Rational Numbers
5
•
Understand that different forms
of numbers are appropriate for
different situations.
•
Select and use appropriate
operations to solve problems
and justify solutions.
•
Uses statistical procedures
to describe data.
Key Vocabulary
common multiples (p. 257)
least common denominator (p. 258)
measures of central tendency
(p. 274)
reciprocals (p. 245)
Real-World Link
Hurricanes A hurricane can be measured by winds
greater than 74 miles per hour, a storm surge greater
than 4 feet, and barometric pressure less than
28.94 inches.
Applying
RationalMake
Numbers
Make this
Foldable
to helpinformation
you record about
information
rational
numbers.
Rational
Numbers
this Foldable
to help
you record
rationalabout
numbers.
Begin
1
_
”
×
11”
paper.
Begin
with
three
sheets
of
8
with three sheets of notebook 2paper.
1 Fold the first two sheets
in half from top to
bottom. Cut along the
fold from edges to
margin.
3 Insert the first two
sheets through the
third sheet and align
the folds.
226 Chapter 5 Rational Numbers
Getty Images
2 Fold the third sheet
in half from top to
bottom. Cut along the
fold from margin to
edge.
4 Label each page with
a lesson number and
title.
…>«ÌiÀÊx\
,>̈œ˜>
ՓLiÀÃ
GET READY for Chapter 5
Diagnose Readiness You have two options for checking Prerequisite Skills.
Option 2
Take the Online Readiness Quiz at pre-alg.com.
Option 1
Take the Quick Check below. Refer to the Quick Review for help.
Find each quotient. Round to the nearest
tenth, if necessary. (Lesson 2-5)
1. 3 ÷ 5
2. -1 ÷ 8
3. 2 ÷ 17
4. -12 ÷ 3
5. -2 ÷ (-9)
6. 4 ÷ (-15)
7. -24 ÷ 14
8. -72 ÷ (-9)
Example 1
Find 3 ÷ 8. Round to the nearest tenth.
3 ÷ 8 = 0.375
≈ 0.4
Find the quotient.
Round to the nearest tenth.
9. LANDSCAPING A hedge of roses is
8.25 meters long. Suppose bricks each
0.25 meter long will make a border along
one side. How many bricks are needed
to make the border? (Prerequisite Skills,
pp. 749–750)
Write each fraction in simplest form. If the
fraction is already in simplest form, write
simplified. (Lesson 4-4)
5
12
14
10. _
11. _
12. _
40
36
13. _
50
20
27
14. _
54
39
32
15. _
85
16. SURVEY Ten of the 25 students in math
class have blue eyes. What fraction of the
students in math class do not have blue
eyes? Write in simplest form. (Lesson 4-4)
Example 2
14
Write _
in simplest form.
35
Find the GCF of 14 and 35.
factors of 14: 1, 2, 7, 14
factors of 35: 1, 5, 7, 35
The GCF of 14 and 35 is 7.
14
14 ÷ 7
_
= _
35
35 ÷ 7
2
= _
Simplest form
5
Find each sum or difference. (Lesson 2-2)
17. 4 + (-9)
18. -10 + 16
19. (-3) + (-8)
20. -1 - (-10)
21. SCUBA DIVING A scuba diver descends
21 feet below the surface of the water. She
then ascends 14 feet. Find an integer that
represents the scuba diver’s position in
relation to the surface of the water.
Divide the numerator and the
denominator by the GCF.
Example 3
Find 14 - 18.
14 - 18 = 14 + (-18)
= -4
To subtract 18, add -18.
Simplify.
(Lesson 2-2)
Chapter 5 Get Ready for Chapter 5
227
5-1
Writing Fractions as Decimals
Main Ideas
• Write fractions as
terminating or
repeating decimals.
• Compare fractions
and decimals.
New Vocabulary
terminating decimal
mixed number
repeating decimal
bar notation
In the 18th century, a silver
dollar contained $1 worth of
silver. The sizes of all other
coins were based on this coin.
Coin
Fraction of
Silver of $1 Coin
quarter-dollar
(quarter)
1
4
10-cent
(dime)
1
10
half-dime*
(nickel)
1
20
a. A half dollar contained half
the silver of a silver dollar.
What was it worth?
b. Write the decimal value of
each coin in the table.
c. Order the fractions in the table
from least to greatest. (Hint: Use
the values of the coins.)
* In 1866, nickels were enlarged
for convenience
Write Fractions as Decimals Any fraction _a , where b ≠ 0, can be
b
written as a decimal by dividing the numerator by the denominator.
So, _a = a ÷ b. If the division ends, or terminates, when the remainder
b
is zero, the decimal is a terminating decimal.
EXAMPLE
Write a Fraction as a Terminating Decimal
3
Write _
as a decimal.
8
Vocabulary Link
Terminating
Everyday Use bringing to
an end
Math Use a decimal
whose digits end
Method 1 Use paper and pencil.
Method 2 Use a calculator.
0.375
8 3.000
-2.4
60
-56
40
-40
0
3 ⫼ 8 ENTER 0.375
_3 = 0.375
8
Division ends when
the remainder is 0.
0.375 is a terminating decimal.
Write each fraction as a decimal.
4
1A. _
5
3
1B. _
16
1
A mixed number such as 3_
is the sum of a whole number and a fraction.
2
Mixed numbers can also be written as decimals.
228 Chapter 5 Rational Numbers
EXAMPLE
Write a Mixed Number as a Decimal
1
Write 3_
as a decimal.
2
Mental Math
It will be helpful to
memorize the
following list of
fraction-decimal
equivalents.
1
2 = 0.5
1
3 = 0.3
1
4 = 0.25
1
5 = 0.2
2
3 = 0.6
3
4 = 0.75
2
5 = 0.4
3
5 = 0.6
4
5 = 0.8
1
1
=3+_
3_
2
2
Write as the sum of an integer and a fraction.
= 3 + 0.5 12 = 0.5
= 3.5
Add.
Write each mixed number as a decimal.
1
2A. 2_
3
2B. 4_
4
4
Not all fractions can be written as terminating decimals.
2→
3
CHECK 2 ⫼ 3
The number
6 repeats.
0.666
3 2.000
-1 8
20
-18
20
-18
2
The remainder after
each step is 2.
.6666666667 The last digit is rounded.
So, 2 = 0.6666666666… . This decimal is called a repeating decimal.
3
Repeating decimals have a pattern in their digits that repeats without end.
You can use bar notation to indicate that a digit or group of digits repeats.
0.6666666666… = 0.
6 The digit 6 repeats, so place a bar over the 6.
The table shows three examples of
repeating decimals.
EXAMPLE
Decimal
Bar Notation
0.13131313. . .
6.855555. . .
0.
13
6.85
19.1724724. . .
19.1
724
Write Fractions as Repeating Decimals
Write each fraction as a decimal. Use a bar to show a repeating decimal.
2
b. _
6
a. -_
11
0.5454. . .
6
-
→
11
11 6.0000. . .
6
So, -_
= -0.
54.
11
7
3A. _
9
Extra Examples at pre-alg.com
The digits 54
repeat.
15
2 →
15
0.1333. . . The digit 3
15 2.0000. . . repeats.
2
So, _
= 0.1
3.
15
5
3B. -_
6
Lesson 5-1 Writing Fractions as Decimals
229
GOLF During the 2005 Masters Tournament, Tiger Woods’ first shot
landed on the fairway 32 of 56 times. To the nearest thousandth, what
part of the time did his shot land on the fairway?
Divide the number of fairways on which he landed, 32, by the total number
of fairways, 56.
32
4
_
=_
≈ 0.571428… or 0.
571428
56
7
Look at the digit to the right of the thousandths place. Round down
since 4 < 5.
Tiger Woods landed on the fairway 0.571 of the time.
4. SOFTBALL The United States women’s softball team had 73 hits out of a
total of 213 at bats in the final round of the 2004 Olympics. To the nearest
thousandth, what part of the time did the team have a hit in the final
round?
Compare Fractions and Decimals It may be easier to compare numbers
when they are written as decimals.
Real-World Link
Tiger Woods became
the youngest golfer to
win The Masters golf
tournament at the age
of 21 years 3 months
and 14 days when he
won in 1997.
Source: masters.org
EXAMPLE
Compare Fractions and Decimals
Replace each
a. 35
with <, >, or = to make a true sentence.
0.75
3
5
0.75
Write the sentence.
0.6
0.75
Write 35 as a decimal.
0.6 < 0.75
In the tenths place, 6 < 7.
0.6
0.75
0.5 0.55 0.6 0.65 0.7 0.75 0.8
3
On a number line, 0.6 is to the left of 0.75, so _
< 0.75.
5
b. -_
8
5
6
-_
9
Write the fractions as decimals and then compare the decimals.
5
6
= -0.625 -_
= -0.6
-_
8
9
On a number line, -0.625 is to the right of -0.
6, so -0.625 is
5
6
greater than -0.
6. The inequality is -_
> -_
.
8
7
5A. _
8
230 Chapter 5 Rational Numbers
Mike Blake/REUTERS/Landov
0.87
9
7
5B. -_
15
5
-_
12
13
17
BREAKFAST In a survey of students, _
of the boys and _
of the girls
20
25
make their own breakfast. Of those surveyed, do a greater fraction of
boys or girls make their own breakfast?
Write the fractions as decimals and then compare the decimals.
13
boys: _
= 0.65
0.65
20
17
girls: _ = 0.68
25
0.68
0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70
13
17
On a number line, 0.65 is to the left of 0.68. Since 0.65 < 0.68, _
<_
.
20
25
So, a greater fraction of girls make their own breakfast.
6. FUND-RAISING Thirty out of thirty-six freshman and 34 out of 40
sophomores participated in the turkey trot. Which class had a greater
fraction participating in the turkey trot?
Personal Tutor at pre-alg.com
Examples 1–3
(pp. 228–229)
Write each fraction or mixed number as a decimal. Use a bar to show a
repeating decimal.
7
1. _
3
2. _
3
4. 1_
20
2
5. 2_
5
2
7. _
5
8. -_
8
3
Example 4
(p. 230)
Example 5
(p. 230)
16
3
6. 6_
4
_
9. 4 2
11
9
10. FOOTBALL In 2004, the San Francisco 49ers led the NFL in converting on
14 of 19 fourth downs. To the nearest thousandth, what part of the time did
the 49ers convert on fourth down?
Replace each
with <, >, or = to make a true sentence.
9
0.90
11. _
10
12. 0.3
_1
3
13. -_
3
4
7
-_
8
(p. 231)
14. WATER USAGE Seven-fiftieths of water usage comes from leaky plumbing
3
come from faucets. Does a greater fraction of water usage come
and _
20
from leaky plumbing or from faucets?
