
after the 7.
7.47
METHOD 2
2
0.83
×9
7.47
Count decimal places.
two decimal places
Count two decimal
places from the right.
Multiply.
a.
3.4 × 5
Extra Examples at tx.msmath1.com
Stephen Marks/Getty Images
b.
11.4 × 8
c.
7 × 2.04
Lesson 11-1 Multiplying Decimals by Whole Numbers
533
If there are not enough decimal places in the product, you need to
annex zeros to the left.
Annex Zeros in the Product
3 Find 2 × 0.018.
1
three decimal places
0.018
× 2
0.036
Annex a zero on the left of 36
to make three decimal places.
Vocabulary Link
Annex
Everyday Use to add
something
Math Use to annex a
zero means to add a
zero at the beginning
or end of a decimal
4 ALGEBRA Evaluate 4c if c = 0.0027.
4c = 4 × 0.0027 Replace c with 0.0027.
2
four decimal places
0.0027
× 4
0.0108 Annex a zero to make four decimal places.
Multiply.
d.
3 × 0.02
g.
ALGEBRA Evaluate 7x if x = 0.03.
e.
0.12 × 8
f.
11 × 0.045
Personal Tutor at tx.msmath1.com
When the number 450 is expressed as the product of 4.5 and 102,
the number is written in scientific notation. You can use order of
operations or mental math to write these numbers in standard form.
Scientific Notation
5 DINOSAURS Write 6.5 × 107 in standard form.
Use order of operations.
Evaluate
first. Then multiply.
6.5 × 107 = 6.5 × 10,000,000
= 65,000,000
METHOD 1
107
Real-World Link
Dinosaurs roamed
Earth until about
6.5 × 107 years ago.
Source: zoomwhales.com
Use mental math.
Move the decimal point to the right the same number of
places as the exponent of the power of 10, or 7 places.
6.5 × 107 = 6.5000000 or 65,000,000
METHOD 2
h.
534
7.9 × 103
i.
4.13 × 104
Chapter 11 Multiplying and Dividing Decimals and Fractions
Photo Network
j.
2.3 × 106
Examples 1, 2
(p. 533)
Examples 3, 4
Multiply.
1.
2.7 × 6
2.
1.3 × 2
3.
0.52 × 3
4.
0.83 × 6
5.
5 × 0.09
6.
4 × 0.012
7.
0.065 × 18
8.
0.015 × 23
9.
ALGEBRA Evaluate 14t if t = 2.9.
(p. 534)
Example 5
10.
(p. 534)
(/-%7/2+ (%,0
For
Exercises
11–18
33, 34
19–22
23–24
25–32,
35, 36
See
Examples
1, 2
3
4
5
MOON The distance from Earth to the Moon is approximately
3.84 × 105 kilometers. Write this number in standard form.
Multiply.
11.
1.2 × 7
12.
1.7 × 5
13.
0.7 × 9
14.
0.9 × 4
15.
2 × 1.3
16.
2.4 × 8
17.
0.8 × 9
18.
3 × 0.5
19.
3 × 0.02
20.
7 × 0.012
21.
0.0036 × 19
22.
0.0198 × 75
23.
ALGEBRA Evaluate 3.05n if n = 27.
24.
ALGEBRA Evaluate 80.44w if w = 2.
Write each number in standard form.
25.
5 × 104
26.
4 × 106
27.
1.5 × 103
28.
9.3 × 105
29.
2.5 × 103
30.
1.26 × 10²
31.
3.45 × 103
32.
2.17 × 106
33.
POSTERS Asher recently bought the poster shown
at the right. What is its area?
34.
SCHOOL SUPPLIES Sharon buys 14 folders for $0.75
each. How much do they cost?
35.
SPACE The diameter of the Sun is about 8.64 × 105
miles. Write the number in standard form.
36.
FT
FT
BIOLOGY The human body contains an average of 2.5 × 1011 blood
vessels. Write this number in standard form.
Find the value of each expression.
%842!02!#4)#%
See pages 688, 705.
Self-Check Quiz at
tx.msmath1.com
37.
2 × 3.8 + 1.5
38.
7 - 4 × 0.8
39.
3 × 2.14 × 104
40.
5 + 2.6 × 103
41.
11 × 7.85 + 33
42.
19 + 0.4 + 105
43.
SOCCER The table shows the prices that
Bernardo found for soccer balls. How
much would Bernardo save by buying
one dozen of the lowest-priced
soccer balls instead of a dozen of the
highest-priced soccer balls? Justify your
answer.
Soccer Ball
Brand A
Brand B
Brand C
Brand D
Brand E
Price ($)
99.99
14.99
6.99
34.99
19.99
Lesson 11-1 Multiplying Decimals by Whole Numbers
Aaron Haupt
535
H.O.T. Problems
47.
44.
OPEN ENDED Create a problem about a real-world situation involving
multiplication by a decimal factor. Then solve the problem.
45.
CHALLENGE Discuss two different ways to find the value of the
expression 5.4 × 1.17 × 102 that do not require you to first multiply
5.4 × 1.17.
46.
*/ -!4( Summarize how to convert a number in scientific
(*/
83 *5*/(
notation to standard form.
A recipe for a batch of cookies calls
for one 5.75-ounce package of
coconut. How many ounces of
coconut are needed for 5 batches
of cookies?
48.
A 20.50 oz
The table shows the admission prices
to Sea World in San Antonio.
Admission
Prices
Adults
Child (ages 3–9)
One-Day
Pass
$39.59
$30.59
B 25.25 oz
Source: seaworld.com
C 28.75 oz
What is the total price of one-day
passes for two adults and three
children?
D 29.75 oz
Find the surface area of each rectangular prism.
49.
50.
F $140.36
H $179.95
G $170.95
J $189.95
(Lesson 10-7)
3.5 ft
7 cm
51.
0.75 cm
4 in.
3 ft
4 in.
5 in.
1 ft
52.
What is the volume of a rectangular prism with a length of 23 meters,
width of 36 meters, and height of 41 meters? (Lesson 10-6)
53.
ENTERTAINMENT The largest Ferris wheel in the United States is
the Texas Star in Dallas. It has a radius of 106 feet. What is the
circumference of the Texas Star? Round to the nearest tenth.
(Lesson 10-2)
PREREQUISITE SKILL Find the value of each expression.
54.
536
Two-Day
Pass
$43.99
$33.99
43 × 25
55.
126 × 13
Chapter 11 Multiplying and Dividing Decimals and Fractions
(Page 658)
56.
18 × 165
3 cm
Explore
11-2
Main IDEA
Use decimal models to
multiply decimals.
Preparation for
TEKS 7.2 The
student adds,
subtracts, multiplies,
or divides to solve problems
and justify solutions.
(A) Represent multiplication
and division situations
involving fractions and
decimals with models,
including concrete objects,
pictures, words, and numbers.
Shading Yellow +
blue = green. So, the
green area is the
region that was
shaded twice.
Math Lab
Multiplying Decimals
In the lab on page 532, you used decimal models to multiply a
decimal by a whole number. You can use similar models to multiply
a decimal by a decimal.
1 Model 0.8 × 0.4 using decimal models.
Draw a 10-by-10 decimal model.
Recall that each small square
represents 0.01.
0.8
Shade eight rows of the model yellow
to represent the first factor, 0.8.
0.8
Shade four columns of the model blue
to represent the second factor, 0.4.
0.4
There are thirty-two hundredths in the region that is shaded green.
So, 0.8 × 0.4 = 0.32.
Use decimal models to show each product.
a. 0.3 × 0.3
b. 0.4 × 0.9
c. 0.9 × 0.5
ANALYZE THE RESULTS
1. Tell how many decimal places are in each factor and in each
product of Exercises a–c above.
2. MAKE A CONJECTURE Use the pattern you discovered in Exercise 1
to find 0.6 × 0.2. Check your conjecture with a model or a
calculator.
3. Find two decimals with a product of 0.24.
Explore 11-2 Math Lab: Multiplying Decimals
537
2 Model 0.7 × 2.5 using decimal models.
Draw three 10-by-10
decimal models.
Shade 7 rows yellow
to represent 0.7.
0.7
Shade 2 large squares
and 5 columns of the
next large square blue
to represent 2.5.
0.7
2.5
Arranging Squares
Arrange the squares
to form as many
whole decimal
models as possible.
Then arrange the
remaining squares
into as many rows of
10 as possible to
make counting them
easier.
Cut off the squares that are
shaded green and rearrange
them to form 10-by-10 grids.
There are one and seventy-five hundredths in the region that is
shaded green. So, 0.7 × 2.5 = 1.75.
Use decimal models to show each product.
d. 1.5 × 0.7
e. 0.8 × 2.4
f. 1.3 × 0.3
ANALYZE THE RESULTS
4. MAKE A CONJECTURE How does the number of decimal places in
the product relate to the number of decimal places in the factors?
5. Analyze each product.
a. Explain why the first
product is less than 0.6.
b. Explain why the second
First
Factor
0.9
1.0
1.5
Second
Factor
×
×
×
product is equal to 0.6.
c. Explain why the third product is greater than 0.6.
538
Chapter 11 Multiplying and Dividing Decimals and Fractions
0.6
0.6
0.6
Product
=
=
=
0.54
0.60
0.90
11-2
Multiplying Decimals
Main IDEA
Multiply decimals by
decimals.
Preparation for
TEKS 7.2 The
student adds,
subtracts, multiplies,
or divides to solve problems
and justify solutions. (B) Use
addition, subtraction,
multiplication, and division to
solve problems involving
fractions and decimals.
SHOPPING A nut store is having
a sale. The sale prices are shown
in the table.
1.
Suppose you fill a bag with
1.3 pounds of cashews. The
product of 1.3 × 2 can be used
to estimate the total cost.
Estimate the total cost.
2.
Multiply 13 by 200.
3.
How are the answers to Exercises 1 and 2 related?
Nuts (Cost per lb)
cashews
$2.07
almonds
$2.21
macadamias $2.79
Repeat Exercises 1–3 for each amount of nuts.
4.
1.7 pounds of almonds
6.
Make a conjecture about how to place the decimal point in the
product of two decimals.
5.
2.28 pounds of macadamias
When multiplying a decimal by a decimal, multiply as with whole
numbers. To place the decimal point, find the sum of the number of
decimal places in each factor. The product has the same number of
decimal places.
Multiply Decimals
1 Find 4.2 × 6.7. Estimate 4.2 × 6.7 → 4 × 7 or 28
4.2 ← one decimal place
× 6.7 ← one decimal place
294
252
28.14 ← two decimal places
The product is 28.14. Compared to the estimate, the product is reasonable.
READING
in the Content Area
For strategies in reading
this lesson, visit
tx.msmath1.com.
2 Find 1.6 × 0.09. Estimate 1.6 × 0.09 → 2 × 0 or 0
1.6 ← one decimal place
× 0.09 ← two decimal places
0.144 ← three decimal places
The product is 0.144. Compared to the estimate, the product is reasonable.
a.
5.7 × 2.8
Extra Examples at tx.msmath1.com
C Squared Studios/Getty Images
b.
4.12 × 0.07
c.
0.014 × 3.7
Lesson 11-2 Multiplying Decimals
539
Evaluate an Expression
3 ALGEBRA Evaluate 1.4x if x = 0.067.
1.4 x = 1.4 × 0.067 Replace x with 0.067.
0.067 ← three decimal places
× 1.4 ← one decimal place
268
67
0.0938 ← Annex a zero to make four decimal places.
Evaluate each expression.
d.
0.04t, if t = 3.2
e.
2.6b, if b = 2.05
f.
1.33c, if c = 0.06
4 TRAVEL Tariq is traveling to Mexico. One U.S. dollar is worth
11.3 pesos. How many pesos would he receive for $75.50?
Estimate 11.3 × 75.50 → 11 × 80 or 880
75.50 ← two decimal places
× 11.3 ← one decimal place
22650
7550
7550
The product has three decimal places. You can drop
853.150
the zero at the end because 853.150 = 853.15.
Tariq would receive 853.15 pesos.
Real-World Link
A 10 centavo Mexican
coin is equal to
0.10 Mexican pesos
and a 50 centavo
Mexican coin is equal
to 0.50 Mexican pesos.
g.
NUTRITION FACTS A nutrition label indicates that one serving of
apple crisp oatmeal has 2.5 grams of fat. How many grams of fat
are there in 3.75 servings?
Source: sanmiguelguide.com
Examples 1, 2
(p. 539)
Example 3
(p. 540)
Example 4
(p. 540)
540
Personal Tutor at tx.msmath1.com
Multiply.
1.
0.6 × 0.5
2.
1.4 × 2.56
3.
27.43 × 1.089
4.
0.3 × 2.4
5.
0.52 × 2.1
6.
0.45 × 0.053
9.
0.02n + 0.016
ALGEBRA Evaluate each expression if n = 1.35.
7.
10.
2.7n
8.
5.343 + 0.5n
MONEY Juanita is buying a video game that costs $32.99. The sales tax is
found by multiplying the cost of the video game by 0.06. How much is
sales tax for the video game?
Chapter 11 Multiplying and Dividing Decimals and Fractions
Spike Mafford/Getty Images
(/-%7/2+ (%,0
For
Exercises
11–22
23–28
29–30
See
Examples
1, 2
3
4
Multiply.
11.
0.7 × 0.4
12.
1.5 × 2.7
13.
0.4 × 3.7
14.
3.1 × 0.8
15.
0.98 × 7.3
16.
2.4 × 3.48
17.
6.2 × 0.03
18.
5.04 × 3.2
19.
14.7 × 11.36
20.
27.4 × 33.68
21.
0.28 × 0.08
22.
0.45 × 0.05
ALGEBRA Evaluate each expression if x = 8.6, y = 0.54, and z = 1.18.
23.
2.7x
24.
6.34y
25.
3.45x + 7.015
26.
1.8y + 0.6z
27.
9.1x - 4.7y
28.
0.096 + 2.28y
29.
TRAVEL A steamboat travels 36.5 miles each day. How far will it travel in
6.5 days?
30.