HELP
Write each fraction or mixed number as a decimal. Use a bar to show a
repeating decimal.
Example 6
HOMEWORK
5
3. _
5
For
See
Exercises Examples
15–26
1–3
27, 28
4
29–40
5
41, 42
6
1
15. _
5
3
19. 7_
10
1
23. _
9
7
16. _
20
1
20. 1_
2
2
_
24.
9
8
17. -_
25
1
21. -5_
8
5
_
25. 11
5
18. -_
8
3
22. -3_
4
4
26. -_
11
Lesson 5-1 Writing Fractions as Decimals
231
27. BUSINESS The customer service department resolved 106 of 120 customer
complaints in a one-hour time span. To the nearest thousandth, find the
resolve rate of the customer service department.
28. HOCKEY During the 2003–2004 season, Kevin Weekes of the Carolina
Hurricanes saved 1506 out of 1652 shots on goal. To the nearest thousandth,
what part of the time did Weekes save shots on goal?
with <, >, or = to make a true sentence.
1.01
WEATHER For Exercises 41 and 42,
use the graph at the right.
41. Did more or less than one-fourth
of an inch of rain fall on Tuesday?
Explain.
9
42. Suppose it rained _
inch on
10
Saturday. How does this compare
to the previous five days?
“œÕ˜ÌʜvÊ,>ˆ˜Ê­ˆ˜°®
ä°Îä
16
1
48. -5_
3
2
47. -2.2 -2_
7
Real-World Link
After being caught, a
marlin can strip more
than 300 feet of line
from a fishing reel in
less than 5 seconds.
Source: Incredible
Comparisons
ä°Ón
ä°Ó£
ä°Óä
ä°£È
ä°£x
ä°£x
ä°£Î
ä°£ä
ä°äx
25
3
-5_
10
>Þ
ˆ`
>Þ
ÕÀ
À
Ã`
Ã`
/…
8
-_
>Þ
Þ
`>
œ˜
5
46. -_
40. 6.18
ä
99
-2.09
12
4
1
6_
5
ä°Óx
1
45. -2_
1
-_
,>ˆ˜v>ÊÕÀˆ˜}Êx‡>ÞÊ*iÀˆœ`
Replace each with <, >, or = to
make a true sentence.
34
7
44. _
0.3
4
43. -0.75 -_
9
2
6
36. -_
25
˜i
1
38. 1_
20
0.
5
1
1
35. -_
-_
8
10
39. 34 3.
4
9
10
_
14
_1
32. 0.3
0.4
5
>Þ
_8
9
2
31. _
0.65
Ã`
7
33. _
8
1
37. _
5
8
5
34. _
7
7
i`
5
30. _
1
29. 0.3 _
4
/Õ
i
Replace each
>Þ
49. AUTOMOBILES Of all the cars sold in the United States in 2003, 2 were
5
imported from Japan and 0.26 were imported from Germany. Are more
Japanese or German cars sold in the United States? Explain.
Order each group of numbers from least to greatest.
7
7
, 0.8, _
50. _
8
2
3 , -_
51. -0.29, -
11
7
1
1
53. -1.
1, -1_, -1_
8
10
9
3
2
52. 2_
, 2.67, 2_
5
3
54. ANALYZE GRAPHS Find a fraction or
mixed number that might represent
each point on the graph at the right.
!"
ä°x
# $
£
%
£°x
Ó
ANIMALS For Exercises 55 and 56, use the following information.
5
mile in one minute.
A marlin can swim _
EXTRA PRACTICE
6
5
as a decimal rounded to the nearest hundredth.
55. Write _
6
See pages 770, 798.
56. Which form of the number is best to use? Explain.
Self-Check Quiz at
pre-alg.com
57. OPEN ENDED Give one example each of real-world situations where it is
most appropriate to give a response in fractional form and in decimal form.
232 Chapter 5 Rational Numbers
Burton McNeely/Getty Images
H.O.T. Problems
58. NUMBER SENSE Find a terminating and a repeating decimal between
1 and 8. Explain how you found them.
6
9
59. CHALLENGE Write the prime factorization of each denominator and the
decimal equivalent of each fraction. Then explain how prime factors of
denominators and the decimal equivalents of fractions are related.
1 _
1 _
_1 , _1 , _1 , _1 , _1 , _1 , _1 , _
, 1,_
, 1
2 3 4 5 6 8 9 10 12 15 20
60. SELECT A TECHNIQUE Luke is making lasagna that calls for 4 pound of
5
mozzarella cheese. The store only has packages that contain 0.75- and
0.85-pound of mozzarella cheese. Which of the following techniques might
Luke use to determine which package to buy? Justify your selection(s).
Then use the technique(s) to solve the problem.
mental math
61.
Writing in Math
number sense
estimation
Explain how 0.5 and 0.
5 are different. Which is
greater?
62. Which decimal represents
the shaded portion of the
figure?
A 0.6
C 0.63
6
B 0.
D 0.6
3
7
63. The fraction _
is found between which
9
pair of fractions on a number line?
3
3
F _
and _
5
4
7
4
G _ and _
5
10
3
7
H _
and _
J
10
4
3
2
_ and _
5
3
Write each number in scientific notation. (Lesson 4-7)
64. 854,000,000
65. 0.077
66. 0.00016
67. 925,000
Write each expression using a positive exponent. (Lesson 4-6)
68. 10-5
69. (-2)-7
70. x-4
71. y -3
72. ALGEBRA Write (a · a · a)(a · a) using an exponent. (Lesson 4-1)
73. TRANSPORTATION A car travels an average of 464 miles on one tank of gas.
If the tank holds 16 gallons, how many miles per gallon does it get? (Lesson 3-8)
74. SCUBA DIVING A scuba diver descends from the surface of the lake at a
rate of 6 meters per minute. Where will the diver be in relation to the lake’s
surface after 4 minutes? (Lesson 2-4)
PREREQUISITE SKILL Simplify each fraction. (Lesson 4-4)
4
75. 30
5
76. 65
77. 36
60
78. 12
18
79. 21
24
80. 16
28
81. 32
48
125
82. 1000
Lesson 5-1 Writing Fractions as Decimals
233
5-2
Rational Numbers
Main Ideas
• Write rational
numbers as fractions.
• Identify and classify
rational numbers.
New Vocabulary
rational number
Animation
pre-alg.com
The solution of 2x = 4 is 2. It is a
member of the set of natural numbers.
N = {1, 2, 3, …}
The solution of x + 3 = 3 is 0. It is a
member of the set of whole numbers.
W = {0, 1, 2, 3, …}
7HOLE
.UMBERS
.ATURAL.UMBERS
.ATURAL.UMBERS
The solution of x + 5 = 2 is -3. It is a member
of the set of integers.
I = { …, -3, -2, -1, 0, 1, 2, 3, …}
)NTEGERS
7HOLE
.UMBERS
The solution of 2x = 3 is 3, which is neither a
2
natural number, a whole number, nor an integer. It
is a member of the set of rational numbers.
.ATURAL.UMBERS
Rational numbers include fractions and decimals
as well as whole numbers and integers.
a. Is 7 a natural number? a whole number?
an integer?
,>̈œ˜>Ê ՓLiÀÃ
b. How do you know that 7 is also a
rational number?
)NTEGERS
7HOLE
.UMBERS
c. Is every whole number a rational
number? Is every rational number a
whole number? Give an explanation
or a counterexample to support your
answers.
.ATURAL.UMBERS
Reading Math
Write Rational Numbers as Fractions A number that can be written as
Ratios Rational comes
from the word ratio. A
ratio is the comparison
of two quantities by
division. Recall that
_a = a ÷ b, where
a fraction is called a rational number. Some examples of rational
numbers are shown below.
3
28
5
1
1
0.75 = _
-0.3 = – _
28 = _
1_
=_
b
b ≠ 0.
4
EXAMPLE
3
1
4
Write Mixed Numbers and Integers as Fractions
Write each rational number as a fraction.
2
a. 5 _
3
17
_
52 =_
3
3
3
1A. 4_
4
234 Chapter 5 Rational Numbers
4
b. -3
Write 5 23 as an
improper fraction.
-3
3
-3 = _
or - _
1
1B. 7
1
Terminating decimals are rational numbers because they can be written as a
fraction with a denominator of 10, 100, 1000, and so on.
EXAMPLE
Write Terminating Decimals as Fractions
a. 0.48
100
12
=_
25
0.48 is 48 hundredths.
Simplify. The GCF of
48 and 100 is 4.
0 4 8
b. 6.375
375
6.375 = 6_
1000
3
= 6_
8
6.375 is 6 and 375
thousandths.
Simplify. The GCF of
375 and 1000 is 125.
2A. 0.56
on
es
ten
ths
hu
nd
red
ths
tho
us
an
dth
ten
s
-th
ou
sa
nd
ths
48
0.48 = _
on
es
ten
ths
hu
nd
red
ths
tho
us
an
dth
ten
s
-th
ou
sa
nd
ths
Write each decimal as a fraction or mixed number in simplest form.
tho
us
an
ds
hu
nd
red
s
ten
s
Decimal Point Use the
word and to represent the
decimal point.
• Read 0.375 as three
hundred seventy-five
thousandths.
• Read 300.075 as three
hundred and seventy-five
thousandths.
tho
us
an
ds
hu
nd
red
s
ten
s
Reading Math
6 3
5
2B. 5.875
Any repeating decimal can be written as a fraction, so repeating decimals are
also rational numbers.
EXAMPLE
Repeating
Decimals
When two digits
repeat, multiply each
side by 100. Then
subtract N from 100N
to eliminate the
repeating part.
Write Repeating Decimals as Fractions
Write 0.
8 as a fraction in simplest form.
N = 0.888…
Let N represent the number.
10N = 10(0.888…) Multiply each side by 10
10N = 8.888…
because one digit repeats.
Subtract N from 10N to eliminate the repeating part, 0.888… .
10N = 8.888…
-(N = 0.888…)
9N = 8
10N - N = 10N - 1N or 9N
9N
8
_
=_
Divide each side by 9.
9
9
8
N=_
9
CHECK
Simplify.
8 ⫼ 9 ENTER .8888888889 3. Write 0.
3 as a fraction in simplest form.
Personal Tutor at pre-alg.com
Lesson 5-2 Rational Numbers
235
READING
in the Content Area
For strategies in reading
this Lesson, visit
pre-alg.com.