WORK Katelyn earns $5.80 an hour. If she works 16.75 hours in a week,
how much will she earn for the week?
Multiply.
31.
25.04 × 3.005
32.
1.03 × 1.005
33.
5.12 × 4.001
ALGEBRA Evaluate each expression if a = 1.3, b = 0.042, and c = 2.01.
34.
ab + c
35.
a × 6.023 - c
36.
3.25c + b
37.
abc
38.
GEOMETRY Find the area of the figure at
the right. Justify your answer.
39.
6.9 in.
3.1 in.
6.1 in.
ALGEBRA Which of the three numbers
9.2, 9.5, or 9.7 is the correct solution of
2.65t = 25.705?
3 in.
40.
GROCERY SHOPPING Pears cost $0.98 per
pound, and apples cost $1.05 per pound.
Mr. Bonilla bought 3.75 pounds of pears
and 2.1 pounds of apples. How much did
he pay for the pears and apples?
41.
FIND THE DATA Refer to the Texas Data File on pages 16–19. Choose
some data and write a real-world problem in which you would multiply
decimals.
Tell whether each sentence is sometimes, always, or never true. Explain
%842!02!#4)#% your reasoning.
See pages 688, 705.
42.
The product of two decimals that are each less than one is also less
than one.
Self-Check Quiz at
43.
The product of a decimal greater than one and a decimal less than
one is greater than one.
tx.msmath1.com
Lesson 11-2 Multiplying Decimals
541
H.O.T. Problems
50.
CHALLENGE Evaluate each expression.
44.
0.3(3 - 0.5)
47.
OPEN ENDED Write a multiplication problem in which the product is
between 0.05 and 0.75.
48.
NUMBER SENSE Place the decimal point in the answer to make it correct.
Explain your reasoning. 3.9853 × 8.032856 = 32013341...
49.
*/ -!4( Describe two methods for determining where to
(*/
83 *5*/(
place the decimal point in the product of two decimals.
45.
0.16(7 - 2.8)
What is the area of the rectangle?
51.
1.4 cm
5.62 cm
A 14.04 cm2
B 10.248 cm2
C 8.992 cm2
46.
1.06(2 + 0.58)
Josefina took her grandmother to
lunch. Josefina’s lunch was $6.70, her
grandmother’s lunch was $7.25, and
they split a dessert that cost $3.50. If
there was an 8.75% tax on the food,
which procedure could be used to
find the amount of tax Josefina paid
for their lunch?
F Add the prices of the food items.
D 7.868 cm2
G Add the prices of the food items
to the tax rate.
H Multiply the tax rate by the price
of the most expensive food item.
J Multiply the tax rate by the sum
of the prices of the food items.
Multiply.
(Lesson 11-1)
52.
45 × 0.27
56.
Find the surface area of the rectangular
prism at the right. (Lesson 10-7)
57.
53.
3.2 × 109
542
27 × 0.45
7 yd
15 yd
FLIGHTS A flight leaves Miami at
11:17 A.M. and arrives in Seattle at
6:02 P.M. How long is the flight? (Lesson 8-7)
PREREQUISITE SKILL Divide.
58.
54.
21 ÷ 3
59.
55.
2.94 × 16
4 yd
(Page 658)
81 ÷ 9
60.
56 ÷ 8
Chapter 11 Multiplying and Dividing Decimals and Fractions
61.
63 ÷ 7
11-3
Dividing Decimals by
Whole Numbers
Main IDEA
Divide decimals by whole
numbers.
Preparation for
TEKS 7.2 The
student adds,
subtracts, multiplies,
or divides to solve problems
and justify solutions.
(A) Represent multiplication
and division situations
involving fractions and
decimals with models,
including concrete objects,
pictures, words, and numbers.
(B) Use addition, subtraction,
multiplication, and division to
solve problems involving
fractions and decimals.
To find 2.4 ÷ 2 using base-ten blocks, model 2.4 as 2 wholes and
4 tenths. Then separate into two equal groups.
There is one whole and two tenths in each group.
So, 2.4 ÷ 2 = 1.2.
Use base-ten blocks to show each quotient.
1.
3.4 ÷ 2
2.
4.2 ÷ 3
3.
5.6 ÷ 4
6.
56 ÷ 4
Find each whole number quotient.
REVIEW Vocabulary
quotient the solution
in division; Example:
the quotient of 16 ÷ 8
is 2
4.
34 ÷ 2
7.
Compare and contrast the quotients in Exercises 1–3 with the
quotients in Exercises 4–6.
8.
MAKE A CONJECTURE Write a rule for dividing a decimal by a
whole number.
5.
42 ÷ 3
Dividing a decimal by a whole number is similar to dividing whole
numbers.
Divide a Decimal by a 1-Digit Number
1 Find 6.8 ÷ 2.
Estimate 6 ÷ 2 = 3
Place the decimal point directly above
3.4
the decimal point in the dividend.
2 6.8
-6
08
-8
0 6.8 ÷ 2 = 3.4 Compared to the estimate, the quotient is reasonable.
Divide.
a.
7.5 ÷ 3
Extra Examples at tx.msmath1.com
b.
3.5 ÷ 7
c.
9.8 ÷ 2
Lesson 11-3 Dividing Decimals by Whole Numbers
543
Divide a Decimal by a 2-Digit Number
2 Find 7.7 ÷ 14.
Checking Your
Answer To check
that the answer is
correct, multiply the
quotient by the
divisor. In Example 2,
0.55 × 14 = 7.7.
Estimate 10 ÷ 10 = 1
Place the decimal point.
0.55
14 7.70
-7 0
Annex a zero and continue dividing.
70
- 70
0
7.7 ÷ 14 = 0.55 Compared to the estimate, the quotient is reasonable.
Divide.
d.
9.48 ÷ 15
e.
3.49 ÷ 4
f.
55.08 ÷ 17
If the answer does not come out evenly, round the quotient to a
specified place-value position.
3 GRIDDABLE Seth purchased 3 video games for $51.79. If each
game costs the same amount, find the price of each game
in dollars. Estimate $51 ÷ 3 = $17
Read the Test Item
To find the price of one game, divide the total cost by the number
of games. Round to the nearest cent, or hundredths place.
Solve the Test Item
Griddable Write the
answer in the answer
boxes on the top line.
Then grid in 17, the
decimal point, and 26.
17.263 Place the decimal point.
3 51.790
-3
21
- 21
07
- 06
19
- 18
10 Divide until you place a digit
- 9 in the thousandths place.
1
Fill in the Grid
1 7
.
2 6
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
9
To the nearest cent, the cost in dollars is 17.26.
g.
GRIDDABLE A bag of 12 bagels costs $7.50. To the nearest cent,
find the cost of each bagel.
Personal Tutor at tx.msmath1.com
544
Chapter 11 Multiplying and Dividing Decimals and Fractions
Examples 1, 2
(pp. 543–544)
Example 3
Divide. Round to the nearest tenth if necessary.
1.
3.6 ÷ 4
2.
9.6 ÷ 2
3.
8.53 ÷ 6
4.
1087.9 ÷ 46
5.
12.32 ÷ 22
6.
69.904 ÷ 34
7.
TEST PRACTICE Four girls on a track team ran the 4-by-100 meter
relay in a total of 46.9 seconds. What was the average time for each
runner to the nearest tenth?
(p. 544)
(/-%7/2+ (%,0
For
Exercises
8–13
20, 21
14–19
31–32
See
Examples
1
2
3
%842!02!#4)#%
Divide. Round to the nearest tenth if necessary.
8.
39.39 ÷ 3
9.
36.8 ÷ 2
10.
118.5 ÷ 5
11.
124.2 ÷ 9
12.
7.24 ÷ 7
13.
6.27 ÷ 4
14.
11.4 ÷ 19
15.
10.22 ÷ 14
16.
55.2 ÷ 46
17.
59.84 ÷ 32
18.
336.75 ÷ 31
19.
751.2 ÷ 25
20.
MONEY Jackson’s father has budgeted $64.50 for his three children’s
monthly allowance. Assuming they each earn the same amount, how
much allowance will Jackson receive?
21.
MUSIC Find the average time of a track on
a CD from the times in the table.
22.
LANDMARKS Each story in an office building is about 4 meters tall. The
Eiffel Tower in Paris, France, is 300.51 meters tall. To the nearest whole
number, about how many stories tall is the Eiffel Tower?
23.
FOOD The Student Council is raising money
by selling bottled water at a band competition.
The table shows the prices for different brands.
Which brand is the best buy? Explain your
reasoning.
24.
TECHNOLOGY A class set of 30 calculators would have cost $4,498.50 in
the early 1970s. However, in 2005, 30 calculators could be purchased for
$284.70. How much less was the average price of one calculator in 2005
than in 1970?
See pages 689, 705.
Self-Check Quiz at
tx.msmath1.com
H.O.T. Problems
Time of Track (minutes)
4.73 3.97 2.93 2.83 3.44
Cost of Bottled Water
(20-oz bottles)
Brand A 6-pack
$3.45
Brand B 12-pack
$5.25
Brand C 24-pack $10.99
STATISTICS Find the mean for each set of data.
25.
22.6, 24.8, 25.4, 26.9
27.
OPEN ENDED Create a set of data for which the mean is 5.5.
28.
CHALLENGE Create a division problem that meets all of the following
conditions.
26.
1.43, 1.78, 2.45, 2.78, 3.25
• The divisor is a whole number, and the dividend is a decimal.
• The quotient is 1.265 when rounded to the nearest thousandth.
• The quotient is 1.26 when rounded to the nearest hundredth.
Lesson 11-3 Dividing Decimals by Whole Numbers
545
29.
FIND THE ERROR Felisa and Tabitha are finding 11.2 ÷ 14. Who is correct?
Explain your reasoning.
0.8
14)11.2
- 112
0
8.
14)11.2
- 112
0
Felisa
30.
31.
Tabitha
*/ -!4( Explain how you can use estimation to place the
(*/
83 *5*/(
decimal point in the quotient 42.56 ÷ 22.
GRIDDABLE Tanner and three
neighborhood friends are buying a
basketball hoop that costs $249.85. If
the cost is divided equally, how
much will each person pay in
dollars?
32.
The table shows the amount Ursalina
made in one week for a variety of jobs.
Jobs
baby-sitting
pet-sitting
lawn work
Pay in a
Week
$50.00
$10.50
$22.50
What was Ursalina’s approximate
average pay for these three jobs?
Multiply.
A $28.00
C $55.00
B $35.00
D $80.00
(Lesson 11-2)
33.
2.4 × 5.7
37.
What is the product of 4.156 and 12?
34.
1.6 × 2.3
35.
0.32(8.1)
39.
8 ft
PREREQUISITE SKILL Divide.
41.
546
25 ÷ 5
42.
(Lesson 10-2)
40.
29 in.
17 cm
(Page 658 and Lesson 11-3)
81 ÷ 9
43.
114.8 ÷ 14
Chapter 11 Multiplying and Dividing Decimals and Fractions
(l)Simon Watson/Stone/Getty Images, (r)CORBIS
2.68(0.84)
(Lesson 11-1)
GEOMETRY Estimate the circumference of each circle.
38.
36.
44.
516.06 ÷ 18
Explore
11-4
Main IDEA
Math Lab
Dividing by Decimals
The model below shows 15 ÷ 3.
Use models to divide a
decimal by a decimal.
Preparation for
TEKS 7.2 The
student adds,
subtracts, multiplies,
or divides to solve problems
and justify solutions.
(A) Represent multiplication
and division situations
involving fractions and
decimals with models,
including concrete objects,
pictures, words, and numbers.
If 15 is divided into three equal
sets, there are 5 in each set.
Dividing decimals is similar to dividing whole numbers. In the Activity
below, 1.5 is the dividend and 0.3 is the divisor.
• Use base-ten blocks to model the dividend.
• Replace any ones block with tenths.
• Separate the tenths into groups represented by the divisor.
• The quotient is the number of groups.
1 Model 1.5 ÷ 0.3.
Place one and 5 tenths in
front of you to show 1.5.
1
0.5
Replace the ones block with
tenths. You should have a
total of 15 tenths.
1
0.5
Separate the tenths into
groups of three tenths
to show dividing by 0.3.
0.3
0.3
0.3
0.3
0.3
5 groups
There are five groups of three tenths in 1.5. So, 1.5 ÷ 0.3 = 5.
Explore 11-4 Math Lab: Dividing by Decimals
547
You can use a similar model to divide by hundredths.
Model 0.4 ÷ 0.05.
Model 0.4 with base-ten blocks.
0.4
Replace the tenths with hundredths
since you are dividing by hundredths.
0.40
0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05
Separate the hundredths into
groups of five hundredths to
show dividing by 0.05.
8 groups
There are eight groups of five hundredths in 0.4.
So, 0.4 ÷ 0.05 = 8.
Use base-ten blocks to find each quotient.
a. 2.4 ÷ 0.6
b. 1.2 ÷ 0.4
c. 1.8 ÷ 0.6
d. 0.9 ÷ 0.09
e. 0.8 ÷ 0.04
f. 0.6 ÷ 0.05
ANALYZE THE RESULTS
1. Explain why the base-ten blocks representing the dividend must be
replaced or separated into the smallest place value of the divisor.
2. Tell why the quotient 0.4 ÷ 0.05 is a whole number. What does the
quotient represent?
3. Determine the missing divisor in the sentence 0.8 = 20. Explain.
4. MAKE A CONJECTURE Tell whether 1.2 ÷ 0.03 is less than, equal to, or
greater than 1.2. Explain your reasoning.
548
Chapter 11 Multiplying and Dividing Decimals and Fractions
11-4
Dividing by Decimals
Main IDEA
Interactive Lab tx.msmath1.com
Divide decimals by
decimals.
Patterns can help you understand how to divide a decimal
by a decimal.
Preparation for
TEKS 7.2 The
student adds,
subtracts, multiplies,
or divides to solve problems
and justify solutions.
(B) Use addition, subtraction,
multiplication, and division to
solve problems involving
fractions and decimals.
Use a calculator to find each quotient.
1.
3.
0.048 ÷ 0.06
2.
0.0182 ÷ 0.13
0.48 ÷ 0.6
0.182 ÷ 1.3
4.8 ÷ 6
1.82 ÷ 13
48 ÷ 60
18.2 ÷ 130
Which of the quotients in Exercises 1 and 2 would be easier to
find without a calculator? Explain your reasoning.
Rewrite each problem so you can find the quotient without
using a calculator. Then find the quotient.
4.
REVIEW Vocabulary
power a number
expressed using
exponents; Example: 24
(Lesson 1-3)
0.42 ÷ 0.7
5.
1.26 ÷ 0.3
6.
1.55 ÷ 0.5
When dividing by decimals, change the divisor into a whole number.
To do this, multiply both the divisor and the dividend by the same
power of 10. Then divide as with whole numbers.
Divide by Decimals
1 Find 14.19 ÷ 2.2.
Estimate 14 ÷ 2 = 7
Multiply by 10 to make
a whole number.
2.2 14.19
→
Multiply by the
same number, 10.
6.45
22 141.90
- 132
99
-88
110
- 110
0
Place the decimal point.
Divide as with whole numbers.
Annex a zero to continue.
14.19 divided by 2.2 is 6.45. Compare to the estimate.
Check
6.45 × 2.2 = 14.19
✔
Divide.
a.
54.4 ÷ 1.7
Extra Examples at tx.msmath1.com
b.
8.424 ÷ 0.36
c.
0.0063 ÷ 0.007
Lesson 11-4 Dividing by Decimals
549
Zeros in the Quotient and Dividend
2 Find 52 ÷ 0.4.
130. Place the decimal point.
4 520.
-4
12
- 12
00 Write a zero in the ones
0.4 52.0
Multiply each by 10.
So, 52 ÷ 0.4 = 130.
Check 130 × 0.4 = 52
place of the quotient
because 0 ÷ 4 = 0.
✔
3 Find 0.09 ÷ 1.8.
0.05
18 0.90
-0
09
- 00
90
- 90
0
1.8 0.09
Multiply each by 10.
Place the decimal point.
18 does not go into 9, so
write a 0 in the tenths place.
Annex a 0 in the dividend
and continue to divide.
So, 0.09 ÷ 1.8 is 0.05.
Check
0.05 × 1.8 = 0.09
✔
Divide.
d.
5.6 ÷ 0.014
e.
62.4 ÷ 0.002
f.
0.4 ÷ 0.0025
Personal Tutor at tx.msmath1.com
Round Quotients
Rounding When
rounding to the
nearest tenth, you
can stop dividing
when there is a digit
in the hundredths
place.
550
4 INTERNET How many times more DSL
DSL Broadband Subscribers
subscribers are there in China than in
in 2004 (millions)
France? Round to the nearest tenth.
China
13.7
Japan
12.7
Find 13.7 ÷ 5.3.
U.S.
12.6
2.58
South Korea
6.7
13.7 → 53 137.00
5.3 - 106
Germany
6.0
France
5.3
310
- 265
Source: internetworldstats.com
450
- 424
26
To the nearest tenth, 13.7 ÷ 5.3 = 2.6. So, there are about 2.6 times
more DSL subscribers in China than in France.
g.
INTERNET How many times more DSL subscribers are there in
the U.S. than in South Korea? Round to the nearest tenth.
Chapter 11 Multiplying and Dividing Decimals and Fractions
Divide.
Example 1
(p. 549)
Examples 2, 3
(p. 550)
Example 4
1.
3.69 ÷ 0.3
2.
9.92 ÷ 0.8
3.
0.45 ÷ 0.3
4.
13.95 ÷ 3.1
5.
0.6 ÷ 0.0024
6.
0.462 ÷ 0.06
7.
0.321 ÷ 0.4
8.
2.943 ÷ 2.7
9.
CRAFTS Alicia bought 5.75 yards of fleece fabric to make blankets for
a charity. She needs 1.85 yards of fabric for each blanket. How many
blankets can Alicia make with the fabric she bought?
(p. 550)
(/-%7/2+ (%,0
For
Exercises
10–13,
22–23
14–21
24, 25
Divide.
See
Examples
1
10.
1.44 ÷ 0.4
11.
0.68 ÷ 3.4
12.
16.24 ÷ 0.14
13.
2.07 ÷ 0.9
14.
0.0338 ÷ 1.3
15.
0.16728 ÷ 3.4
16.
96.6 ÷ 0.42
17.
1.08 ÷ 2.7
18.
13.5 ÷ 0.03
2, 3
4
19.
8.4 ÷ 0.02
20.
0.12 ÷ 0.15
21.
0.242 ÷ 0.4
22.
CARPENTRY If a board 4.5 feet long is cut into 0.75-foot pieces, how
many pieces will there be?
23.
EXERCISE The average person’s stride length, the distance covered by one
step, is approximately 2.5 feet long. How many steps would the average
person take to travel 50 feet?
24.
PARKS Refer to the table. How
many times more people
attended Big Bend National Park
than Lyndon B. Johnson National
Park? Round to the nearest tenth
if necessary.
25.
Real-World Link
Bats use the echoes
of their own cries to
determine the size and
shape of their prey and
where it is going.
Source: bats.org
26.
TRAVEL The Vielhaber family
drove 315.5 miles for a soccer
tournament and used 11.4 gallons
of gas. How many miles did they
get per gallon of gas to the nearest
hundredth? Estimate the answer
before calculating.
2004 Attendance at National Parks
in Texas
National Park
Big Bend
Fort Davis
Gaudalupe Mountains
Lyndon B. Johnson
Padre Island
Attendance
(thousands)
357.7
54.0
182.4
80.7
643.8
Source: National Park Service
BATS Sound travels through air at 330 meters per second. How long will
it take a bat’s cry to reach its prey and echo back if the prey is 1 meter
away from the bat?
Lesson 11-4 Dividing by Decimals
Stephen Dalton/Animals Animals
551
ALGEBRA Evaluate each expression if m = 88.2, n = 3, and p = 17.5. Round
to the nearest tenth if necessary.
27. _
n
m
28.
mp
_
29.
n
m
-p
32. _
n
p
31. _
n
mn
_
30. _
p
m
p
p
+n
33. _
n
34.
m+n+p
_
p
35.
RESEARCH Use the Internet or another source to find the average speed
of Jeff Gordon’s car in the 2005 Daytona 500 race. If a passenger car
averages 55.5 miles per hour, how many times as fast was Gordon’s car,
to the nearest hundredth?
36.
REASONING Determine whether the following statement is true or false.
If false, provide a counterexample.
If a decimal greater than 0 and less than 1 is divided by a
lesser decimal, the quotient will be less than 1.
37.
TUNNELS The longest vehicle tunnel in the world is the Laerdal Tunnel
located in Norway with a length of 15.2 miles. How many vehicles
could fit in the tunnel bumper to bumper if the average vehicle length is
0.004 mile?
38.
FIND THE DATA Refer to the Texas Data File on pages 16–19. Choose
some data and write a real-world problem in which you would divide
decimals.
39.
CHALLENGE Replace each below with different digits in order to make
a true sentence. .8 3 ÷ 0.82 = 4.6 40.
OPEN ENDED Write a division problem with decimals in which it is
necessary to annex one or more zeros to the dividend. Then solve the
problem. Round to the nearest tenth if necessary.
41.
NUMBER SENSE Use the number line below to determine if the quotient
of 1.92 ÷ 0.5 is closest to 2, 3, or 4. Do not calculate. Explain your
reasoning.
%842!02!#4)#%
See pages 689, 705.
Self-Check Quiz at
tx.msmath1.com
H.O.T. Problems
1.92
0
42.
552
1
1.5
2
Which One Doesn’t Belong? Identify the problem that does not have the
same quotient as the other three. Explain your reasoning.
0.35 ÷ 0.5
43.
0.5
3.5 ÷ 5
0.035 ÷ 0.05
35 ÷ 5
*/ -!4( Refer to the table in Exercise 24 on 2004 national
(*/
83 *5*/(
park attendance in Texas. Write and solve a problem in which you
would divide decimals. Include instructions for rounding in
your problem.
Chapter 11 Multiplying and Dividing Decimals and Fractions
44.
To the nearest tenth, how many
times as many people in the U.S.
own dogs than own birds?
45.
.UMBEROF0EOPLEMILLIONS
/WNING0ETS
To the nearest tenth, how many
times greater was the average
gasoline price on January 9, 2006
than on January 9, 2005?
Date
January 9, 2005
January 9, 2006
Source: Energy Information Administration
$OGS #ATS
Average
Price
(per gal)
$1.79
$2.33
F 0.5
"IRDS
G 1.3
3OURCE3TATISTICAL!BSTRACTOFTHE
5NITED3TATES
H 1.5
A 6.8
J 1.8
B 12.2
C 26.6
D 35.8
46.
Find the quotient when 68.52 is divided by 12.
Multiply.
(Lesson 11-3)
(Lesson 11-2)
47.
19.2 × 2.45
50.
RESTAURANTS At a restaurant, Miles ordered an appetizer for $2.49,
an entrée for $11.99, a beverage for $1.29, and a dessert for $3.49. If
he decides to tip 18%, about how much will Miles leave as a tip?
48.
8.25 × 12.42
49.
9.016 × 51.9
(Lesson 7-8)
GEOMETRY Find the value of x in each quadrilateral.
51.
67
108
x
52.
75
53.
115
x
113
85
(Lesson 9-5)
91
x
47
130
54.
BIOLOGY The average human heart has 72 beats per minute. At this
rate, how many beats will there be in one week? (Lesson 7-7)
55.
PREREQUISITE SKILL A number is multiplied by 8. Next, 4 is subtracted
from the product. Then, 12 is added to the difference. If the result is
32, what is the number? Use the work backward strategy. (Lesson 3-6)
Lesson 11-4 Dividing by Decimals
553
11-5 Problem-Solving Investigation
MAIN IDEA: Solve problems by determining reasonable answers.
Preparation for TEKS 7.2 The student adds, subtracts, multiplies, or divides to solve problems and justify
solutions. (G) Determine the reasonableness of a solution to a problem. Also addresses TEKS 7.13(B), 7.13(C).
I-M:
REASONABLE ANSWERS
YOUR MISSION: Determine if an answer is reasonable.
THE PROBLEM: What is the weight in pounds of an
average gray whale?
▲
EXPLORE
PLAN
SOLVE
CHECK
▲
ANGEL: For our science
project we need to know
how much a gray whale
weighs in pounds. I found
a table that shows the
weights of whales in tons.
STEPHANIE: Well, I know there are
2,000 pounds in one ton.
You know the weight in tons. You need to find a
reasonable weight in pounds.
One ton equals 2,000 pounds. So, estimate the product of
38.5 and 2,000 to find a reasonable weight.
2,000 × 38.5 → 2,000 × 40 or 80,000
A reasonable weight is 80,000 pounds.
Since 2,000 × 38.5 = 77,000, the answer of
80,000 pounds is reasonable.
Whale
Blue
Bowhead
Fin
Gray
Humpback
Source: Top 10 of Everything
Describe a situation where determining a reasonable answer would help
you solve a problem.
*/ -!4( Write and solve a problem that can be solved by
(*/
2.
83 *5*/(
determining a reasonable answer. Then tell the steps you would take
to find the solution of the problem.
1.
554
John Evans
Chapter 11 Multiplying and Dividing Decimals and Fractions
Weight
(tons)
151.0
95.0
69.9
38.5
38.1
Determine reasonable answers for
Exercises 3–5.
3.
4.
BASEBALL In a recent year, a total of
820,590 people attended 25 of the Atlanta
Braves home games. Which is a more
reasonable estimate for the number of
people that attended each game: 30,000
or 40,000?
8.
AGES Erin’s mother is 4 times as old as
Erin. Her grandmother is twice as old as
Erin’s mother. The sum of their three ages
is 104. How old is Erin, her mother, and
her grandmother?
9.
GRADUATION A high school gym will hold
2,800 guests and the 721 seniors who are
graduating. Is it reasonable to offer each
graduate three, four, or five tickets for
family and friends? Explain your
reasoning.
POPULATION Use the graph below to
determine whether 1,000, 1,300, or
1,500 is a reasonable prediction of the
population of Dorsey Intermediate
School in 2008.
Dorsey Intermediate Enrollment
1,500
For Exercises 10–12, select the appropriate
operation(s) to solve the problem. Justify
your selection(s) and solve the problem.
Students
1,200
900
600
10.
300
0
’02 ’03 ’04 ’05 ’06 ’07
Year
5.
MONEY Brandy wants to buy 2 science
fiction books for $3.95 each, 3 magazines
for $2.95 each, and 1 bookmark for $0.39
at the book fair. Does she need to bring
$20 or will $15 be enough? Explain your
reasoning.
RIVERS The graphic below shows the
lengths in miles of the longest rivers in the
world. About how many total miles long
would these three rivers be if they were
laid end to end?
Lengths of World's Longest Rivers
(in thousands of miles)
4.0
Nile
Use any strategy to solve Exercises 6–9.
Some strategies are shown below.
Amazon
Chang Jiang
G STRATEGIES
PROBLEM-SOLVIN
tep plan.
• Use the four-s
r problem.
• Solve a simple
m.
• Draw a diagra
4.16
3.96
Source: The World Almanac
6.
CONCERT In how many ways can 4 people
stand in line at a concert if Terrez and
Missy must stand next to each other?
7.
CHICKENS The most eggs a chicken has
ever laid in a single day is 7. At this rate,
how many eggs will a chicken lay in
8 years? Assume that there are 365 days
in a year.
11.
MUSIC In entertainment, a gold album
award is presented to an artist who has
sold at least 500,000 units of a single album
or CD. If an artist has 16 gold albums, what
is the minimum number of albums that
have been sold?
12.
BIRTHDAYS A relative doubles your age
with dollars each birthday. You are
13 years old. How much money have
you been given over the years by this
relative?
Lesson 11-5 Problem-Solving Investigation: Reasonable Answers
555
CH
APTER
11
Multiply.