Identify And Classify Rational Numbers All rational numbers can be written
as terminating or repeating decimals. Decimals that are neither terminating
nor repeating, such as the numbers below, are called irrational. You will learn more
about irrational numbers in Chapter 9.
= 3.141592654…
→ The digits does not repeat.
4.232232223…
→ The same block of digits does not repeat.
Rational Numbers
A rational number is any
number that can be expressed
a
as the quotient _ of two
b
integers, a and b, where b ≠ 0.
Words
Model
Rational
Numbers
1.8
0.7
EXAMPLE
2
45
Integers
⫺12
⫺5 Whole
2
Numbers
3
6 0 15
2
⫺3.2222...
Classify Numbers
Identify all sets to which each number belongs.
a. -6
-6 is an integer and a rational number.
4
b. 2 _
5
4
14
Because 2_
=_
, it is rational. It is neither a whole number nor an integer.
5
5
c. 0.914114111…
This is a nonterminating, nonrepeating decimal. So, it is not rational.
6
4A. -_
9
Example 1
(p. 234)
4B. 1.414213562…
4C. 0
Write each number as a fraction.
1
1. -2 _
3
5
2. 1 _
6
3. 10
(p. 235)
4. MEASUREMENT A micron is a unit of measure that is approximately
0.000039 inch. Express this as a fraction.
Examples 2, 3
Write each decimal as a fraction or mixed number in simplest form.
Example 2
(p. 235)
5. 0.8
7
8. -0.
Example 4
(p. 236)
6. 6.35
9. 0.45
7. 3.16
10. 0.06
Identify all sets to which each number belongs.
11. -5
236 Chapter 5 Rational Numbers
12. 6.05
1
13. 0.
Extra Examples at pre-alg.com
HOMEWORK
HELP
For
See
Exercises Examples
14–17
1
18–25
2
26–31
3
32–39
4
Write each number as a fraction.
2
14. 4_
4
15. -1_
3
16. -21
7
17. 60
Write each decimal as a fraction or mixed number in simplest form.
18. 0.4
19. 0.09
20. 5.22
21. 1.68
22. 3.625
23. 8.004
24. WHITE HOUSE The White House covers an area of 0.028 square mile. What
fraction of a square mile is this?
25. RECYCLING Use the information at the left to find the fraction of all recycled
newspapers that were used to make tissues in 2004.
Write each decimal as a fraction or mixed number in simplest form.
26. 0.2
27. -0.333…
28. 4.
5
6
29. 5.
30. 0.
32
31. 2.
05
Identify all sets to which each number belongs.
32. 4
33. -7
5
34. -2_
6
35. _
36. 15.8
37. 9.0202020…
38. 1.7345…
39. 30.151151115…
8
3
40. TRACK AND FIELD During the women’s 100-meter final in the 2004 Olympics,
the eight finalists finished within twenty-five hundredths of a second of
each other. Write this number as a fraction in simplest form.
Real-World Link
The portions of recycled
newspapers used for
other purposes are
shown below.
Newsprint: 0.31
Exported for
Recycling: 0.28
Paperboard: 0.13
Tissues: 0.08
Other products: 0.18
Source: American Forest and
Paper Association, Newspaper
Association of America
41. ANALYZE TABLES The city of Heath makes
1
up _
of the population in Rockwall County.
10
Use the table to find the fraction of Rockwall
County’s population that lives in other cities.
Write each fraction in simplest form.
Replace each
42. -0.23
PRACTICE
Decimal Part of
Rockwall County’s
Population
Fate
0.018
McLendonChisholm
Rockwall
Royse City
0.02
0.42
0.07
with <, >, or = to make a true statement.
-0.3
1
-0.9
45. -1_
11
EXTRA
City
43. 8
9
0.888…
5
4_
46. 4.63
8
44. 0.714
_5
47. -5.
3
5.333…
7
3
48. MACHINERY Will a steel peg 2.37 inches in diameter fit in a 2 _
-inch
8
diameter hole? How do you know?
See pages 770, 798.
Self-Check Quiz at
pre-alg.com
H.O.T. Problems
49. GEOMETRY Pi () to six decimal places has a value of 3.141592. Pi is often
estimated as 22
. Is the estimate for greater than or less than the actual
7
value of ? Explain.
50. OPEN ENDED Give an example of a number that is not rational. Explain
why it is not rational.
Lesson 5-2 Rational Numbers
Lester Lefkowitz/CORBIS
237
51. CHALLENGE Show that 0.999… = 1.
REASONING Determine whether each statement is sometimes, always, or
never true. Explain by giving an example or a counterexample.
52. An integer is a rational number.
53. A rational number is an integer.
54. A whole number is not a rational number.
55.
Writing in Math
Explain how rational numbers are related to other
sets of numbers. Illustrate your reasoning with examples of numbers
that belong to more than one set and examples of numbers that are
only rational.
56. There are infinitely many
between S and T on the number line.
S
57. Which fraction is between 0.12
and 0.15?
3
F _
T
25
1
G_
8
⫺2⫺1 0 1 2 3 4 5 6 7 8
A rational numbers
3
H_
20
1
J _
5
B integers
C whole numbers
D negatives
Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. (Lesson 5-1)
2
58. _
4
59. -7 _
5
5
13
60. -_
5
61. 2_
64. 7.4 × 10-4
65. 1.681 × 10-2
9
20
Write each number in standard form. (Lesson 4-7)
62. 2 × 103
63. 3.05 × 106
2
12n
66. ALGEBRA Write _
in simplest form. (Lesson 4-4)
3an
Find the perimeter and area of each rectangle. (Lesson 3-8)
67.
68.
{°xÊV“
7 in.
™ÊV“
16 in.
69. COIN COLLECTING Jada has 156 coins in her collection. This is 12 more than
8 times the number of nickels in the collection. How many nickels does
Jada have in her collection? (Lesson 3-6)
PREREQUISITE SKILL Estimate each product. (page 752)
2
1
70. 1_
· 4_
3
8
1
4
71. -5_
· 3_
238 Chapter 5 Rational Numbers
3
5
1
1
72. 2_
· 2_
4
9
9
7
73. 6_
· 1_
8
10
5-3
Multiplying Rational Numbers
Main Ideas
• Multiply positive and
negative fractions.
• Use dimensional
analysis to solve
problems.
New Vocabulary
To find 2 · 3, use an area model
3
4
3 4
2
to find of 3.
3
4
2
3
a. The overlapping green area
2
represents the product of _
3
3
and _. What is the product?
4
dimensional analysis
Draw a rectangle
and shade three
fourths of it yellow.
Then shade two
thirds of the
rectangle blue.
Use an area model or another manipulative to model each product.
Explain how the model shows the product.
3 1
c. _ · _
1 _
·1
b. _
2
5
3
3 1
d. _ · _
4
4
3
e. What is the relationship between the numerators and denominators
of the factors and the numerator and denominator of the product?
Multiply Fractions This model suggests a rule for multiplying fractions.
Multiplying Fractions
To multiply fractions, multiply the numerators and multiply the
denominators.
Words
Symbols
a _
a·c
_
·c=_
, where b, d ≠ 0
b
d
b·d
1 2
2
1·2
Example _ · _ = _
or _
3 5
15
3·5
EXAMPLE
Multiply Fractions
2 _
Find _
· 3 . Write the product in simplest form.
3
Review
Vocabulary
GCF (greatest common
factor) the greatest
number that is a factor of
two or more numbers;
Example: for 12 and 20, the
GCF is 4. (Lesson 4-4)
4
←Multiply the numerators.
←Multiply the denominators.
3·4
6
1
= _ or _ Simplify. The GCF of 6 and 12 is 6.
2
12
2·3
_2 · _3 = _
3
4
Find each product. Write in simplest form.
1 _
1A. _
· 4
2
10
5 _
1B. _
· 6
12
10
Lesson 5-3 Multiplying Rational Numbers
239
If the fractions have common factors in the numerators and denominators,
you can simplify before you multiply.
EXAMPLE
Multiply Negative Fractions
5 _
Find -_
· 3 . Write the product in simplest form.
Negative
Fractions
12 8
1
5 _
5 _
-_
· 3 = -_
·3
12 8
12 8
Divide 3 and 12 by their GCF, 3.
4
5 can be written as
-
12
-5 · 1
=_
Multiply the numerators and multiply the denominators.
4·8
5
= -_
32
-5
5
or as .
12
-12
Simplify.
Find each product. Write in simplest form.
3 _
6
3
2A. _
· 9
2B. _
· -_
4
9
12
EXAMPLE
11
Multiply Mixed Numbers
Find 125 · 21. Write the product in simplest form. Estimate 1 · 3 = 3
2
12
1
7 _
_
· 2_
=_
·5
5
2
5
Rename 125 as 75 and rename 212 as 52 .
2
1
7 _
=_
·5
Estimation
5
You can justify your
answer by using
estimation.
7·1
=_
1·2
7
1
= _ or 3_
2
2
• 125 is close to 1.
• 212 is close to 3.
So, 125 · 212 ≈ 1 · 3
or 3.
Divide by the GCF, 5.
2
1
Multiply.
Simplify.
Find each product. Write in simplest form.
3
5
1
1
3A. 3_
· 2_
3B. -1_
· 4_
8
3
6
8
ROLLER COASTERS The first drop on one roller coaster at a theme park
is 255 feet. The first drop on another roller coaster at the park is about
11
_
as high. Find the height of the drop on the second roller coaster.
20
11 .
To find the height of the drop on the second roller coaster, multiply 255 by _
20
255 _
11
255
=_
· 11
Rename 255 as _ .
255 · _
20
1
1
20
51
255 _
=_
· 11
1
20
Divide by the GCF, 5.
4
51 · 11
=_
1·4
561
1
_
=
or 140_
4
4
240 Chapter 5 Rational Numbers
Multiply.
Simplify. The height of the drop is about 140 feet.
Extra Examples at pre-alg.com
4. SKYSCRAPERS The Sears Tower in Chicago is about 1450 feet. The Empire
State Building in New York City is about 4 as tall. About how tall is the
5
Empire State Building?
Algebraic fractions are multiplied in the same manner as numeric fractions.
EXAMPLE
Multiply Algebraic Fractions
b2
2a _
Find _
· . Write the product in simplest form.
b
d
b2
1
2a _ _
b·b
_
· = 2a · _
b
d
b
1
d
2ab
=_
The GCF of b and b2 is b.
Simplify.
d
Find each product. Write in simplest form.
x2 _
5A. _
· z
3y
2
7 _
5B. _
· rs
2
2x
r
10
Dimensional Analysis Dimensional analysis is the process of including units
of measurement as factors when you compute. You can use dimensional
analysis to check whether your answers are reasonable.