Mid-Chapter Quiz
Lessons 11-1 through 11-5
(Lesson 11-1)
1.
4.3 × 5
2.
0.78 × 9
3.
3 × 0.006
4.
44 × 0.035
ALGEBRA Evaluate each expression if p = 5.7,
m = 0.08, and n = 1.7. (Lesson 11-1)
5.
26p
8.
MONEY EXCHANGE If the Japanese yen is
worth 0.0078 U.S. dollar, what is the value
of 3,750 yen in U.S. dollars? (Lesson 11-1)
9.
6.
15m
7.
Divide. Round to the nearest tenth if
necessary. (Lesson 11-3)
20.
19.752 ÷ 24
21.
48.6 ÷ 6
22.
54.45 ÷ 55
23.
2.08 ÷ 5
24.
TEST PRACTICE Mr. Dillon will pay
a total of $9,100.08 for his car lease over
36 months. Which of the following
equations can be used to find m, his
monthly payment? (Lesson 11-3)
9n
TEST PRACTICE Yoko wants to buy
3 necklaces that cost $12.99 each and
4 T-shirts that are on sale for $11.95 each.
Which of the following equations can be
used to find a, the total amount Yoko will
pay? (Lesson 11-1)
A a = 3(12.99) + 11.95
F m = 9,100.08 × 36
G m = 9,100.08 ÷ 12
H m = 9,100.08 × 12
J m = 9,100.08 ÷ 36
ALGEBRA Evaluate each expression if
w = 26.8, y = 0.01, and z = 3.4. Round to
the nearest tenth if necessary.
25.
B a = 3(12.99) + 4(11.95)
C a = 12.99 + 11.95
D a = 4(12.99) + 3(11.95)
Write each number in standard form.
w
_
y
26.
5.1w
_
z
27.
wy
_
z
Divide. Round to the nearest tenth if
necessary. (Lesson 11-4)
28.
3.29 ÷ 1.4
29.
93.912 ÷ 4.3
30.
0.015 ÷ 0.02
31.
0.008 ÷ 0.01
32.
SPORTS At the 1996 Olympics,
Michael Johnson set a world record
of 19.32 seconds for the 200-meter dash.
A bee can fly 200 meters in 40.752 seconds.
About how many times faster was
Michael Johnson? (Lesson 11-4)
33.
MONEY Guillermo bought a new baseball
glove for $49.95. The sales tax is found by
multiplying the price of the glove by
0.075. Is about $3, $4, or $5 a reasonable
answer for the amount of sales tax that
Guillermo paid? (Lesson 11-5)
34.
PIZZA Six friends will split the cost of
2 large pizzas. If each pizza costs $14.99,
is $4, $5, or $6 a reasonable answer for
the amount that each friend will pay?
(Lesson 11-1)
10.
3 × 104
11.
7.7 × 106
12.
1.21 × 105
13.
4.09 × 102
14.
SPACE The distance from Earth to the
Moon is about 3.83 × 105 kilometers. Write
this distance in standard form. (Lesson 11-1)
Multiply.
(Lesson 11-2)
15.
5.72 × 6.9
16.
18.55 × 0.002
17.
0.08 × 1.26
18.
33.71 × 1.9
19.
GEOMETRY Find the area of the rectangle.
(Lesson 11-2)
2.2 cm
4.2 cm
556
(Lesson 11-5)
Chapter 11 Multiplying and Dividing Decimals and Fractions
11-6
Estimating Products
of Fractions
Main IDEA
Estimate products of
fractions using compatible
numbers and rounding.
Preparation for
TEKS 7.2 The
student adds,
subtracts, multiplies,
or divides to solve problems
and justify solutions.
(A) Represent multiplication
and division situations
involving fractions and
decimals with models,
including concrete objects,
pictures, words, and
numbers. (B) Use addition,
subtraction, multiplication,
and division to solve
problems involving fractions
and decimals.
1
SPORTS Gracia made about _
of the
3
14 shots she attempted in a basketball
game.
1.
For the shots attempted, what is the
nearest multiple of 3?
2.
How many basketballs should be
added to reflect the nearest multiple of 3?
3.
Divide the basketballs into three groups each having the same
number. How many basketballs are in each group?
4.
About how many shots did Gracia make?
One way to estimate products involving fractions is to use
compatible numbers, or numbers that are easy to divide mentally.
Estimate Using Compatible Numbers
NEW Vocabulary
compatible numbers
_
1 Estimate 1 × 13. _14 × 13 means _14 of 13.
Find a multiple of 4 close to 13.
_1 × 13 → _1 × 12
REVIEW Vocabulary
multiple of a number
the product of the number
and any whole number;
Example: 6, 9, and 12 are all
multiples of 3 (Lesson 4-4)
1
4
4
12 and 4 are compatible numbers
12
4
4
since 12 ÷ 4 = 3
_1 × 12 = 3
12 ÷ 4 = 3.
4
1
So, _
× 13 is about 3.
4
_
2 Estimate 2 of 11.
1
5
5
1
_
Estimate × 11 first.
5
1
_ × 11 → _1 × 10
Use 10 since 10 and 5
5
5
are compatible numbers.
_1 × 10 = 2 10 ÷ 5 = 2
5
1
2
If _ of 10 is 2, then _
of 10 is 2 × 2 or 4.
5
5
2
× 11 is about 4.
So, _
5
1
5
10
Estimate each product.
a.
_1 × 16
5
Extra Examples at tx.msmath1.com
b.
_5 × 13
6
c.
_3 of 23
4
Lesson 11-6 Estimating Products of Fractions
557
Estimate by Rounding to 0,
_1 , or 1
2
_ _
3 Estimate 1 × 7 .
3
8
_1 × _7 → _1 × 1
1
1
is about 2 .
3
8
3
2
1
1
_×1=_
2
2
1
7
1
×_
is about _
.
So, _
3
8
2
7
is about 1.
8
1
3
7
8
1
2
0
1
Estimate each product.
d.
9
_5 × _
8
e.
10
9
_5 × _
6
f.
10
_5 of _1
6
9
Personal Tutor at tx.msmath1.com
Estimate With Mixed Numbers
4 GARDENING Estimate the area of the
flower bed.
Round each mixed number to
the nearest whole number.
Look Back
You can review
rounding fractions
in Lesson 5-1.
3
1
× 8_
→
11_
4
6
FT
12 × 8 = 96
Round 11_ to 12.
3
4
FT
Round 8_ to 8.
1
6
So, the area is about 96 square feet.
g.
2
1
BRICK WALLS A wall is made up of 32_
bricks that are 1_
feet
long. About how long is the wall?
Examples 1–4
(pp. 557–558)
(p. 558)
1.
_1 × 15
2.
_3 × 21
4
5
1
_
6.
×_
8
9
3.
_2 of 26
7.
2
1
6_
× 4_
5
3
5
4.
1
_
of 68
8.
9
3
_
× 10_
10
10
4
3
9. PORCHES Hakeem’s front porch measures 9_ feet by 4 feet. Estimate the
4
area of his front porch.
10.
558
6
Estimate each product.
8
8
1
_
5.
×_
9
4
Example 4
3
1
2
KITCHENS A kitchen measures 24_
feet by 9_
feet. Estimate the area of
6
3
the kitchen.
Chapter 11 Multiplying and Dividing Decimals and Fractions
(/-%7/2+ (%,0
For
Exercises
11–20
21–28
29, 30
See
Examples
1, 2
3
4
Estimate each product.
11.
_1 × 21
12.
19.
20.
_1 × 26
13.
5
2
16. _ of 88
9
4
5
15. _ of 22
7
_1 of 41
14.
3
2
17. _ × 10
3
_1 of 17
6
3
18. _ × 4
8
PIZZA Cyrus is inviting 11 friends over for pizza. He would like to have
1
enough pizza so each friend can have _
of a pizza. About how many
4
pizzas should he order?
2
BOOKS Tara would like to finish _
of her book by next Friday. If
5
the book has 203 pages, about how many pages does she need
to read?
Estimate each product.
21.
_5 × _1
25.
3
1
4_
× 2_
7
22.
9
3
4
1
7
_
×_
23.
8
10
4
1
26. 6_ × 4_
5
9
3
11
_
×_
8
12
1
1
27. 5_ × 9_
8
12
24.
9
_2 × _
28.
9
5
2_
× 8_
5
10
6
10
Estimate the area of each rectangle.
29.
30.
3
5 4 ft
1
3 4 in.
5
9 8 in.
1
8 8 ft
31.
32.
33.
A cup of walnuts weighs about
8 ounces. About how many ounces
are called for in the recipe?
34.
If Angelina wants to make 3 cakes,
about how many cups of milk will
she need?
See pages 690, 705.
35.
Self-Check Quiz at
tx.msmath1.com
1
5
3
1
PAINTING A wall measures 8_
feet by 12_
feet.
2
4
If a gallon of paint covers about 150 square
feet, will one gallon of paint be enough to
cover the wall? Explain.
BAKING For Exercises 33 and 34, use
the recipe shown that Angelina is
using to make the cake.
%842!02!#4)#%
Teens Volunteering
VOLUNTEER The circle graph shows how
many teens volunteer. Suppose 104 teens
were surveyed. About how many teens
do not volunteer?
No
4
5 Yes
Source: 200M and Research &
Consulting
2ECIPE 4URTLE#AKE
CUPSMILK
CUPSFLOUR
CUPSCHOCOLATE
CUPCARAMEL
CUPS
WALNUTS
1
HOCKEY In the 2003–2004 season, the Dallas Stars tied about _
of the
5
games in which Marty Turco was the recorded goalie. If he was the
recorded goalie in 71 games, about how many games did the Stars tie?
Lesson 11-6 Estimating Products of Fractions
559
H.O.T. Problems
36.
SELECT THE TECHNIQUE Which of the following techniques could
you use to easily determine whether an answer is reasonable to the
10
1
by 7_
? Justify your response.
multiplication of 4_
11
13
mental math
37.
38.
number sense
estimation
CHALLENGE Determine which point on the
number line could be the graph of the
product of the numbers graphed at C and D.
Explain your reasoning.
0
M
N
CR D
1
*/ -!4( Summarize how you would use compatible
(*/
83 *5*/(
2
numbers to estimate _
of 8.
3
39.
Which is the best estimate of the area
of the rectangle?
40.
3
5 16 in.
7
3 8 in.
A total of 133 sixth-grade students
went to a local museum. Of these,
between one half and three fourths
packed their lunch. Which of the
following ranges could represent the
number of students who packed
their lunch?
F Less than 65
A 24 in2
C 16 in2
in2
in2
B 20
D 15
G Between 65 and 100
H Between 100 and 130
J More than 130
41.
ATTENDANCE After 5 performances, the total attendance of a play was
39,963. Which is a more reasonable estimate for the number of people
that attended each performance 7,000 or 8,000? (Lesson 11-5)
Divide. Round to the nearest hundredth, if necessary.
42.
94.3 ÷ 16.4
46.
FOOD Three people equally share 7.5 ounces of juice. How much
juice does each receive? (Lesson 11-3)
47.
WEATHER Would 60°C or 10°C be a more reasonable temperature if
you need to wear a jacket? Justify your response. (Lesson 8-8)
43.
14.798 ÷ 4.9
(Lesson 11-4)
44.
95.5 ÷ .05
PREREQUISITE SKILL Find the GCF of each set of numbers.
48.
560
6, 9
49.
10, 4
50.
15, 9
Chapter 11 Multiplying and Dividing Decimals and Fractions
45.
112 ÷ 5.2
(Lesson 4-1)
51.
24, 16
Explore
11-7
Main IDEA
Math Lab
Multiplying Fractions
In the lab on pages 537 and 538, you used decimal models to
multiply decimals. You can use a similar model to multiply fractions.
Multiply fractions using
models.
Preparation for
TEKS 7.2 The
student adds,
subtracts, multiplies,
or divides to solve problems
and justify solutions.
(A) Represent multiplication
and division situations
involving fractions and
decimals with models,
including concrete objects,
pictures, words, and numbers.
_ _
1 Find 1 × 1 using a model.
3
2
1
1
1
1
×_
, find _
of _
.
To find _
3
2
3
2
Begin with a square
to represent 1.
Shade _ of the
1
2
square yellow.
1
2
Shade _ of the
1
3
1
2
square blue.
1
3
1
1
1
One sixth of the square is shaded green. So, _
×_
=_
.
3
2
6
Find each product using a model.
a.
_1 × _1
4
2
b.
_1 × _1
3
c.
4
_1 × _1
2
5
ANALYZE THE RESULTS
1.
1
1
Describe how you would change the model to find _
×_
.
1
1
Is the product the same as _
×_
? Explain.
3
2
3
2
Explore 11-7 Math Lab: Multiplying Fractions
561
_ _
2 Find 3 × 2 using a model. Write in simplest form.
5
3
3
3
2
2
, find _
of _
.
To find _ × _
5
3
5
3
Begin with a square
to represent 1.
Shade _ of the
2
3
square yellow.
2
3
Shade _ of the
3
5
2
3
square blue.
3
5
3
6
2
2
Six out of 15 parts are shaded green. So, _
×_
=_
or _
.
5
3
5
15
Find each product using a model. Then write in simplest form.
d.
_3 × _2
4
e.
3
_2 × _5
5
f.
6
_4 × _3
5
8
ANALYZE THE RESULTS
2.
5
10
2
Draw a model to show that _
×_
=_
. Then explain how the
3
6
10
5
model shows that _
simplifies to _
.
18
562
18
9
3.
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?
4.
MAKE A CONJECTURE Write a rule you can use to multiply fractions.
Chapter 11 Multiplying and Dividing Decimals and Fractions
11-7
Multiplying Fractions
Main IDEA
Multiply fractions.
Preparation for
TEKS 7.2 The
student adds,
subtracts, multiplies,
or divides to solve problems
and justify solutions.
(A) Represent multiplication
and division situations
involving fractions and
decimals with models,
including concrete objects,
pictures, words, and
numbers. (B) Use addition,
subtraction, multiplication,
and division to solve
problems involving fractions
and decimals.