SPACE TRAVEL The landing speed of the space shuttle is about 216 miles
per hour. How far does the shuttle travel in 13 hour during landing?
Words
Distance equals the rate multiplied by the time.
Variable
Let d represent the distance.
Equation
1 hour
d = 216 miles per hour · _
3
216 miles _
· 1 hour distance = rate · time
d=_
1 hour
3
72
216 miles _
=_
· 1 hour
1 hour
3
Divide out the common factors and units.
1
Simplify. The space shuttle travels 72 miles in _ hour
3
during landing.
1
Multiplying by _
is the same as dividing by 3.
3
1
216 · = 216 ÷ 3
3
1
= 72 miles
CHECK
= 72
6. SPEED RECORD The record for the fastest land car speed is about 760 miles
1
per hour. How far would the car travel in _
hour?
Personal Tutor at pre-alg.com
4
Lesson 5-3 Multiplying Rational Numbers
241
Examples 1– 3
(pp. 239–240)
Find each product. Use an area model if necessary.
1 _
1. _
·3
4 5
1
_
4. 3 · _
7 6
1 _
7. _
-5
2
6
1
2
10. 3 _
· -_
4
11
Example 4
(pp. 240–241)
Example 5
(p. 241)
3 5
2
_
5. 5 · _
10 9
2 _
8. -_
-1
3
6
3
1
11. -5 _
· -3 _
3
8
3 _
3. _
·1
8
4
4 _
6. _
·5
5 8
6 _
9. -_
·1
10 8
1
2
12. -2 _
· 5_
2
3
13. GEOGRAPHY “Midway” is the name of 252 towns in the United States.
“Pleasant Hill” occurs 5 as many times. How many towns named “Pleasant
9
Hill” are there in the United States?
ALGEBRA Find each product. Write in simplest form.
2 _
14. _
· 3x
x
7
5b
15. _a · _
b
4t _
16. _
· 18r
2
c
t
9r
(p. 241)
1
17. TRAVEL A car travels 65 miles per hour for 3_
hours. What is the distance
2
traveled? Use the formula d = rt to solve the problem and show how you
can divide by the common units.
HELP
Find each product. Use an area model if necessary.
Example 6
HOMEWORK
1 _
2. _
·2
For
See
Exercises Examples
18–27
1–2
28–35
3
36, 37
4
38–43
5
44, 45
6
6 _
18. -_
·2
7 7
3 _
21. -_
·3
4 5
2 _
24. _
·5
5 6
4 _
19. _
·2
1
1
20. _
-_
9 3
5 _
22. _
· 8
9 25
8 _
25. _
· 27
9 28
5
8
1
2
23. -_
-_
2
7
3
1
26. _
-_
4
3
27. -7 · 2
8 5
7
28. 2 · 6
29. (-3)
15
2 _
·1
30. 6 _
5
1
31. -_
· 3_
1
2
32. 2_
· 6_
3 2
5
1
33. 3 _
· 2_
3
8
12
9
12
2
1
34. -6_
-1_
3
2
3
7
3
4
35. 1_
-9 _
7
5
36. BREAD The average person living in Slovakia consumes about 320 pounds
of bread per year. The average person living in the United States consumes
1
about _
as much. How many pounds of bread does the average American
5
consume every year?
37. BRIDGES The Golden Gate Bridge in San Francisco is 4200 feet long.
19
The Brooklyn Bridge in New York City is _
as long. How long is the
50
Brooklyn Bridge?
ALGEBRA Find each product. Write in simplest form.
4a _
38. _
·3
5 a
2
8 _
· c
41. _
c 11
242 Chapter 5 Rational Numbers
3x _
9y
39. _
y · x
n _
42. _
· 64
18
n
12 _
40. _
· 3k
4
jk
2
x _
43. _
· 2z
2z
3
44. HYBRID CARS Hybrid cars can get up to 52 miles per gallon of gas. How far
3
can the car travel on _
gallon of gas?
4
2
1
45. LAWN CARE Dexter’s lawn is _
of an acre. If 7_
bags of fertilizer are needed
3
2
for 1 acre, how much will he need to fertilize his lawn?
Real-World Link
An improvement of
5 miles per gallon in
fuel economy saves
55 million metric tons
of carbon emissions
per day.
Source: hybridcars.com
ANALYZE TABLES For Exercises 46–48,
use the table that shows statistics
from the last election for class
&RACTIONOFSTUDENTBODYTHATVOTED ?
president.
?
&RACTIONOFVOTESFOR(ECTOR
46. What fraction of the student
?
body voted for Hector?
&RACTIONOFVOTESFOR.ORA
47. What fraction of the student
body voted for Nora?
48. Was there another candidate for class president? How do you know?
Explain your reasoning. If there was another candidate, what fraction of
the student body voted for this person?
49. FILMS The table shows the number of sports films
created with different themes. Which theme occurs
5
as many times as boxing?
12
Sport Theme
Boxing
Horse Racing
Football
Baseball
3
4
50. ALGEBRA Evaluate (xy)2 if x = _
and y = -_
.
4
5
Films
204
139
123
85
Source: Top 10 of Everything
MEASUREMENT Complete.
5
mile
51. ? feet = _
3
52. ? ounces = _
pound
6
8
(Hint: 1 mile = 5280 feet)
(Hint: 1 pound = 16 ounces)
2
hour = ? minutes
53. _
3
54. _
yard = ? inches
3
4
CONVERTING MEASURES Use dimensional analysis
and the fractions in the table to find each missing
measure.
EXTRA
PRACTICE
See pages 771, 798.
Self-Check Quiz at
pre-alg.com
55. 5 in. = _____
? cm
56. 10 km = _____
? mi
? in.
57. 26.3 cm = _____
2 2
58. 8_
ft = _____
? m2
Customary→
Metric
3
3
60. _ cm = _____
? in.
4
59. 72 m2 = _____
? ft2
Metric→
Customary
2.54 cm
1 in.
1.61 km
1 mi
0.39 in.
1 cm
0.62 mi
1 km
0.09 m2
1 ft2
10.76 ft2
1 m2
62. 130.5 km = _____
? mi
? km
61. 130.5 mi = _____
H.O.T. Problems
Conversion Factors
63. OPEN ENDED Choose two rational numbers whose product is a number
between 0 and 1.
5 _
64. FIND THE ERROR Terrence and Marie are finding _
· 18 . Who is correct?
24 25
Explain your reasoning.
Terrence
1
5
3
18
Marie
3
·
25 = 24
20
4
5
1
9
5
18
4
5
9
· =
24 25
20
Lesson 5-3 Multiplying Rational Numbers
Ford Motor Company
243
CHALLENGE Use the digits 3, 4, 5, 6, 8, or 9 no more than once to make true
sentences.
6
□ _
× □ =_
65. _
□
67.
□
5
□ _
66. _
× □ =_
□
5
□
8
Writing in Math
Use the information about fractions on page 239
to explain how multiplying fractions is related to areas of rectangles.
Illustrate your reasoning with an area model.
68. What is the equivalent length of a
chain that is 52 feet long?
8
3
and _
is a
69. The product of _
8
15
number
.
A 4 yd 5 ft
F between 0 and 1
B 4.5 yd
G between 1 and 2
C 17 yd 1 ft
H between 2 and 3
D 17.1 yd
J greater than 3
Write each decimal as a fraction or mixed number in simplest form. (Lesson 5-2)
70. 0.18
71. -0.2
73. 0.
7
72. 3.04
1
74. FOOD In an online survey, about _
of teenagers go to sleep between 9 and
4
13
_
10 P.M., while
of teenagers go to sleep at 12 A.M. or later. Which group is
50
larger? (Lesson 5-1)
Express each situation with a number in scientific notation. (Lesson 4-7)
75. The number of possible ways that a player can play the first four moves in
a chess game is 3 billion.
76. A particle of dust floating in the air weighs 0.000000753 gram.
77. ALGEBRA What is the product of x2 and x4? (Lesson 4-5)
GEOMETRY Find the perimeter and area of each rectangle. (Lesson 3-8)
78.
79.
5 in.
3.5 m
12 in.
4.9 m
ALGEBRA Solve each equation. Check your solution. (Lesson 3-5)
80. 2x - 1 = 9
81. 14 = 8 + 3n
k
82. 7 + _
= -1
5
83. ALGEBRA Simplify 4(y + 2) - y. (Lesson 3-2)
PREREQUISITE SKILL Find the GCF of each pair of monomials. (Lesson 4-3)
84. 8n, 16n
85. 5ab, 8b
244 Chapter 5 Rational Numbers
86. 9k, 27
87. 4p2, 6p
5-4
Main Ideas
Dividing Rational Numbers
1
1
The model shows 4 ÷ _
. Each of the 4 circles is divided into _
-sections.
3
• Divide positive and
negative fractions
using multiplicative
inverses.
• Use dimensional
analysis to solve
problems.
New Vocabulary
multiplicative inverses
reciprocals
3
1
2
4
3
5
7
6
8
10
9
11
12
1
1 = 12. Another way to find the
There are twelve _
-sections, so 4 ÷ _
3
3
number of sections is to multiply 4 × 3 = 12.
Use a circle model or another manipulative to model each quotient.
Explain how the model shows the quotient.
1
a. 2 ÷ _
1
b. 4 ÷ _
3
1
c. 3 ÷ _
2
4
d. MAKE A CONJECTURE Write about how dividing by a fraction is
related to multiplying.
Reading Math
Synonyms Multiplicative
inverse and reciprocal are
different terms for the same
concept. They may be used
interchangeably.
Divide Fractions Rational numbers have all of the properties of integers.
3
1 _
Another property is shown by _
· = 1. Two numbers whose product
3
1
is 1 are called multiplicative inverses or reciprocals.
Inverse Property of Multiplication
Words
The product of a number and its multiplicative inverse is 1.
Symbols
a
For every number _ , where a, b ≠ 0, there is exactly one
b
a _
b
b
_
number _
a such that b · a = 1.
EXAMPLE
Find Multiplicative Inverses
Find multiplicative inverse of each number.
1
b. 2 _
3
a. - _
8
8
_
- 3 -_
=1
8
3
( )
5
The product is 1.
The multiplicative inverse
3
8
or reciprocal of -_
is -_
.
8
3
7
1A. -_
9
1 _
2_
= 11
5
5
5
11 _
_
· =1
5 11
Write as an improper fraction.
The product is 1.
1 _
The reciprocal of 2_
is 5 .
5
11
1
1B. 6 _
3
Lesson 5-4 Dividing Rational Numbers
245
1
Dividing by 2 is the same as multiplying by _
,
2
its multiplicative inverse. This is true for any
rational number.
reciprocals
1
6·_
=3
6÷2=3
2
same result
Dividing Fractions
Words
To divide by a fraction, multiply by its multiplicative inverse.
Symbols
a
a d
c
_
÷_
=_·_
, where b, c, d ≠ 0
Example
1
7
5
1 _
_
÷_
=_
· 7 or _
d
b
4
b
7
EXAMPLE
4
c
5
20
Divide by a Fraction or Whole Number
Find each quotient. Write in simplest form.
Dividing By a
Whole Number
When dividing by a
whole number, always
rename it as an
improper fraction first.
Then multiply by its
reciprocal.
5
1 _
a. _
÷
5
b. _
÷6
3
9
_1 ÷ _5 = _1 · _9
3
9
3 5
3
1 _
=_
·9
3 5
8
_5 ÷ 6 = _5 ÷ _6
Multiply by the
5 9
reciprocal of _, _.
6
Write 6 as _.
1
8
1
5 _
1 Multiply by the
_
= ·
6 1
8 6 reciprocal of _1 , _6 .
5
=_
Multiply.
48
8
9 5
Divide by the GCF, 3.
1
3
=_
Simplify.
5
3
5
3
2B. _ ÷ -_
1 _
7
2A. _
÷
8
15
3
2C. _
÷ 11
6
2D. -_
÷ 12
4
4
7
To divide by a mixed number, you can rewrite the divisor as an
improper fraction.
EXAMPLE
Divide by a Mixed Number
1
1
Find -7_
÷ 2_. Write the quotient in simplest form.
2
10
1
1
15 _
21
-7_ ÷ 2_ = -_
÷
2
2
10
10
15
10
= -_ · _
2 21
5
5
2
21
Rename the mixed numbers as improper fractions.
Multiply by the multiplicative inverse of _, _.
21 10
10 21
15 _
= -_
· 10
1
Divide out common factors.
7
25
4
= -_
or -3_
7
7
Simplify.
Find each quotient. Write in simplest form.
3
1
÷ -4_
3A. 6_
8
246 Chapter 5 Rational Numbers
4
4
2
3B. -6_
÷ -2_
5
5
Extra Examples at pre-alg.com
You can divide algebraic fractions in the same way that you divide
numerical fractions.
EXAMPLE
Divide by an Algebraic Fraction
3xy
2x
Find _ ÷ _
. Write the quotient in simplest form.
8
4
3xy
3xy 8
2x
8
2x _
_
÷ _ = _ · _ Multiply by the multiplicative inverse of _
, .
8 2x
4
8
2x
4
1
3xy
4
2
8
=_·_
1
2x
Divide out common factors.
1
6y
= _ or 3y
2
Simplify.
Find each quotient. Write in simplest form.
mn2
m2n
4B. _ ÷ _
5ab _
10b
÷
4A. _
6
4
7
8
Dimensional Analysis Dimensional analysis is a useful way to examine the
solution of division problems.
Real-World Link
Organized cheerleading
is over a hundred years
old. In November of
1898, John Campbell
led the crowd at the
University of Minnesota
football game in the
first-ever organized
cheer.
Source: Official Cheerleader
Handbook
CHEERLEADING How many cheerleading uniforms can be made with
3
7
74 _
yards of fabric if each uniform requires 2_
yards?
8
4
3
7
by 2_
. Think: How many 2 _7 s are in 74_3 ?
To find how many uniforms, divide 74_
3
299
23
7
74_
÷ 2_ = _ ÷ _
8
4
8
4
299 _
8
_
=
·
4
23
13
2
4
23
299 _
=_
· 8
1
= 26
8
8
4
4
Write 74_ and 2_ as improper fractions.
3
4
7
8
Multiply by the reciprocal of _, _.
23 8
8 23
Divide out common factors.
1
Simplify.
So, 26 uniforms can be made.
CHECK
Use dimensional analysis to examine the units.
yards
uniforms
uniforms
yards
yards ÷ _ = yards · _ Divide out the units.
= uniforms
Simplify.
The result is expressed as uniforms.
3
2
5. BREAKFAST A box of cereal contains 15_
ounces. If a bowl holds 2_
ounces
5
5
of cereal, how many bowls of cereal are in one box?
Personal Tutor at pre-alg.com
Lesson 5-4 Dividing Rational Numbers
Tony Anderson/Getty Images
247
Example 1
(p. 245)
Find the multiplicative inverse of each number.
4
1. _
Examples 2– 3
(p. 246)
(p. 247)
Example 5
(p. 247)
HOMEWORK
HELP
For
See
Exercises Examples
17–22
1
23–36
2
37–40
3
41–46
4
47, 48
5
8
Find each quotient. Use an area model if necessary.
5
2
5. -_
÷ -_ 6
1 _
÷
4. _
2
7
7
7. _ ÷ (-14)
9
8
1
10. -_
÷ 3_
5
9
Example 4
1
3. 3 _
2. -16
5
3
6
4
8. _
÷ (-2)
5
1
1
11. 2_
÷ -1_
6
5
8
4 _
6. -_
÷
9
5
1
9. 7_ ÷ 5
3
2
1
12. -5 _
÷ 2_
7
7
ALGEBRA Find each quotient. Write in simplest form.
14 _
1
13. _
n ÷n
x2 _
ax
15. _
÷
ab _
b
14. _
÷
6
4
5
2
16. CARPENTRY How many boards, each 2 feet 8 inches long, can be cut from
a board 16 feet long if there is no waste?
Find the multiplicative inverse of each number.
6
17. _
1
18. -_
19. -7
20. 24
1
21. 5_
2
22. -3 _
11
5
9
4
Find each quotient. Use an area model if necessary.
3
1 _
÷
23. _
2 _
1
24. _
÷
5
1 _
25. -_
÷
6
4
26. _
÷ -_
8 _
4
27. _
÷
7 _
14
28. _
÷
3 _
3
29. _
÷
2
2
30. _
÷ -_
3
5
31. -_
÷ -_
3
1
32. -_
÷ -_
4
33. 12 ÷ _
4
34. -8 ÷ _
5
35. -_
÷ (-4)
2
36. 6_
÷5
3
5
37. 3_
÷ 1_
5
4
( 5)
11
4
9
9
1
1
38. 7_
÷ -1_
(
( 9)
5
)
9
5
6
10
1
2
39. -6_
÷ 3_
9
15
5
3
8
2
8
9
5
6
2
3
9
4
10
4
3
3
2
40. -10_
÷ -2_
(
5
5
)
ALGEBRA Find each quotient. Write in simplest form.
a
a
÷_
41. _
7
EXTRA
PRACTICE
See pages 771, 798.
Self-Check Quiz at
pre-alg.com
42
5s _
6rs
÷
44. _
t
t
10 _
5
42. _
÷
3x
cd
43. _c ÷ _
2x
8
k3
2s _
st3
46. _
÷
2
k
45. _ ÷ _
9
5
24
t
8
3
1
-pound hamburgers can be made from 2_
pounds of
47. FOOD How many _
4
4
ground beef?
1
48. COOKING A batch of cookies requires 1_
cups of sugar. How many batches
2
1
cups of sugar?
of cookies can be made from 7_
2
248 Chapter 5 Rational Numbers
ALGEBRA Evaluate each expression.
8
7
and n = _
49. m ÷ n if m = -_
3
1
50. r2 ÷ s2 if r = -_
and s = 1_
18
9
3
4
1
51. BABY-SITTING Barbara baby-sat for 3_
hours and earned $19.50. What was
4
her hourly rate?
H.O.T. Problems
52. OPEN ENDED Write a division expression that can be simplified by using
7
the multiplicative inverse of _
.
5
3
1 _
1
1
53. CHALLENGE Divide _
by _
, 1, _
, and _
. What happens to the quotient as
4
2 4 8
12
the value of the divisor decreases? Make a conjecture about the quotient
3
by fractions that increase in value. Test your conjecture.
when you divide _
4
54.
Writing in Math
Explain how dividing by a fraction is related to
multiplying. Illustrate your reasoning by including a model of a whole
number divided by a fraction.
1
A promotional poster is printed on 16-inch by 24 _
-inch
2
posterboard and the space between the three sections and
both top and bottom of the poster are shown.
55. If the total length of the three sections is 18 1 inches,
2
how long are each of the three equal sections?
1
A 2_
in.
3
5
C 4_
in.
8
4
in.
B 3_
1
D 6_
in.
15
IN
IN
IN
IN
6
1
inches, what is the
56. GRIDDABLE If the width of one section is 14 _
4
area of one section? Round your answer to the nearest hundredth.
Find each product. Write in simplest form. (Lesson 5-3)
3 _
57. _
·1
5
3
15
2
58. -_
· -_
9
16
4 _
59. -2_
·3
5
8
5
1
60. -_
· 1_
12
7
Identify all sets to which each number belongs. (Lesson 5-2)
61. 16
62. -2.8888 …
63. 0.
8
64. 5.121221222 …
65. COMPUTERS In a survey, 17 students out of 20 said they use a computer as a
reference source for school. Write 17 out of 20 as a decimal. (Lesson 5-1)
PREREQUISITE SKILL Write each improper fraction as a mixed number in
simplest form. (Lesson 4-4)
9
66. _
4
8
67. _
7
17
68. _
2
24
69. _
5
Lesson 5-4 Dividing Rational Numbers
249
5-5
Adding and Subtracting
Like Fractions
Main Ideas
• Add like fractions.
• Subtract like fractions.
Measures of different parts of an insect
are shown. The sum of the parts is
_6 inch. Use a ruler to find each
8
measure.
3
1
a. _
in. + _
in.
5
in.
8
3
4
b. _
in. + _
in.
8
8
4
4
_
_
c. in. + in.
8
8
1
in.
8
6
in.
8
8
8
6
_
_
d. in. - 3 in.
8
8
Add Like Fractions Fractions with the same denominator are called
like fractions.
Adding Like Fractions
Words
To add fractions with like denominators, add the numerators and
write the sum over the denominator.
a+b
a
b
_
_
Symbols _
c + c = c , where c ≠ 0
EXAMPLE
1+2
3
1
2
Example _ + _ = _ or _
5
5
5
5
Add Fractions
Find each sum. Write in simplest form.
3
5
a. _
+_
7
7
3+5
3
_ + _5 = _
7
7
7
8
1
= _ or 1_
7
7
Estimate 0 + 1 = 1
The denominators are the same. Add the numerators.
Simplify and rename as a mixed number.
Compared to the estimate, the answer is reasonable.