EARTH SCIENCE The model shows that
7
about _
of Earth is covered by water.
10
1
Of that part, about _
is covered by the
2
Pacific Ocean. The overlapping area
represents the part covered by the Pacific
1
7
Ocean, which is _
of _
or
2
7
_1 × _
.
2
1.
2.
10
About 7 of Earth’s
10
surface is water.
10
What part of Earth’s
surface is covered by the
Pacific Ocean?
7
10
1
2
What is the relationship
between the numerators and
denominators of the factors
and the numerator and
denominator of the product?
About 1 of Earth’s water
2
surface is the Pacific Ocean.
+%9#/.#%04
Words
Multiply Fractions
Multiply the numerators and multiply the denominators.
Examples
Numbers
Algebra
2×1
2
1
_
×_=_
a×c
c
a
_
× _ = _, where b and d are not 0.
5
2
5×2
b
d
b×d
Multiply Fractions
_ _
1 Find 1 × 1 .
3
4
1×1
_1 × _1 = _
4
3
3×4
1
=_
12
Multiply the numerators.
Multiply the denominators.
1
4
Simplify.
1
3
Multiply. Write in simplest form.
a.
_1 × _3
2
5
Extra Examples at tx.msmath1.com
Stocktrek/CORBIS
b.
_1 × _3
3
4
c.
_2 × _5
3
6
Lesson 11-7 Multiplying Fractions
563
Interactive Lab tx.msmath1.com
To multiply a fraction and a whole number, first write the whole
number as a fraction.
Multiply Fractions and Whole Numbers
_
2 Find 3 × 4.
_3 × 4 = _3 × _4
5
1
_
×4=2
Estimate
5
5
1
3×4
=_
5×1
2
_
= 12 or 2_
5
5
3
5
2
Write 4 as _.
4
1
Multiply.
4
Simplify. Compare to the estimate.
Multiply. Write in simplest form.
d.
REVIEW Vocabulary
factor two or more
numbers that are multiplied
together to form a product;
Example: 1, 2, 3, and 6 are
all factors of 6 (Lesson 1-2)
_2 × 6
e.
3
_3 × 5
f.
4
1
3×_
2
If the numerators and the denominators have a common factor, you
can simplify before you multiply.
Simplify Before Multiplying
_ _
3 Find 3 × 5 .
6
4
Estimate
1
1
_
×1=_
2
2
Divide both the
numerator and the
denominator by 3.
1
The numerator and
the denominator
have a common
factor, 3.
3×5
_3 × _5 = _
6
4
4×6
2
5
=_
Simplify. Compare to the estimate.
8
Multiply. Write in simplest form.
g.
_3 × _4
h.
9
4
9
_5 × _
6
i.
10
_3 × 10
5
Personal Tutor at tx.msmath1.com
Evaluate Expressions
_
_
4 ALGEBRA Evaluate ab if a = 2 and b = 3 .
Mental Math You
can multiply some
fractions mentally.
For example,
3
2
ab = _
×_
3
1
3
8
1
1
4
1
=_
Simplify.
4
j.
8
The GCF of 2 and 8 is 2. The GCF of 3 and 3 is 3.
Divide both the numerator and the denominator
by 2 and then by 3.
3×8
3
8
8
3
2
2
1
_
_
_
of = or _.
3
4
8
8
8
3
2
Replace a with _ and b with _.
2×3
=_
3
1
1
_
of _ = _. So,
564
3
3
2
Evaluate _
c if c = _
.
4
5
Chapter 11 Multiplying and Dividing Decimals and Fractions
k.
3
Evaluate 5a if a = _
.
10
Examples 1–3
(pp. 563–564)
Multiply. Write in simplest form.
1.
_1 × _1
8
2
4
3. _ × 10
5
3
5
5. _ × _
6
10
Example 2
7.
(p. 564)
Example 4
8.
(p. 564)
2.
_2 × _4
4.
_3 × 12
3
5
4
3
5
6. _ × _
5
6
HOBBIES Suppose you are building a
1
model car that is _
the size of the
12
actual car. How long is the model if
the actual car is shown at the right?
16 ft
1
ALGEBRA Evaluate xy if x = _
4
5
and y = _
.
6
(/-%7/2+ (%,0
For
Exercises
9–12
13–16,
25–28
17–20
21–24
See
Examples
1
2
3
4
Multiply. Write in simplest form.
9.
_1 × _2
3
5
3
2
12. _ × _
5
7
5
15. _ × 15
6
3
5
18. _ × _
5
7
10.
_1 × _3
13.
_3 × 2
8
11.
4
4
3
16. _ × 11
8
3
4
19. _ × _
9
8
_
_3 × _5
8
4
2
14. _ × 4
3
1
2
17. _ × _
4
3
5
2
20. _ × _
6
5
_
_
ALGEBRA Evaluate each expression if a = 3 , b = 1 , and c = 1 .
5
21.
25.
26.
27.
28.
ab
22.
bc
23.
_1 a
3
2
3
24.
_6 c
7
7
LIFE SCIENCE About _
of the human body is water. How many pounds
10
of water are in a person who weighs 120 pounds?
MALLS The South Coast Plaza in Orange County, California, is one of the
larger shopping malls in the U.S. It has an area of about 2,700,000 square
1
feet. If _
of the area is for stores that are food-related, about how many
9
square feet in the mall are for food-related stores?
2
WEATHER In a recent year, the weather was partly cloudy _
of the days.
5
Assuming there are 365 days in a year, how many days were partly
cloudy?
5
PIZZA Alvin ate _
of a pizza. If there were 16 slices of pizza, how many
8
slices did Alvin eat?
Lesson 11-7 Multiplying Fractions
Ron Kimball/Ron Kimball Stock
565
Multiply.
29.
_1 × _1 × _1
30.
_2 × _3 × _2
31.
15
_1 × _2 × _
32.
9
5
_2 × _
×_
2
3
2
5
4
16
3
4
3
3
_
10
_
9
_
ALGEBRA Evaluate each expression if x = 4 , y = 3 , and z = 7 .
33.
37.
Real-World Link
The Great Lakes have
enough water to cover
the entire continental
United States with over
9.5 feet of water.
38.
Source: michigan.gov
39.
_2 xz
34.
3
xyz
35.
7
_3 x + z
10
36.
4
_7 y + _5 z
8
7
GEOGRAPHY Michigan’s area is
96,810 square miles. Water makes up
2
about _
of the area of the state. About
5
how many square miles of water does
Michigan have?
4
FLAGS In a recent survey, _
of Americans
5
said they display the American flag. Of
5
these, _
display the flag on their homes.
8
What fraction of Americans display a flag
on their homes?
1
FRANCE In a poll of the students in Lily’s French class, _
have been
6
to France. Of these, 4 have been to Paris. Would 18, 26, or 30 be a
reasonable number of students in Lily’s French class? Explain your
reasoning.
%842!02!#4)#%
See pages 690, 705.
40.
Self-Check Quiz at
tx.msmath1.com
H.O.T. Problems
5
41.
13
RECYCLING In a community survey, _
of residents claim to recycle. Of
20
1
these, _ recycle only paper products. If there are 720 residents, how
4
many residents recycle only paper products?
2
1
1
OPEN ENDED Create a model to explain why _
×_
=_
.
3
2
3
REASONING State whether each statement is true or false. If the statement
is false, provide a counterexample.
42.
The product of two fractions that are each between 0 and 1 is also
between 0 and 1.
43.
The product of a mixed number between 4 and 5 and a fraction between
0 and 1 is less than 4.
44.
The product of two mixed numbers that are each between 4 and 5 is
between 16 and 25.
45.
46.
47.
2
NUMBER SENSE Natalie multiplied _
and 22 and got 33. Decide if this
3
answer is reasonable. Explain why or why not.
3
99
1
2
4
CHALLENGE Find _
×_
×_
×_
× ... × _
without actually
2
3
5
4
100
multiplying. Justify your answer.
*/ -!4( Without multiplying, decide whether the product
(*/
83 *5*/(
5
4
of _ and _
is a fraction or a mixed number. Explain your reasoning.
9
566
7
Chapter 11 Multiplying and Dividing Decimals and Fractions
Planetary Visions Ltd/Photo Researchers
48.
6
On a warm day in May, _
of the
7
students at West Middle School wore
2
short-sleeved T-shirts, and _
of
3
those students wore shorts. How
would you find what fraction of the
students wore T-shirts and shorts to
school?
49.
6
2
A Add _
and _
7
3
6
2
B Subtract _
and _
7
3
6
2
_
C Multiply and _
7
6
2
and _
D Divide _
7
50.
6
G 18
H 27
3
J 72
3
Estimate each product.
_1 of 29
There are 150 students in the band
and 90 students in the chorus.
One-half of the band members
4
and _
of the chorus members
5
participated in a charity concert.
How many more band members
than chorus members participated
in the concert?
F 3
51.
(Lesson 11-6)
8
1
1_
× 5_
9
6
52.
35
_1 × _
7
53.
6
_4 × _8
9
9
54.
GASOLINE Benita spent $31.20 at the gas station to fill up her car’s gas
tank. If she pumped 15 gallons of gasoline into her car, is about $1.50,
$2, or $2.50 a more reasonable answer for the cost of each gallon of
gasoline? (Lesson 11-5)
55.
GEOMETRY Find the volume of the rectangular prism.
56.
(Lesson 10-6)
3
1
HEALTH Nicolás is 65_
inches tall. Stuart is 61_
inches tall.
2
4
How much taller is Nicolás than Stuart? (Lesson 5-6)
4.3 ft
1.7 ft
57.
SPORTS The table shows the finishing times for four
runners in a 100-meter race. At this rate, how far
could Aisha run in one minute? Round to the nearest
tenth. (Lesson 6-1)
Runner
Silvia
Aisha
Fala
Debbie
2.1 ft
Time
14.31 s
13.84 s
13.97 s
13.79 s
PREREQUISITE SKILL Write each mixed number as an improper fraction.
(Lesson 4-3)
58.
1
3_
4
59.
2
5_
3
60.
5
2_
7
61.
5
6_
8
Lesson 11-7 Multiplying Fractions
567
11- 8
Multiplying Mixed Numbers
Main IDEA
Multiply mixed numbers.
Preparation for
TEKS 7.2 The
student adds,
subtracts, multiplies,
or divides to solve problems
and justify solutions. (B) Use
addition, subtraction,
multiplication, and division to
solve problems involving
fractions and decimals.
1
EXERCISE Jasmine walks 3 days a week, 2_
miles each day. The
2
number line shows the miles she walks in a week.
Day 1
Day 2
Day 3
1
2 2 mi
1
2 2 mi
2 2 mi
0
1
2
3
4
1
5
6
7
8
1.
How many miles does Jasmine walk in a week?
2.
Write a multiplication sentence that shows the total miles
walked in a week.
3.
Write the multiplication sentence using improper fractions.
Use a number line and improper fractions to find each product.
4.
1
2 × 1_
5.
3
1
2 × 2_
4
6.
3
3 × 1_
4
Multiplying mixed numbers is similar to multiplying fractions.
+%9#/.#%04
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
_
_
1 Find 1 × 4 4 .
Estimate Use compatible numbers 5
4
4
1
24
1
_ × 4_ = _ × _
5
5
4
4
1
_
×4=1
4
Write 4_ as _.
4
5
24
5
6
1 × 24
=_
4×5
Divide 24 and 4 by their GCF, 4.
1
6
1
=_
or 1_
5
5
Simplify. Compare to the estimate.
Multiply. Write in simplest form.
a.
568
_2 × 2_1
3
2
b.
_3 × 3_1
8
Chapter 11 Multiplying and Dividing Decimals and Fractions
3
c.
1
1
3_
×_
2
3
Multiply Mixed Numbers
_
2 BAKING Alejandro is making 3 1 times the recipe for Cowboy
2
Corn Bread. If the recipe calls for 1 3 cups of cornmeal, how
4
much cornmeal will he need?
_
3
3
1
The recipe calls for 1_
cups of cornmeal. Multiply 3_
by 1_
.
2
4
3 _
1
7
3_
× 1_
= 7 ×_
2
4
4
First, write mixed numbers as improper fractions.
2
4
7 _
_
= ×7
2
4
49
1
=_
or 6_
8
8
Then, multiply the numerators and multiply the
denominators.
Simplify.
1
Alejandro will need 6_
cups of plain cornmeal.
Real-World Link
For most of the state’s
first 100 years, Texans
lived on beans and
cornbread.
8
d.
Source:
awesome-tex-mex-recipes.com
CONSTRUCTION Mr. Wilkins is laying bricks to make a
rectangular patio. The area he is covering with bricks is
3
1
15_
feet by 9_
feet. What is the area of the patio?
2
4
Personal Tutor at tx.msmath1.com
Evaluate Expressions
_
_
3 ALGEBRA If c = 1 7 and d = 3 1 , what is the value of cd?
8
7
1
cd = 1_
× 3_
8
3
5
5
8
3
15 _
=_
× 10
4
e.
Examples 1, 2
(pp. 568–569)
Example 2
(p. 569)
Example 3
(p. 569)
Divide the numerator and denominator by 3 and by 2.
4
Simplify.
3
1
ALGEBRA If a = 3_
and b = 2_
, what is the value of ab?
5
4
Multiply. Write in simplest form.
1.
_1 × 2_3
2
8
2.
1
2
1_
×_
2
3
3.
3
4
1_
× 2_
5
4
1
4. 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
9
1
5. ALGEBRA If x = _ and y = 1_, find xy.
3
10
Extra Examples at tx.msmath1.com
Mark Thomas/FoodPix/Getty Images
3
8
1
25
1
=_
or 6_
4
3
7
1
_
Replace c with 1 and d with 3_.
Lesson 11-8 Multiplying Mixed Numbers
569
(/-%7/2+ (%,0
For
Exercises
6–11, 23
12–17, 22
18–21
See
Examples
1
2
3
Multiply. Write in simplest form.
6.
_1 × 2_1
7.