5
-7
b. _
+ _
( )
( )
8
8
5 + (-7)
5
-7
_+ _ =_
8
8
8
-2
1
= _ or -_
8
4
1
1
Estimate _ + (-1) = -_
2
2
The denominators are the same. Add the numerators.
Simplify.
Compare your answer to the estimate. Is it reasonable?
5
4
1A. _
+_
6
250 Chapter 5 Rational Numbers
John Cancalosi/Stock Boston
6
6
4
1B. _
+ -_
7
( 7)
Extra Examples at pre-alg.com
EXAMPLE
Add Mixed Numbers
5
1
Find 6 _
+ 1_
. Write the sum in simplest form. Estimate 7 + 1 = 8
Alternative
Method
You can also stack
the mixed numbers
vertically to find
the sum.
6_
8
8
5
5
1
1
6_
+ 1_
= (6 + 1) + _
+_
8
8
8
8
5+1
_
=7+
8
6
3
_
= 7 or 7_
8
4
(
)
Add the whole numbers and fractions separately.
Add the numerators.
Simplify. Compared to the estimate, the answer is reasonable.
5
8
_1
+1
8
_
3
6
7 _ or 7 _
8
4
Find each sum. Write in simplest form.
3
1
4
1
2A. 3_
+ 7_
2B. 6 _
+ 9_
5
5
10
10
Subtract Like Fractions The rule for subtracting fractions with like
denominators is similar to the rule for addition.
Subtracting Like Fractions
To subtract fractions with like denominators, subtract the numerators and
write the difference over the denominator.
Words
Symbols
a-b
a
b
_
- _ = _, where c ≠ 0
c
c
EXAMPLE
5
5-1
1
4
Example _ - _ = _ or _
c
7
7
7
7
Subtract Fractions
9
13
Find _
-_
. Write the difference in simplest form.
20
20
9
- 13
9
13
_-_=_
20
20
20
-4
1
= _ or -_
5
20
1
1
Estimate _ - 1 = -_
2
2
The denominators are the same. Subtract the numerators.
Simplify.
Find each difference. Write in simplest form.
5
10
3
6
3A. _
-_
3B. _
-_
15
EXAMPLE
Alternative
Method
You can check your
answer by subtracting
the whole numbers and
fractions separately.
2
9_ - 5_
= (9 - 5)
6
6
2
1
+ _
-_
1
(6 6)
1
= 4 + (-_)
6
= 3_
5
6
9
15
9
Subtract Mixed Numbers
1
2
Evaluate a - b if a = 9 _
and b = 5 _
.
6
1
2
a - b = 9_
- 5_
6
6
55
32
_
_
=
6
6
23
5
_
=
or 3 _
6
6
6
Estimate 9 - 5 = 4
2
Replace a with 9_ and b with 5_
.
1
6
6
Write the mixed numbers as improper fractions.
Subtract the numerators. Simplify.
3
7
4. Evaluate x - y if x = 5 _
and y = 9 _
.
8
8
Personal Tutor at pre-alg.com
Lesson 5-5 Adding and Subtracting Like Fractions
251
You can use the same rules for adding or subtracting like algebraic fractions
as you did for adding or subtracting like numerical fractions.
EXAMPLE
Add Algebraic Fractions
n
5n
Find _
+_
. Write the sum in simplest form.
8
8
n
+ 5n
5n
n
_+_=_
8
8
8
6n
3n
= _ or _
8
4
The denominators are the same. Add the numerators.
Add the numerators. Simplify.
Find each sum. Write in simplest form.
y
5y
5B. _ + _
4d
2d
+_
5A. _
10
Examples 1–3
(pp. 250–251)
8
Find each sum or difference. Write in simplest form.
5
1
+_
1. _
9
10
10
2
4
3. _ + -_
3
3
2. _
+_
7
7
9
3
1
4. 2_
+ 8_
4
2
5. -2_
+ -_
3
5
6. 3_
+ 6_
3
11
7. _
-_
5
1
8. -_
-_
4
10
9. _ - _
6
6
5
14
14
Example 4
8
10
8
5
12
8
8
8
12
1
inches tall at the end of school in June. He
10. MEASUREMENT Hoai was 62_
8
7
inches tall in September. How much did he grow during the
was 63_
(p. 251)
8
summer?
3
1
ALGEBRA Evaluate each expression if u = 7_
and v = 6_
.
7
11. u - v
Example 5
(p. 252)
ALGEBRA Find each sum or difference. Write in simplest form.
6r
2r
13. _ + _
11
HOMEWORK
HELP
For
See
Exercises Examples
15–18
1
19–22
2
23–26
3
27–36
4
37–40
5
7
12. v - u
19
12
_
14. _
a - a ,a≠0
11
Find each sum or difference. Write in simplest form.
2
1
+_
15. _
5
5
13
9
+ -_
18. -_
21.
24.
27.
30.
16
16
5
7
+ 3_
5_
9
9
5
17
_
-_
18
18
3
5
- 1_
2_
8
8
6
5
- -2_
-8_
11
11
252 Chapter 5 Rational Numbers
3
3
17. -_
+ -_
19.
20.
22.
25.
28.
3
7
16. _
+_
31.
10
10
2
2
7_ + 4_
5
5
5
7
2_
+ 2_
12
12
1
7
_
- -_
12
12
9
1
8_
- 6_
10
10
5
3
-4_
-_
8
8
23.
26.
29.
32.
4
4
9
17
5_ + 5_
20
20
10
8
_
-_
11
11
9
7
_
- -_
20
20
5
4
7_
- 2_
7
7
3
2
12_
- 13 _
6
6
8
1
11
ALGEBRA Evaluate each expression if x = _
, y = 2_
, and z = _
. Write in
12
12
12
simplest form.
33. x + y
34. z + y
35. z - x
36. y - x
ALGEBRA Find each sum or difference. Write in simplest form.
x
4x
+_
37. _
8
3r
3r
38. _
+_
8
10
4
1
39. 5_
c - 3_
c
7
10
5
1
40. -2_
y + 8_
y
7
6
6
41. CARPENTRY A 3-foot long shelf is to be installed between two walls that
5
inches apart. How much of the shelf must be cut off so that it fits
are 32_
8
between the walls?
Real-World Career
Carpenter
Carpenters must be
able to make precise
measurements and
know how to add and
subtract fractional
measures.
For more information, go
to pre-alg.com.
PETS The table shows the weight of Leon’s dog during
its first five years.
42. How much weight did Leon’s dog lose between
Age
Weigh
(years) (pounds)
ages 3 and 4?
43. How much weight did Leon’s dog gain between
1
2
17 _
2
5
18 _
8
8
4
8
3
_
18
8
7
_
20
8
19 _
3
years 1 and 5?
7
44. Suppose Leon’s dog gained 2_
pounds between
4
8
years 5 and 6. How much does it weigh now?
5
Find each sum or difference. Write in simplest form.
3
5
7
- 7_
+ 2_
45. 12_
8
EXTRA
PRACTICE
See pages 771, 798.
Self-Check Quiz at
pre-alg.com
H.O.T. Problems
8
5
5
1
46. 5_
+ 3_
- 2_
8
6
6
6
47. GARDENING Tate’s flower garden has a perimeter of 25 feet. He plans to
add 2 feet 9 inches to the width and 3 feet 9 inches to the length. What is
the new perimeter in feet?
48. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose
some data and write a real-world problem in which you would add or
subtract like fractions.
49. OPEN ENDED Write a subtraction expression in which the difference of two
18
fractions is _
.
25
3
1
50. FIND THE ERROR Kayla and Ethan are adding -2_
and -4_
. Who is
8
8
correct? Explain your reasoning.
Kayla
-17
35
3
-2 _1 + -4 _
= _ + -_
8
8
8
8 Ethan
3
35
17
1
-2_
+ (-4_) = _ + (- _)
8
52
= -_
or -6 _1
8
2
CHALLENGE The 7-piece square puzzle at the right is
called a tangram.
51. If the value of the entire puzzle is 1, what is the
value of each piece?
52. How much is A + B?
53. How much is F + D?
54. How much is C + E?
55. Which pieces each equal the sum of E and G?
8
8
8
18
1
_
= - or -2_
4
8
B
C
D
A
E
F
Lesson 5-5 Adding and Subtracting Like Fractions
Tony Freeman/PhotoEdit
G
253
56.
Writing in Math
Explain how fractions are important when taking
measurements. Include in your answer some real-world examples in which
fractional measures are used.
57. The average times it takes Miguel to
cut his lawn and his neighbor’s lawn
are given in the table. Last summer,
he cut his lawn 10 times and his
neighbor’s 5 times. About how many
hours did he spend cutting both
lawns?
Lawn
_3
Neighbor’s
_2
2
16
15
inch thick is
layer of padding _
16
placed on top. What is the total
thickness of the wood and the
padding?
3
F 1_
in.
8
1
in.
G 1_
2
24
in.
H 1_
16
1
in.
J 2_
2
Time of Cut
(hours)
Miguel’s
1
A 8_
h
9
inches thick. A
58. A piece of wood is 1_
4
4
B 9h
1
C 9_
h
D 10 h
2
Find each quotient. Write in simplest form. (Lessons 5-4)
3
1
59. _
÷_
6
4
5
1
60. -_
÷_
3
8
2
1
61. _
÷ 1_
5
2
4
62. 8_
÷ -12 5
15
Find each product. Write in simplest form. (Lesson 5-3)
2 _
63. _
·3
5
4
8
1
64. _
· -_
6
9
4
1
65. _
· 2_
7
1
1
66. -1_
· 1_
3
7
3
67. ALGEBRA Find the product of 4y2 and 8y5. (Lesson 4-5)
EXERCISE The table shows the amount of time Craig spends
jogging every day. He increases the time he jogs every
week. (Lesson 3-7)
68. Write an equation to show the number of minutes
spent jogging m for each week w.
69. How many minutes will Craig jog during week 9?
7EEK
PREREQUISITE SKILL Use exponents to write the prime factorization of each
number or monomial. (Lesson 4-2)
70. 60
73. 12n
254 Chapter 5 Rational Numbers
71. 175
74.
24s2
72. 112
75. 42a2b
4IME*OGGING
MIN
Factors and Multiples
Many words used in mathematics are also used in everyday language. You
can use the everyday meaning of these words to better understand their
mathematical meaning. The table shows both meanings of the words factor
and multiple.
Term
Everyday Meaning
Mathematical Meaning
factor
something that contributes to the production
of result
• The weather was not a factor in the
decision.