2
3
5
4
9. 1_ × _
5
6
1
1
12. 1_ × 1_
3
4
5
1
15. 4_ × 2_
2
6
_3 × 2_5
8.
6
4
1
7
10. _ × 3_
4
8
1
1
13. 3_ × 3_
5
6
3
2
16. 6_ × 3_
3
10
_
7
4
1_
×_
8
5
5
3
11. _ × 2_
6
10
3
2
14. 3_ × 2_
5
4
3
5
17. 3_ × 5_
5
12
_
_
ALGEBRA Evaluate each expression if a = 2 , b = 3 1 , and c = 1 3 .
3
_1 c
18.
ab
22.
ART A reproduction of Claude Monet’s Water1
1
Lilies has dimensions 34_
inches by 36_
inches.
2
2
Find the area of the painting.
23.
ANIMALS A three-toed sloth can travel at a
3
speed of _
mile per hour. At this rate, how far
20
1
can a three-toed sloth travel in 2_
hours?
19.
20.
2
bc
2
21.
_1 a4
8
2
Multiply. Write in simplest form.
24.
_3 × 2_1 × _4
26.
2
1
2
3_
× 4_
× 2_
28.
2
4
5
25.
5
2
3
1
2
1_
×_
×_
3
5
5
1
1
27. _ × 5_ × 1_
6
4
7
3
2
7
CEREAL The box of a popular brand of cereal measures 1_
inches
8
3
1
_
_
by 11 inches by 8 inches. Find the volume of the box. Write in
3
5
simplest form.
MUSIC For Exercises 29–32, use the table and
the following information.
A dot following a music note (•) means that
1
the note gets 1_
times as many beats as the
2
same note without a dot (). How many beats
does each note get?
%842!02!#4)#%
tx.msmath1.com
570
Whole Note
Half Note
Quarter Note
Eighth Note
Number
of Beats
4
2
_1
1
2
dotted whole note
30.
dotted quarter note
31.
dotted eighth note
32.
dotted half note
33.
FIND THE DATA Refer to the Texas Data File on pages 16–19. Choose
some data and write a real-world problem in which you would multiply
mixed numbers.
_
_
_
ALGEBRA Evaluate each expression if g = 5 3 , k = 2 1 , and h = 1 7 .
34.
gk + h
35.
gkh
Chapter 11 Multiplying and Dividing Decimals and Fractions
Archivo Iconografico, S.A./CORBIS
Note
29.
See pages 690, 705.
Self-Check Quiz at
Name
4
3
36.
gh - k
8
H.O.T. Problems
37.
OPEN ENDED Write a problem that can be solved by multiplying mixed
numbers. Explain how to find the product.
38.
NUMBER SENSE Without multiplying,
1
2
determine whether the product 2_
×_
2
A
0
3
B
1
C
2
3
is located on the number line at point A, B,
or C. Explain your reasoning.
39.
40.
41.
CHALLENGE Determine if the product of two positive mixed numbers is
always, sometimes, or never less than 1. Explain your reasoning.
*/ -!4( Summarize how to multiply mixed numbers.
(*/
83 *5*/(
The table shows some ingredients
needed to make Tex-Mex lasagna.
cheddar
cheese
chopped
onion
pinto
beans
3_ cups
_1 cup
2
2_
cups
1
2
4
42.
3
5
F 13_
lb
8
7
lb
G 17_
8
1
lb
H 35_
4
1
lb
J 36_
4
If you make four times the recipe,
how many cups of pinto beans are
needed?
3
A 9_
c
2
C 10_
c
1
c
B 10_
1
D 5_
c
4
2
3
3
Multiply. Write in simplest form.
43.
_5 × _3
7
4
44.
_2 × _1
3
Dario buys a bag of lawn fertilizer
3
that weighs 35_
pounds. He wants to
4
1
_
use of the bag on his front lawn.
2
How many pounds of fertilizer will
he use on the front lawn?
6
(Lesson 11-7)
45.
_3 × _2
8
46.
5
_1 × _4
2
47.
RECREATION There are about 300 million people who visit a national
2
park in the United States. If about _
come from overseas, about how
5
many visitors come from abroad? (Lesson 11-6)
48.
OLYMPICS The table shows the winners of
the Women’s Marathon in the 2000 and 2004
Summer Olympics. How much faster was
Takahashi’s time than Noguchi’s time?
(Lesson 8-7)
Olympic
Year
2000
Runners
Time
Naoko Takahashi
2 h 23 min 14 s
2004
Mizuki Noguchi
2 h 26 min 20 s
Source: The World Almanac
PREREQUISITE SKILL Multiply. Write in simplest form.
49.
_1 × _3
4
8
7
50.
_2 × _3
7
4
51.
_1 × _1
2
6
(Lesson 11-7)
52.
_2 × _5
5
6
Lesson 11-8 Multiplying Mixed Numbers
571
Explore
11-9
Main IDEA
Divide fractions using
models.
Math Lab
Dividing Fractions
There are 8 small prizes that are given
away 2 at a time. How many people will
get prizes?
1.
Preparation for
TEKS 7.2 The
student adds,
subtracts, multiplies,
or divides to solve problems
and justify solutions.
(A) Represent multiplication
and division situations
involving fractions and
decimals with models,
including concrete objects,
pictures, words, and numbers.
How many 2s are in 8? Write as a
division expression.
Suppose there are two granola bars
divided equally among 8 people. What
part of a granola bar will each person get?
2.
What part of 8 is in 2? Write
as a division expression.
_
1 Find 1 ÷ 1 using a model.
5
Make a model of the dividend, 1.
THINK How many _s
5
are in 1?
1
Rename 1 as _ so the numbers have common
5
5
1
denominators. So, the problem is _ ÷ _
. Redraw
5
5
the model to show _.
5
5
5
How many _s are in _?
5
5
1
5
Circle groups that are the size of the divisor _.
1
5
There are five _s in _.
1
5
1
So, 1 ÷ _
= 5.
5
Find each quotient using a model.
a.
572
1
2÷_
5
b.
1
3÷_
3
Chapter 11 Multiplying and Dividing Decimals and Fractions
Gary Rhijnsburger/Masterfile
c.
2
3÷_
3
d.
3
2÷_
4
5
5
A model can also be used to find the quotient of two fractions.
_ _
2 Find 3 ÷ 3 using a model.
8
4
Rename _ as _ so the fractions have common
3
4
6
8
6
3
denominators. So, the problem is _ ÷ _. Draw a
8
6
model of the dividend, _ .
8
8
THINK
How many _s are in _?
3
8
6
8
Circle groups that are the size of the divisor, _.
3
8
There are two _s in _.
3
8
6
8
3
3
÷_
= 2.
So, _
4
8
Find each quotient using a model.
e.
1
4 ÷_
_
10
5
f.
_3 ÷ _1
4
2
g.
_4 ÷ _1
5
5
h.
_1 ÷ _1
6
3
ANALYZE THE RESULTS
Use greater than, less than, or equal to to complete each sentence.
Then give an example to support your answer.
1. When the dividend is equal to the divisor, the quotient is 1.
2. When the dividend is greater than the divisor, the quotient
is 1.
3. When the dividend is less than the divisor, the quotient
is 1.
4. MAKE A CONJECTURE You know that multiplication is commutative
because the product of 3 × 4 is the same as 4 × 3. Is division
commutative? Give examples to explain your answer.
Explore 11-9 Math Lab: Dividing Fractions
573
11-9
Dividing Fractions
Main IDEA
Divide fractions.
Preparation for
TEKS 7.2 The student
adds, subtracts,
multiplies, or divides
to solve problems and justify
solutions. (B) Use addition,
subtraction, multiplication, and
division to solve problems
involving fractions and
decimals.
Kenji and his friend Malik ordered three
one-foot submarine sandwiches. They
1
of a sandwich will serve
estimate that _
2
one person.
1
1. How many _-sandwich servings
2
are there?
2.
reciprocal
1
1
The model shows 3 ÷ _
. What is 3 ÷ _
?
2
2
Draw a model to find each quotient.
3.
NEW Vocabulary
1
3÷_
4.
4
1
2÷_
5.
6
1
4÷_
2
1
1
The Mini Lab shows that 3 ÷ _
= 6. Notice that dividing by _
gives
2
2
the same result as multiplying by 2.
1
=6
3÷_
3×2=6
2
Notice that _ × 2 = 1.
1
2
1
The numbers _
and 2 have a special relationship. Their product is 1.
2
Any two numbers whose product is 1 are called reciprocals.
Find Reciprocals
_
2 Find the reciprocal of 2 .
1 Find the reciprocal of 5.
1
Since 5 × _
= 1, the
3
2
Since _
×_
= 1, the
1
.
reciprocal of 5 is _
2 _
reciprocal of _
is 3 .
5
3
5
3
2
3
2
Find the reciprocal of each number.
a.
11
b.
_3
c.
5
_1
3
You can use reciprocals to divide fractions.
+%9#/.#%04
Words
Examples
574
Divide Fractions
To divide by a fraction, multiply by its reciprocal.
Numbers
Algebra
3
1
2
1
_
÷_=_×_
3
2
2
2
c
d
a
a
_
÷ _ = _ × _, where b, c, and d ≠ 0
d
c
b
b
Chapter 11 Multiplying and Dividing Decimals and Fractions
Divide by a Fraction
_ _
3 Find 1 ÷ 3 .
8
4
_1 ÷ _3 = _1 × _4
8
8
4
Mental Math To
find the reciprocal
of a fraction, invert
the fraction. That is,
switch the numerator
and denominator.
3
Multiply by the reciprocal,
_4 .
3
1
1×4
=_
8×3
Divide 8 and 4 by the GCF, 4.
2
1
=_
Multiply numerators.
Multiply demonimators.
6
_
4 Find 3 ÷ 1 .
2
3
1
2
=_
×_
3÷_
2
1
1
6
= _ or 6
1
Multiply by the reciprocal of
_1 .
2
Simplify.
Divide. Write in simplest form.
d.
_1 ÷ _3
4
e.
8
_2 ÷ _3
3
f.
8
_
3
4÷_
4
Divide by a
Whole Number
5 CAMP At a summer day camp, 3 of each day is spent in group
4
activities. There are 6 camp counselors who split their time
equally as the activity leaders. What fraction of the day is each
counselor the activity leader?
3
into 6 equal parts.
Divide _
4
_3 ÷ 6 = _3 × _1
6
4
4
Multiply by the reciprocal.
1
3
1
=_
×_
Divide 3 and 6 by the GCF, 3.
4
1
=_
8
6
2
Simplify.
1
Each camp counselor spends _
of his or her day at camp as the
8
activity leader.
g.
2
GARDENING A neighborhood garden that is _
of an acre is to be
3
divided into 4 equal-size areas. What is the size of each area?
Personal Tutor at tx.msmath1.com
Extra Examples at tx.msmath1.com
Lesson 11-9 Dividing Fractions
575
Examples 1, 2
(p. 574)
Examples 3–5
(p. 575)
Find the reciprocal of each number.
1.
_2
3
2.
_1
7
5.
11.
(p. 575)
_1 ÷ _1
6.
_5 ÷ _1
4.
5
4
3
2
8. 5 ÷ _
7
5
10. _ ÷ 3
6
4
6
2
FOOD Mrs. Perkins has _
of a pan of lasagna left for dinner. She wants
3
to divide the lasagna into 6 equal pieces for her family. What part of the
original pan of lasagna will each person get?
(/-%7/2+ (%,0
Find the reciprocal of each number.
For
Exercises
12–17
18–21,
33
22–25, 30
26–29, 31,
32
12.
See
Examples
1, 2
3
_2
Divide. Write in simplest form.
2
1
7. 2 ÷ _
3
4
_
9.
÷2
5
Example 5
3.
_1
4
7
15. _
9
13.
1
_
14.
_2
16.
8
17.
1
10
5
Divide. Write in simplest form.
4
5
18.
_1 ÷ _1
2
3
22. 3 ÷ _
4
3
26. _ ÷ 6
5
30.
31.
8
19.
_1 ÷ _2
2
3
3
23. 2 ÷ _
5
5
27. _ ÷ 5
6
20.
_3 ÷ _2
21.
3
3
24. 5 ÷ _
4
5
28. _ ÷ 2
8
4
9
_3 ÷ _
4
10
4
25. 8 ÷ _
7
8
29. _ ÷ 4
9
DOGS Aida works at a kennel and uses 30-pound bags of dog food to
2
feed the dogs. If each dog gets _
pound of food each day, how many
5
dogs can she feed with one bag?
8
BUILDING Jamar has a piece of plywood that is _
yard long. He wants
9
to cut this into 3 equal-size pieces to use as small shelves in his
bedroom. What will be the length of each of these shelves?
32.
3
TIME Chelsea devoted _
of her day to run errands, exercise, visit with
8
her friends, and go shopping. If she devotes an equal amount of time
to each of these four activities, what fraction of her day is spent on each
activity?
33.
MEASUREMENT A piece of string is to be cut into equal-size pieces. If the
11
1
length of the string is _
foot long and each piece is to be _
foot long,
12
how many pieces can be cut?
576
Chapter 11 Multiplying and Dividing Decimals and Fractions
24
GEOGRAPHY For Exercises 34 and 35, use the table below.
The table shows the approximate fraction of the total boundary of Texas
occupied by each segment.
34.
How many times longer is the Rio Grande boundary than the Red River
boundary?
35.
How many times longer is the Coastline boundary than the boundary
along the 32nd parallel?
Texas’ Boundary Lines
Rio Grande
8
_
Coastline
3
_
Red River
3
_
25
20
20
Panhandle line (East,
North, and West)
_1
5
Sabine River
1
_
Along 32nd parallel
2
_
10
25
Source: Texas Almanac
_
_
_
ALGEBRA Find the value of each expression if a = 2 , b = 3 , and c = 1 .
36.
%842!02!#4)#%
40.
a÷b
37.
b÷c-a
38.
3
a÷c
4
39.
c÷b+a
2
3
1
PAINTING Lena has painted _
of a room. She has used 1_
gallons of
2
4
paint. How much paint will she need to finish the job?