• The type of wood is one factor that
contributes to the cost of the table.
one of two or more numbers that
are multiplied together to form a
product
multiple
involving more than one or shared by many
• multiple births
• multiple ownership
the product of a quantity and a
whole number
Source: Merriam Webster’s Collegiate Dictionary
When you count by 2, you are listing the multiples of 2. When you count by 3,
you are listing the multiples of 3, and so on, as shown in the table below.
Number
Factors
Multiples
2
1, 2
2, 4, 6, 8, . . .
3
1, 3
3, 6, 9, 12, . . .
4
1, 2, 4
4, 8, 12, 16, . . .
Notice that the mathematical meaning of each word is related to the everyday
meaning. The word multiple means many, and in mathematics, a number has
infinitely many multiples.
Reading to Learn
1. Write your own rule for remembering the difference between factor and
multiple.
2. RESEARCH Use the Internet or a dictionary to find the everyday meaning of
each word listed below. Compare them to the mathematical meanings of
factor and multiple. Note the similarities and differences.
a. factotum
b. multicultural
c. multimedia
3. Make lists of other words that have the prefixes fact- or multi-. Determine
what the words in each list have in common.
Reading Math Factors and Multiples
255
CH
APTER
5
Mid-Chapter Quiz
Lessons 5-1 through 5-5
Write each fraction or mixed number as a
decimal. Use a bar to show a repeating
decimal. (Lesson 5-1)
4
1. _
25
1
3. 3_
8
5 _
13. _
· 4
18
15
1 _
·2
15. -1_
2 3
2
2. -_
9
5
4. 1_
6
5. MULTIPLE CHOICE Which fraction is between
_5 and _7 ? (Lesson 5-1)
7
Find each product or quotient. Write in
simplest form. (Lessons 5-3 and 5-4)
8
C _
9
10
D _
11
10
3
B _
4
6. MANUFACTURING A garbage bag has a
thickness of 0.8 mil, which is equal to
0.0008 inch. What fraction of an inch is
this? (Lesson 5-2)
Write each decimal as a fraction or mixed
number in simplest form. (Lesson 5-2)
7. -6.75
8. 0.12
9. -0.5555 . . .
3
10. 3.08
1
11. GEOGRAPHY Africa makes up _
of Earth’s
5
entire land surface. Use the table to find the
fraction of Earth’s land surface that is made
up by each of the other continents. Write each
fraction in simplest form. (Lesson 5-2)
Antarctica
Decimal Portion
of Earth’s Land
0.095
Asia
0.295
Europe
0.07
North America
0.16
Source: Incredible Comparisons
12. TRAVEL One of the fastest commuter trains is
the Japanese Nozomi, which averages 162 miles
per hour. About how many minutes would it
take to travel 119 miles from Hiroshima to
Kokura on the train? (Lesson 5-3)
256 Chapter 5 Rational Numbers
4
1
16. 3 _
÷ (-4)
3
2
18. MULTIPLE CHOICE If
the newsletter is
4
printed on 8_
-inch by
8
11-inch paper and the
space between the
columns and both
ends of the page are
shown, how wide are
the three equal
columns? (Lesson 5-4)
1
F 2_
in.
6
5
in.
G 2_
12
Î
ˆ˜°
n
Ó
ˆ˜°
n
Ó
ˆ˜°
n
Î
ˆ˜°
n
.%73
1
H 2_
in.
2
3
J 2_
in.
4
Find each sum or difference. (Lesson 5-5)
8
2
19. _
+_
6
11
20. _
-_
1
2
-_
21. -2_
3
3
5
22. -3 + -4_
3
6
+ 2_
23. 5 _
5
7
24. 2_
- 8_
15
7
Continent
8
17. SEWING How many 9-inch ribbons can be cut
1
from 1_
yards of ribbon? (Lesson 5-4)
8
7
A _
7
1
14. _
÷ -_
15
7
12
12
12
8
12
25. MULTIPLE CHOICE A pitcher of lemonade
9
was _
full at the beginning of the party. It
10 _
was only 1 full after the party ended. How
10
much lemonade was drunk during the
party? (Lesson 5-5)
1
A _
10
3
B _
10
3
C _
5
_
D 4
5
5-6
Least Common Multiple
Main Ideas
Interactive Lab pre-alg.com
• Find the least
common multiple of
two or more
numbers.
A voter voted for both president and a
senate seat in the year 2004.
a. List the next three years
in which the voter can
vote for president.
• Find the least
common
denominator of two
or more fractions.
b. List the next three years
in which the voter can
vote for the same senate
seat.
New Vocabulary
multiple
common multiples
least common multiple
(LCM)
least common
denominator (LCD)
Candidate
Length of
Term (years)
President
4
Senator
6
c. What will be the next year in which the voter has a chance to vote
for both president and the same senate seat?
Least Common Multiple A multiple of a number is a product of that
number and a whole number. Sometimes numbers have some of the
same multiples. These are called common multiples.
multiples of 4: 0, 4, 8, 12, 16, 20, 24, 28, …
multiples of 6:
0, 6, 12, 18, 24, 30, 36, 42, …
Some common multiples of 4 and 6
are 0, 12, and 24.
The least of the nonzero common multiples is called the least common
multiple (LCM). So, the LCM of 4 and 6 is 12.
When numbers are large, an easier way of finding the least common
multiple is to use prime factorization. The LCM is the smallest product
that contains the prime factors of each number.
EXAMPLE
Find the LCM
Find the LCM of 108 and 240.
Number Prime Factorization
Prime Factors
If a prime factor
appears in both
numbers, use the
factor with the greatest
exponent.
Exponential Form
108
2·2·3·3·3
22 · 33
240
2·2·2·2·3·5
24 · 3 · 5
The prime factors of both numbers are 2, 3, and 5. Multiply the
greatest power of 2, 3, and 5 appearing in either factorization.
LCM = 24 · 33 · 5
= 2160
1. Find the LCM of 120 and 180.
Extra Examples at pre-alg.com
Lesson 5-6 Least Common Multiple
257
The LCM of two or more monomials is found in the same way as the LCM of
two or more numbers.
The LCM of Monomials
EXAMPLE
Find the LCM of 18xy2 and 10y.
18xy2 = 2 · 32 · x · y2
10y = 2 · 5 · y
Find the prime factorization of each monomial.
Highlight the greatest power of each prime factor.
LCM = 2 · 32 · 5 · x · y2
= 90xy2
Multiply the greatest power of each prime factor.
2. Find the LCM of 24 a3b and 30a.
Least Common Denominator The least common denominator (LCD) of two
or more fractions is the LCM of the denominators.
EXAMPLE
Find the LCD
5
11
Find the LCD of _
and _
.
9
21
Write the prime factorization of 9 and 21.
9 = 32
21 = 3 · 7
Highlight the greatest power of each prime factor.
LCM = 32 · 7
= 63
Multiply.
5
11
and _
is 63.
The LCD of _
9
21
3
7
3. Find the LCD of _
and _
.
8
10
One way to compare fractions is to write them using the LCD. We can
multiply the numerator and the denominator of a fraction by the same
number, because it is the same as multiplying the fraction by 1.
EXAMPLE
Replace
Compare Fractions
1
with <, >, or = to make _
7
_
a true statement.
6
15
The LCD of the fractions is 2 · 3 · 5 or 30. Rewrite the fractions using the
LCD and then compare the numerators.
5
1·5
5
_1 = _
=_
Multiply the fraction by _ to make the denominator 30.
5
6
2·3·5
30
7
·
2
14
7
2
_ = _ = _ Multiply the fraction by _ to make the denominator 30.
2
3·5·2
30
15
5
14
1
7
Since _ < _, then _ < _.
30
30
6
15
4. Replace
258 Chapter 5 Rational Numbers
2
with <, >, or = to make _
3
_5 a true statement.
9
Order Rational Numbers
TRAVEL The table shows the arrival
times of four flights compared to
their scheduled arrival times into
Greensboro, North Carolina. Order
the flights from least delayed to most
delayed. (Hint: A negative fraction
indicates a flight that arrived earlier
than its scheduled arrival time.)
$EPARTURE#ITY
Step 1 Order the negative fractions
first. The LCD of 6 and 8 is 24.
3
9
-_
= -_
8
1
4
-_
= -_
6
24
$IFFERENCEBETWEEN
!RRIVAL4IMEAND
3CHEDULED4IMEH
!TLANTA
?
n
$ALLAS
?
-IAMI
?
n
3AN
&RANCISCO
?
24
9
3
4
1
Compare the negative fractions. Since -_
< -_, then -_ < -_.
Ordering Rational
Numbers
You can break ordering
rational numbers into
two steps since
negative numbers are
always less than
positive numbers.
24
8
24
6
Step 2 Order the positive fractions. The LCD of 2 and 7 is 14.
5
10
1_
= 1_
7
1
7
1_
= 1_
2
14
14
10
5
7
1
Compare the positive fractions. Since 1_
< 1_, then 1_ < 1_.
14
2
14
7
5
3
1
Since -_
< -1 < 1_ < 1_, the order of the flights from least delayed to most
8
6
2
7
delayed are Atlanta, Miami, San Francisco, and Dallas.
Order the fractions from least to greatest.
1 3 1 2
3
3
4
1
5A. -7_
, -6_
, -6_
, -7_
5B. _, _, _, _
5
5
4
4
6 20 10 7
Personal Tutor at pre-alg.com
Example 1
(p. 257)
Example 2
(p. 258)
Example 3
(p. 258)
Find the least common multiple (LCM) of each pair of numbers.
1. 6, 8
2. 7, 9
Find the least common multiple (LCM) of each pair of monomials.
4. 36ab, 4b
(p. 258)
Example 5
(p. 259)
5. 5x2, 12y2
6. 14e3, 8e2
Find the least common denominator (LCD) of each pair of fractions.
1 _
,3
7. _
2 8
Example 4
3. 10, 14
Replace each
1
10. _
4
3
_
16
3 _
9. _
,5
2 _
8. _
, 7
3 10
5 8
with <, >, or = to make a true statement.
10
11. _
45
_2
5
12. _
9
7
13. WEATHER The table shows the amount of rain
that fell during a rainstorm in four Kentucky
cities. Order the cities from least to greatest
amount of rainfall.
City
_7
9
Rainfall (in.)
Bowling Green
7
1
10
Frankfort
15
Lexington
17
Louisville
2
2
3
3
8
Lesson 5-6 Least Common Multiple
259
HOMEWORK
HELP
For
See
Exercises Examples
14–23, 29
1
24–28
2
30–37
3
38–43
4
44–47
5
Find the least common multiple (LCM) of each pair of numbers
or monomials.