See pages 691, 705.
41.
Self-Check Quiz at
tx.msmath1.com
H.O.T. Problems
FIND THE DATA Refer to the Texas Data File on pages 16–19. Choose
some data and write a real-world problem in which you would divide
fractions.
42.
3
3
1
2
1
OPEN ENDED Create a model to explain why _
÷_
=_
×_
=_
.
43.
2
FIND THE ERROR Jason and Max are solving _
÷ 4. Who is correct?
3
Explain your reasoning.
2
_2 ÷ 4 = _2 × _4
3
Jason
1
3
8
2
=_
or 2_
3
3
3
2
2
4
_2 ÷ 4 = _2 × _1
3
3
4
2
=_
or _1
12
6
Max
Lesson 11-9 Dividing Fractions
(t)Getty Images, (bl)Kevin Peterson/Getty Images, (br)Brad Wilson/Getty Images
577
CHALLENGE Analyze each expression to find the solution mentally.
44.
46.
47.
2,345
2,345
12
_
×_
÷_
1,015
11
45.
1,015
2,345
2,345
12
_
×_
÷_
11
1,015
1,015
*/ -!4( Create two real-world problems that involve the
(*/
83 *5*/(
1
fraction _
and the whole number 3. One problem should involve
2
multiplication, and the other should involve division.
The table shows the weight factors
of other planets relative to Earth.
For example, an object on Jupiter
is 3 times heavier than on Earth.
48.
_1
Venus
9
_
Jupiter
3
2
2
F _
8
7
G _
12
2
H _
3
5
J _
24
Planets’ Weight Factors
Planet
Weight Factor
Mercury
Which of the following numbers,
1
when divided by _
, gives a result
2
1
_
less than ?
3
10
Source: factmonster.com
About how many times heavier is an
object on Venus than on Mercury?
3
A 1_
7
C 2_
5
1
B 2_
16
10
1
D 3_
8
Multiply. Write in simplest form.
49.
53.
2
1
2_
× 3_
5
(Lesson 11-8)
5
3
50. 1_ × 2_
6
4
3
51.
3
3
3_
× 2_
7
52.
8
4
1
4_
× 5_
9
4
VOLUNTEERING According to a survey, 9 in 10 teens volunteer at
1
least once a year. Of these, about _
help clean up their communities.
3
What fraction of teens volunteer by helping clean up their
communities? (Lesson 11-7)
54.
ARCHITECTURE The main entrance of the Rock and Roll Hall of Fame
is a triangle with a base of about 241 feet and a height of about
165 feet. Find the approximate area of this triangle. (Lesson 10-4)
PREREQUISITE SKILL Write each mixed number as an improper fraction.
Then find the reciprocal of each. (Lesson 4-3)
55.
578
2
1_
3
56.
5
1_
9
57.
1
4_
2
Chapter 11 Multiplying and Dividing Decimals and Fractions
58.
3
3_
4
59.
4
6_
5
11-10
Dividing Mixed Numbers
Main IDEA
Divide mixed numbers.
Preparation for
TEKS 7.2 The
student adds,
subtracts, multiplies,
or divides to solve problems
and justify solutions. (B) Use
addition, subtraction,
multiplication, and division to
solve problems involving
fractions and decimals.
FABRIC Suppose you are going to
3
cut pieces of fabric 1_
yards long
4 1
from a bolt containing 5_ yards
2
of fabric.
1.
3
1 4 yd
To the nearest yard, how long is
each piece?
2.
To the nearest yard, how long is
the fabric on the bolt?
3.
About how many pieces can
you cut?
4.
3
1 4 yd
1
5 2 yd
3
1 4 yd
1
About how many pieces of fabric 2_
yards long could you cut
4
3
from a bolt containing 7_
yards of fabric?
4
Dividing mixed numbers is similar to dividing fractions. To divide
mixed numbers, write the mixed numbers as improper fractions and
then divide as with fractions.
Divide by a Mixed Number
_
_
1 Find 5 1 ÷ 1 3 .
2
4
Estimate 6 ÷ 2 = 3
3
1
11
7
5_
÷ 1_
=_
÷_
2
4
Write mixed numbers as improper fractions.
2
4
11
4
=_
×_
2
7
Multiply by the reciprocal.
2
11
4
=_
×_
2
1
Divide 2 and 4 by the GCF, 2.
7
22
1
=_
or 3_
7
7
Simplify.
Check for Reasonableness 3_ ≈ 3 ✔
1
7
Divide. Write in simplest form.
a.
1
1
4_
÷ 2_
5
3
b.
1
8 ÷ 2_
2
c.
5
1
1_
÷ 2_
9
3
Personal Tutor at tx.msmath1.com
Extra Examples at tx.msmath1.com
Lesson 11-10 Dividing Mixed Numbers
579
Evaluate Expressions
_
_
2 ALBEBRA Find m ÷ n if m = 1 3 and n = 2 .
5
4
3
2
÷_
m ÷ n = 1_
5
4
7
2
=_
÷_
5
4
5
7
=_×_
2
4
35
3
=_
or 4_
8
8
d.
Replace m with 1_ and n with _.
3
4
2
5
Write the mixed number as an improper fraction.
Multiply the reciprocal.
Simplify.
3
1
ALGEBRA Find c ÷ d if c = 2_
and d = 1_
.
8
4
_
3 WEATHER A tornado traveled 100 miles in 1 1 hours. How many
2
miles per hour did it travel? Estimate 100 ÷ 2 = 50
100
3
1
100 ÷ 1_
=_
÷_
2
Real-World Career
How Does a Tornado
Tracker Use Math?
Tornado trackers calculate
the speed and direction of
tornadoes. They also
calculate the intensity of
the storm.
(p. 579)
Example 2
(p. 580)
Multiply by the reciprocal.
Simplify.
Compare to the estimate.
3
e.
1
FUND-RAISING The soccer team has 16_
boxes of wrapping
2
paper to sell to raise money for team T-shirts. If selling the
wrapping paper is split equally among the 12 players, how
many boxes should each player sell?
Divide. Write in simplest form.
1.
1
3_
÷2
4.
3
1
ALGEBRA What is the value of c ÷ d if c = _
and d = 1_
?
5.
3
BAKING Jay is cutting a roll of biscuit dough into slices that are _
inch
(p. 580)
Example 3
Write the mixed number as an improper fraction.
2
So, the tornado traveled 66_
miles per hour.
For more information,
go to tx.msmath1.com.
Example 1
2
1
100
2
=_
×_
3
1
200
_
=
3
2
= 66_
3
2.
2
1
8 ÷ 1_
8
Aaron Haupt
1
2
3_
÷_
5
7
2
1
thick. If the roll is 10_
inches long, how many slices can he cut?
2
580
3.
3
Chapter 11 Multiplying and Dividing Decimals and Fractions
8
(/-%7/2+ (%,0
For
Exercises
6–17
18–23
24–27
See
Examples
1
2
3
Divide. Write in simplest form.
6.
1
5_
÷2
7.
2
1
4_
÷ 10
8.
6
3
1
10. 6_ ÷ _
2
4
3
1
_
13. 8 ÷ 2_
6
4
2
2
16. 4_ ÷ 2_
3
9
1
9. 6 ÷ 2_
4
1
1
_
12. 6 ÷ 3_
2
4
3
5
15. 3_ ÷ 5_
8
4
1
3 ÷ 4_
2
4
1
11. 7_ ÷ _
5
5
3
4
_
14. 3 ÷ 1_
5
5
3
3
17. 6_ ÷ 2_
5
4
_
_
_
ALGEBRA Evaluate each expression if a = 4 4 , b = 2 , c = 6, and d = 1 1 .
5
3
2
18.
12 ÷ a
19.
2
b ÷ 1_
21.
a÷c
22.
c÷d
24.
1
1
FOOD How many _
-pound hamburgers can be made from 2_
pounds
2
4
of ground beef?
25.
26.
9
20.
a÷b
23.
c ÷ (ab)
MEASUREMENT Suppose you are designing the layout for your school
3
yearbook. If a student photograph is 1_
inches wide, how many
8 7
photographs will fit across a page that is 6_ inches wide? Assume
8
there is no spacing between photographs.
3
CASHEWS How many _
-pound bags of cashews can be made from
8
3
6_
pounds of cashews?
8
27.
2
BORDERS The length of a kitchen wall is 24_
feet long. A border will be
3
placed along the wall of the kitchen. If the border comes in strips that
3
are each 1_
feet long, how many strips of border are needed?
4
28.
3
days. How many
9_
Iditarod Race Trail
White
Mountain
Nome
Safety
Koyuk
Nulato
Elim
Shaktoolik
Kaltag
Unalakleet
Golovin
Norton
Sound
Grayling
Y uk
SLED DOG RACING In 2005,
Robert Sorlie won the
Iditarod Trail Sled Dog
Race for the second
time. He completedthe
1,100-mile course in
on
Anvik
Eagle
Island
Shageluk
Iditarod
4
miles did he average
each day?
Northern Route
(even numbered years)
Galena
Ruby
Cripple
Landing
Ophir
Takotna Nikolai
McGrath
Rohn
Southern Route
(odd numbered years)
Rainy Pass
Skwentna
Finger
Lake
Yentna
Knik
Anchorage
Wasilla
Eagle
River
HURRICANES For Exercises 29 and 30, use the following information.
%842!02!#4)#%
See pages 691, 705.
Suppose a hurricane traveled 130 miles from a point in the Atlantic Ocean
1
to the Florida coastline in 6_
hours.
2
29.
How many miles per hour did the hurricane travel?
30.
1
How far would the hurricane travel in 1_
hours at the same speed?
Self-Check Quiz at
tx.msmath1.com
2
Lesson 11-10 Dividing Mixed Numbers
581
H.O.T. Problems
31.
OPEN ENDED Create a problem about a real-world situation that is
3
1
represented by 12_
÷ 2_
. Then solve your problem.
2
4
32.
Which One Doesn’t Belong? Select the expression that has a quotient less
than 1. Explain your reasoning.
1
1
2_ ÷ 1_
33.
3
1
1
2_ ÷ 3_
5
8
3
1
3_ ÷ 1_
3
2
5
8
2
CHALLENGE Decide whether _
÷ 1_
is greater than or less than
3
10
8
3
_
_
÷ 1 . Write a convincing explanation to support your decision.
10
34.
2
1
4_ ÷ 2_
3
2
4
*/ -!4( In your own words, explain how to find the
(*/
83 *5*/(
2
quotient of 12 and 2_
.
3
The widths of 10 blooms in a test of a
new marigold variety are given.
36.
Marigold Bloom
Width (in.)
3
3
F _
c
What is the average (mean) bloom
width?
3
A 2_
in.
4
_
B 3 1 in.
20
1
C 3_
in.
4
2
D 3_
in.
5
10
1
c
G _
2
5
c
H _
9
2
c
J _
3
MEASUREMENT For Exercises 37 and 38, use the
graphic at the right and the information below.
(Lesson 11-9)
9
One U.S. ton equals _
metric ton. So, you can
10
9
use t ÷ _ to convert t metric tons to U.S. tons.
10
37.
Write a division expression to represent the U.S.
tons of gold that were produced in South Africa.
Then simplify.
38.
How many U.S. tons of gold were produced in
Russia?
Multiply. Write in simplest form.
39.
582
_4 × 1_3
5
4
40.
7
2EGION
3OURCE7ORLD'OLD#OUNCIL
Chapter 11 Multiplying and Dividing Decimals and Fractions
41.
(Lesson 11-8)
5
2
2_
×_
8
7ORLD'OLD0RODUCTION
RA
LIA
#H
IN
A
2U
SS
IA
3
3!
3
4
1
3_
4
ST
2_
5
1
2
1
3_
4
!U
2_
!F
RIC
A
3
4
1
3_
2
H
2_
UT
1
4
1
3_
4
3O
3_
1
Lola used 1_
cups of dried apricots
5 2
_
to make of her trail mix. How
6
many more cups of dried apricots
does she need to finish making her
trail mix?
4OTAL0RODUCTION
METRICTONS
35.
1
1
1_
× 5_
8
3
CH
APTER
11
Study Guide
and Review
Download Vocabulary
Review from tx.msmath1.com
Key Vocabulary
Be sure the following Key
Concepts are noted in your
Foldable.
Key Concepts
Multiplying Decimals
compatible numbers (p. 557)
reciprocal (p. 574)
££‡£
Õ
Ìˆ«
Þ
>˜ ˆ˜}
ÊˆÛ `
ˆ
iV `ˆ˜}
ˆ“>
>˜ Ã
À> `
V̈œ
˜Ã
££‡Ó
££‡Î
££‡{
£‡x
£‡È
scientific notation (p. 534)
(Lessons 11-1 and 11-2)
• When multiplying decimals,
multiply with whole numbers. The product has
the same number of decimal places as the sum of
the number of decimal places in each factor.
Round to the nearest tenth if necessary.
Dividing Decimals
(Lessons 11-3 and 11-4)
• When dividing a decimal by a whole number,
place the decimal point directly above the decimal
point in the dividend. Then divide as with whole
numbers. Round to the nearest tenth if necessary.
• When dividing a decimal by a decimal, change the
divisor into a whole number by multiplying both
the dividend and the divisor by the same power
of ten. Then divide as with dividing a decimal by
a whole number. Round to the nearest tenth if
necessary.
Multiplying Fractions
(Lessons 11-6 to 11-8)
• To multiply fractions, multiply the numerators and
multiply the denominators. Write the result in
simplest form.
Vocabulary Check
State whether each sentence is true or
false. If false, replace the underlined word
or number to make a true sentence.
1.
Any two numbers whose product is 1
are called compatible numbers.
2.
To annex a zero means to add a zero.
3.
Reciprocals can be used when
estimating products and are numbers
that are easy to divide mentally.
4.
The number 3.85 × 102 is written in
scientific notation.
5.
The product of 9.12 × 0.8 will have two
decimal places.
6.
When rounding the quotient of a
division problem to the nearest tenth,
stop dividing when there is a digit in the
hundredths place.
• To multiply mixed numbers, write the mixed
numbers as improper fractions and then multiply
as with fractions. Write the result in simplest form.
7.
Dividing Fractions
(Lessons 11-9 to 11-10)
• To divide by a fraction, multiply by its reciprocal.
Write the result in simplest form.
• To divide mixed numbers, write the mixed
numbers as improper fractions. Then divide as
with fractions. Write the result in simplest form.
Vocabulary Review at tx.msmath1.com
5
2
The product of _
×_
has a denominator
6
7
of 13 when simplified.
8.
8
The fraction _
is an improper fraction.
9.
1
The reciprocal of _
is 4.
10.
3
4
When dividing fractions, multiply the
reciprocal of the first fraction.
Chapter 11 Study Guide and Review
583
CH
APTER
11
Study Guide and Review
Lesson-by-Lesson Review
11-1
Multiplying Decimals by Whole Numbers
(pp. 533–536)
Example 1
Multiply.
11-2
Estimate 6.45 × 7 → 6 × 7 or 42
11.
1.4 × 6
12.
3 × 9.95
13.
0.082 × 17
14.
12.09 × 19
15.
5 × 0.048
16.
24.7 × 31
17.
16 × 6.65
18.
2.6 × 38
19.
SHOPPING Three pairs of shoes are
priced at $39.95 each. Find the total
cost for the shoes.
20.
MONEY If you work 6 hours at $6.35
an hour, how much would you
make?
Multiplying Decimals
Find 6.45 × 7.
3 3
6.45 There are two decimal places to the
× 7 right of the decimal in 6.45.
45.15 Count the same number of places from
right to left in the product.
(pp. 539–542)
Multiply.
21.
0.6 × 1.3
22.
8.74 × 2.23
23.
0.04 × 5.1
24.
2.6 × 3.9
25.
0.002 × 50
26.
0.04 × 0.0063
27.
GEOMETRY Find the area of the
rectangle.
Example 2 Find 38.76 × 4.2.
38.76 two decimal places
× 4.2 one decimal place
7752
15504
162.792 three decimal places
5.4 in.
1.3 in.
11-3
Dividing Decimals by Whole Numbers
(pp. 543–548)
Divide.
584
28.
12.24 ÷ 36
29.
203.84 ÷ 32
30.
136.5 ÷ 35
31.
37.1 ÷ 14
32.
4.41 ÷ 5
33.
26.96 ÷ 8
34.
SPORTS BANQUET The cost of the
Spring Sports Banquet is to be
divided equally among the 62 people
attending. If the cost is $542.50, find
the cost per person.
Example 3 Find the quotient of
16.1 ÷ 7.
2.3 Place the decimal point.
7 16.1 Divide as with whole numbers.
- 14
21
-2 1
0
Chapter 11 Multiplying and Dividing Decimals and Fractions
Mixed Problem Solving
For mixed problem-solving practice,
see page 705.
11-4
Dividing by Decimals
(pp. 549–553)
Divide.
11-5
35.
0.96 ÷ 0.6
36.
11.16 ÷ 6.2
37.
0.276 ÷ 0.6
38.
5.88 ÷ 0.4
39.
18.45 ÷ 0.5
40.
0.155 ÷ 0.25
41.
SPACE The Aero Spacelines Super
Guppy, a converted Boeing C-97, can
carry 87.5 tons of cargo. Tanks that
weigh 4.5 tons each are loaded onto
the Super Guppy. What is the most
number of tanks it can transport?
PSI: Reasonable Answers
43.
11-6
11
HEIGHT Evan is 5_
feet tall. His
12
4
_
sister, Cindy, is of his height.
5
Which is a reasonable height for
Cindy: about 4 feet, 5 feet, or 6 feet?
Explain your reasoning.
MONEY Derek has $23.80 in his
2
pocket. He spent about _
of this
3
amount on a CD. Would $8, $16, or
$20 be a reasonable price of the CD?
Estimating Products of Fractions
44.
45.
3
10 × 2_
47.
3
1
7_
×_
48.
5
3
11
3
4_
× 8_
49. _ × _
50.
4
4
6
10
Example 5 There are 24 students in
the Spanish club. If the number of
_
students in the school is 19 7 times this
8
amount, would about 400, 500, or 600
be a reasonable number of students in
the school?
7
24 × 19_
is about 25 × 20 or 500. So, 500
8
is a reasonable number of students in
the school.
Example 6
_1 × 21
4
1.4 Place the decimal point.
82 114.8 Divide as with whole numbers.
-82
32 8
-32 8
0
(pp. 557–560)
Estimate each product.
5
dividend by 10 to move the
decimal point one place to
the right so that the divisor
is a whole number.
(pp. 554–555)
Determine reasonable answers for
Exercises 42 and 43.
42.
Example 4 Find 11.48 ÷ 8.2.
8.2 11.48 Multiply the divisor and the
46.
_5 × 13
6
7
_1 × 41 _1 × 42
7
7
7
42 and 7 are compatible
numbers since
42 ÷ 7 = 6.
_1 × 42 = 6 _1 of 42 is 6.
12
AMUSEMENT PARKS The average wait
time to ride the Super Coaster is
55 minutes. If Joy and her friends
5
have waited _
of that time, estimate
6
how long they have waited.
_
Estimate 1 × 41.
7
7
1
So, _
× 41 is about 6.
7
Chapter 11 Study Guide and Review
585
CH
APTER
11
Study Guide and Review
11-7
Multiplying Fractions
(pp. 563–567)
51.
54.
11-8
_1 × _1
52.
4
3
4
_7 × _
53.
21
8
_5 × 9
6
55.
_2 × 4_1
58.
ART Find the area of the painting.
2
3
5
6_
×4
57.
8
5
3
2
=_
Simplify.
15
1
2
2_
× 6_
4
3
_
3
2
3
7
49
1
=_
or 16_
3
FT
_2 ÷ _4
60.
5
_1 ÷ _3
61.
4
5÷_
9
3
62. COOKING Ashanti uses _ cup of oats
4
1
to make cookies. This is _
the amount
3
3
8
4
Simplify.
3
(pp. 574–578)
Divide. Write in simplest form.
59.
Divide 2 and 14
by their GFC, 2.
3
1
3
Write the numbers as
improper fractions.
7
14
=_
×_
FT
Dividing Fractions
2
1
2
7
14
× 4_
=_
×_
3_
2
_
Find 3 1 × 4 2 .
Example 8
2
11-9
Divide the numerator and
denominator by the GCF.
10 · 9
9
(pp. 568–571)
Multiply. Write in simplest form.
56.
2
3
3·4
4
_
×_
=_
10
9
10
1
SCHOOL Half of Mr. Carson’s class
play a sport. Of these, two-thirds are
male. What fraction of the class is
male and play a sport?
Multiplying Mixed Numbers
_ _
Find 3 × 4 .
Example 7
Multiply. Write in simplest form.
_ _
Find 3 ÷ 2 .
Example 9
8
_3 ÷ _2 = _3 × _3
3
8
2
8
3
Multiply by the reciprocal
2
of _.
3
9
=_
16
Multiply the numerators and
multiply the denominators.
called for in the recipe. How many
cups of oats are called for in the recipe?
11-10
Dividing Mixed Numbers
(pp. 579–582)
Divide. Write in simplest form.
63.
65.
3
4
2_
÷ 5_
5
5
64.
1
8 ÷ 2_
2
1
PIZZA Daniela has 1_
pizzas to
2
divide evenly among 6 friends. How
much of a pizza will each friend get?
_
2
5
1
11
11
÷ 1_
=_
÷_
5_
2
6
2
6
6
11
=_
×_
2
11
1
3
2
11
6
11
=_
×_
1
3
=_
or 3
1
586
_
Example 10 Find 5 1 ÷ 1 5 .
Chapter 11 Multiplying and Dividing Decimals and Fractions
6
Rewrite as improper
fractions.
Multiply by the
reciprocal.
Divide by the GCF.
1
Simplify.
CH
APTER
11
Practice Test
Multiply.
21.
1.
7.8 × 6
2.
0.92 × 4
3.
12 × 0.034
4.
4.56 × 9.7
5.
ALGEBRA Evaluate 4.5h + gk if g = 6.5,
h = 0.2, and k = 7.
6.
TEST PRACTICE Armando and his
3 friends ordered a 4-foot sub for $25.99,
4 large drinks for $1.79 each, and a salad
for $5.89. Which of the following represents
the total cost, not including tax?
A $134.68
B $39.04
7.
C $37.25
D $33.67
2
RECIPES A recipe calls for 2_
cups of flour.
3
If Tremayne triples the recipe, how many
cups of flour does he need?
Multiply. Write in simplest form.
22.
_1 × _4
9
4
3
2
_
24.
× 2_
8
3
26.
27.
PHYSICS The speed of light is 1.86 × 105
miles per second. Write this speed in
standard form.
3.24 × 103
9.
12.
0.45 ÷ 15
13.
36.08 ÷ 8.2
14.
10.79 ÷ 4.15
15.
16.
Estimate each product.
17.
_1 × 22
3
7
19. _ × 39
8
18.
2
1
3_
× 5_
3
9
4
1
20. 6_ × 8_
5
7
Chapter Test at tx.msmath1.com
8
3
2
2 ft
28.
TEST PRACTICE Pearl works at a kite
1
store. To make a kite tail, she needs 2_
feet
4
1
of fabric. If Pearl has 29_
feet of fabric,
ALGEBRA Evaluate a ÷ b if a = 8.62 and
b = 3.79. Round to the nearest tenth if
necessary.
ANIMALS The greyhound can run as fast as
39.35 miles per hour. Without calculating,
would about 12, 14, or 16 be a reasonable
answer for the number of miles a
greyhound could run at this rate in
0.4 hour? Explain your reasoning.
9
1 3 ft
Divide. Round to the nearest tenth if
necessary.
7.2 ÷ 3
25.
7
1
7_
× 5_
5
GEOMETRY To find the area of a
parallelogram, use the formula A = bh,
where b is the length of the base and
h is the height. Find the area of the
parallelogram.
0.175 × 105 10. 12.5 × 102
11.
_3 × _2
5
1
ALGEBRA Evaluate 2_
m if m = 4_
.
3
6
Write in simplest form.
Write in standard form.
8.
23.
4
how many kite tails can she make?
F
G
H
J
26
15
14
13
Divide. Write in simplest form.
29.
_1 ÷ _3
8
4
4
31. 6 ÷ 1_
5
33.
30.
_2 ÷ 4
32.
3
1
5_
÷ 1_
5
4
2
2
ALGEBRA Evaluate x ÷ y if x = 7_
3
4
and y = 1_
. Write in simplest form.
5
Chapter 11 Practice Test
587
CH
APTER
11
Texas
Test Practice
Cumulative, Chapters 1–11
Read each question. Then fill in the
correct answer on the answer document
provided by your teacher or on a sheet
of paper.
1.
3.
A
A garden in the shape of a regular polygon
is made of congruent equilateral triangles.
The table shows the relationship between
the area of the triangle and the area of the
garden it is part of. Which expression can
be used to find the area of a similar garden
made of triangles with an area of n square
units each?
Area of Triangle
(square units)
7
8
9
10
n
B
C
Area of Garden
(square units)
49
56
63
70
A 1n
Each figure below is divided into sections
of equal size. Which figure has 87.5% of its
total area shaded?
D
C 7n
If one of the angles of a trapezoid
measures 72°, what type of angle
is it?
D n + 42
F Obtuse
4.
B n+7
G Straight
2.
The drawing shows 2 circles that share a
common center point. Which expression
can be used to find the circumference of
the outer circle in inches?
4 in.
F π(5 + 4)
G 2(5 + 4)
H 2π(5 + 4)
1
J _
(5 + 4π)
2
588
H Right
J Acute
5.
GRIDDABLE Alicia made a circle for
an art project. The radius of the
circle was 5.2 centimeters. Find the
circumference of the circle to the nearest
hundredth. Use π = 3.14.
6.
What is the prime factorization of 2,100
using exponents?
5 in.
A 2 · 3 · 52 · 7
B 2 2 · 3 · 52 · 7
C 22 · 33 · 52 · 7
D 2 · 32 · 5 · 72
Chapter 11 Multiplying and Dividing Decimals and Fractions
Get Ready
for the Texas Test
For test-taking strategies and more practice,
see pages TX1–TX23.
7.
To solve a number puzzle, Sally needs to
find 2 numbers that have a difference of
3 and a sum of 33. She thinks numbers
are 21 and 18. Why is Sally’s answer
incorrect?
10.
F The difference between 21 and 18 is
not 3.
11.
Find the greatest common factor of 12, 32,
and 36.
F 2
H 6
G 4
J 8
Find the area of the rectangle.
G The difference between 21 and 18
is 3.
3.7 yd
H The sum of 21 and 18 is 33.
6.2 yd
J The sum of 21 and 18 is not 33.
8.
A student in Rob’s math class chose
a letter at random from the letter
cards shown below. What is the
probability that the letter chosen
was a vowel?
A 6.5 yd2
C 19.8 yd2
B 9.4 yd2
D 22.94 yd2
Question 11 If you are not permitted
to write in your test booklet, copy the
figure onto paper.
Pre-AP
Record your answers on a sheet of paper.
Show your work.
12.
1
A _
2
C _
3
4
B _
11
9.
9
1
D _
2
a. About how many of Ms. Ling’s
students signed up to go to the
museum? How many of Mr. Williams’?
GRIDDABLE Marta drew a triangle on the
sidewalk for a game. Find the perimeter of
the triangle in inches.
24 in.
Mr. Williams and Ms. Ling each teach
science to 50 students. Two-fifths of
Mr. Williams’ students signed up for a
field trip to the museum. About two-thirds
of Ms. Ling’s students signed up for the
same trip.
b. One fourth of the students who
attended the trip from Mr. Williams’
class bought souvenirs. What fraction
of his total students attended the trip
and bought souvenirs?
13 in.
29 in.
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Chapters 1–11 Texas Test Practice
589