14. 4, 10
15. 20, 12
16. 2, 9
17. 16, 3
18. 15, 75
19. 21, 28
20. 14, 28
21. 20, 50
22. 18, 32
23. 24, 32
24. 20c, 12c
25. 16a2, 14ab
26. 7x, 12x
27. 75n2, 25n4
28. 20ef, 52f 3
29. AUTO RACING One driver can circle a one-mile track in 30 seconds. Another
driver takes 20 seconds. If they both start at the same time, in how many
seconds will they be together again at the starting line?
Find the least common denominator (LCD) of each pair of fractions.
1 _
30. _
,7
4 8
4 _
, 5
34. _
9 12
Replace each
8 _
31. _
,1
4 _
32. _
,1
15 3
3 _
35. _
,5
8 6
2 _
33. _
,6
5 2
1 _
36. _
,4
3 7
5 7
5 _
37. _
,8
6 9
with <, >, or = to make a true statement.
1
38. _
5
_
2 12
21
_1
41. _
100
5
7
39. _
_5
9 6
17 _
1
42. _
34 2
3
40. _
_4
5 7
12 _
36
43. _
17 51
Order the fractions from least to greatest.
5 _
1 _
, 3, _
,5
44. _
12 4 3 6
1
4
11
1
, -2_
, -2_
, -2_
46. -2_
2
9
6
18
23
4
2
7
45. -_
, -_
, -_
, -_
30
5
3
10
5 _
1 _
47. 1_
, 1 3 , 1_
, 11
24 4 8 3
48. PETS In Brady’s math class, approximately 3 of the students have pets.
5
About 41 out of every 50 students in his school have pets. Do a greater
fraction of students have pets in Brady’s math class or in his school?
49. ANALYZE TABLES The table shows the number of
children who signed up to play soccer in the park
district. Would you use the GCF or LCM to find the
greatest number of teams that can be formed if each
team must have the same number of 6-year-olds,
7-year-olds, and 8-year-olds? Explain your reasoning
and then find the answer. How many 6-year-olds,
7-year-olds, and 8-year-olds are on each team?
Age
Number
6
60
7
96
8
24
Find the least common multiple (LCM) of each set of numbers.
50. 7, 21, 84
EXTRA
PRACTICE
See pages 772, 798.
Self-Check Quiz at
pre-alg.com
51. 9, 12, 15
52. 45, 30, 35
53. FITNESS Suppose you run on the treadmill every other day and lift
weights every third day. After you add pushups to your routine, you do all
three exercises every thirtieth day. How often do you do pushups?
54. Find two composite numbers between 10 and 20 whose least common
multiple (LCM) is 36.
260 Chapter 5 Rational Numbers
55. FIND THE DATA Refer to the United States Data File on pages 18–21. Choose
some data and write a real-world problem in which you would compare
fractions.
56. OPEN ENDED Write two fractions whose least common denominator (LCD)
is 35.
H.O.T. Problems
CHALLENGE Determine whether each statement is sometimes, always, or
never true. Give an example or explanation to support your answer.
57. The LCM of three numbers is one of the numbers.
58. If two numbers do not contain any factors in common, then the LCM of the
two numbers is 1.
59. The LCM of two numbers, except 1, is greater than the GCF of the numbers.
60. The LCM of two whole numbers is a multiple of the GCF of the same
two numbers.
61.
Writing in Math
Use the information about prime factors on page 257
to explain how to use them to find the LCM of two or more numbers.
62. A radio station is giving away
two concert tickets to every sixteenth
caller and a dinner for two to every
twentieth caller. Which caller will
receive both the concert tickets and
the dinner?
63. A party goods store sells the party
supplies in the table. In order to have
the same number of cups, plates, and
napkins, what is the least number of
each that must be purchased?
Quantity in
One Package
A 32th
Supply
B 40th
cups
15
C 56th
plates
30
D 80th
napkins
20
F 40
G 48
H 60
J 64
Find each sum or difference. Write in simplest form. (Lesson 5-5)
3
7
64. _
-_
8
8
9
5
65. 3_
-_
11
11
13
3
66. _
+_
14
14
5
1
67. 2_
+ 4_
6
6
ALGEBRA Find each quotient. Write in simplest form. (Lesson 5-4)
3
_1
68. _
n÷n
x
x
69. _
÷_
8
6
ac
70. _
÷ _c
5
d
6k
3
71. _
÷_
7m
14m
72. ALGEBRA Translate the sum of 7 and two times a number is 11 into an equation.
Then find the number. (Lesson 3-6)
PREREQUISITE SKILL Estimate each sum. (page 751)
3
3
73. _ + _
8
4
9
14
74. _ + _
10
15
4
1
75. _
+ 2_
7
5
7
2
76. 5 _
+_
8
3
Lesson 5-6 Least Common Multiple
261
EXTEND
5-6
Algebra Lab
Juniper Green
Juniper Green is a game that was invented by a
teacher in England.
GETTING READY
This game is for two people, so students should
divide into pairs.
RULES OF THE GAME
• The first player selects an even number from the
hundreds chart and circles it with a colored marker.
• The next player selects any remaining number that
is a factor or multiple of this number and circles it.
• Players continue taking turns circling numbers, as
shown below.
• When a player cannot select a number or circles a number
incorrectly, then the game is over and the other player wins.
2nd move
Player 2 circles 7
because it is a
factor of 42.
1
1st move
Player 1
circles 42.
2
3
4
5
6
7
8
9
10
11 12
13 14
15 16
17 18
19 20
21 22
23 24
25 26
27 28
29 30
31 32
33 34
35 36
37 38
39 40
41 42
43 44
45 46
47 48
49 50
51 52
53 54
55 56
57 58
59 60
61 62
63 64
65 66
67 68
69 70
71 72
73 74
75 76
77 78
79 80
81 82
83 84
85 86
87 88
89 90
91 92
93 94
95 96
97 98
99 100
3rd move
Player 1 circles
70 because it is
a multiple of 7.
ANALYZE THE RESULTS
Play the game several times and then answer the following questions.
1. Why do you think the first player must select an even number? Explain.
2. Describe the kinds of moves that were made just before the game was over.
Reprinted with permission from Mathematics Teaching in the Middle School, copyright (c) 1999, by the National Council of Teachers of Mathematics. All rights reserved.
262 Chapter 5 Rational Numbers
Geoff Butler
5-7
Main Ideas
• Add unlike fractions.
• Subtract unlike
fractions.
Adding and Subtracting
Unlike Fractions
1
1
The sum _
+_
is modeled at the
2
3
right. We can use the LCM to find
the sum.
£
Ó
a. What is the LCM of the
denominators?
b. If you divide the model into
six parts, what fraction of the
model is shaded?
£
Î
1 _
c. How many parts are _
? 1?
2 3
d. Describe a model that you could
1
1
use to add _
and _
. Then use it to
3
4
find the sum.
£
Ó
£
Î
Add Unlike Fractions Fractions with different denominators are called
unlike fractions. In the activity, you used the LCM of the denominators to
rename the fractions. You can use any common denominator.
Adding Unlike Fractions
To add fractions with unlike denominators, rename the fractions with
a common denominator. Then add and simplify as with like fractions.
Words
1 _
2
1 5
2 3
_
+ =_·_+_·_
Example
3
5
3
5
5
3
5
6
11
= _ + _ or _
15
EXAMPLE
LCD
You can rename unlike
fractions using any
common denominator.
However, it is usually
simpler to use the least
common denominator.
15
15
Add Unlike Fractions
Find 1 + 23.
4
_1 + _2 = _1 · _3 + _2 · _4
3
4
3
4 3
3
8
_
_
=
+
12
12
11
_
=
12
Find each sum.
1
1
1A. _
+_
2
5
4
Use 4 · 3 or 12 as the least common denominator.
Rename each fraction with the common denominator.
Add the numerators.
2
1
1B. _
+_
3
8
Lesson 5-7 Adding and Subtracting Unlike Fractions
263
EXAMPLE
Add Fractions and Mixed Numbers
Find each sum. Write in simplest form.
3
-7
1
1
a. _
+ _
Estimate _ - _ = 0
Negative Signs
When adding or
subtracting a negative
fraction, place the
negative sign in the
numerator.
2
2
8
12
3 _
-7
-7 _
_3 + _
=_
·3+ _
· 2 The LCD is 23 · 3 or 24.
2
8 3
12
12
8
9
-14
=_
+ _
Rename each fraction with the LCD.
24
24
5
= -_
Add. Compare to the estimate. Is the answer reasonable?
24
1
2
b. 1_
+ -2_
Estimate 1 + (-2) = -1
3
9
2
1
11
7
1_
+ -2_
=_
+ -_
Write the mixed numbers as improper fractions.
9
3
9
3
11
7 _
=_
+ -_
· 3 Rename -_73 using the LCD, 9.
3
9
3
11
-21
=_+ _
Simplify.
9
9
-10
1
=_
or -1_
Add. Compared to the estimate, the answer is reasonable.
9
9
3
5
2A. _
+_
4
9
3
5
2C. 3 _
+ -4 _
5
8
2B. -_
+_
14
5
12
6
Subtract Unlike Fractions The rule for subtracting fractions with unlike
denominators is similar to the rule for addition.
Subtracting Unlike Fractions
To subtract fractions with unlike denominators, rename the fractions with a
common denominator. Then subtract and simplify as with like fractions.
EXAMPLE
Reasonableness
Use estimation to
check whether your
answer is reasonable.
6
2
- ≈0-1
21 7
≈ -1
17 is close to -1.
-
21
Subtract Fractions and Mixed Numbers
Find each difference. Write in simplest form.
6
1
a. _
-_
7
21
6 _
1
1
_
_
- 6 =_
-_
·3
7
7 3
21
21
18
1
=_
-_
21
21
-17
17
_
=
or -_
21
21
3
8
3A. _
-_
4
264 Chapter 5 Rational Numbers
9
1
1
b. 6 _
- 4_
The LCD is 21.
Rename using
LCD.
5
2
5
Write as
improper
fractions.
2
5
13 _
21 _
· 5 -_
· 2 Rename
=_
using LCD.
2 5
5
65
42
=_-_
10
10
23
3
= _ or 2 _
10
10
Subtract.
5
1
3B. 7_
- 6_
6
2
13
1
1
21
6_
- 4_
=_
-_
8
2
Simplify.
Subtract.
5
1
3C. 5 _
- -4 _
3
9
COMPUTERS To set up a computer
network in an office, a 100-foot
cable is cut and used to connect 3
computers to the server as shown.
How much cable is left to connect
the third computer?
You know that the 100-foot
cable was used to connect
two computers to the server.
Plan
Add the measures of the cables that were
already used and subtract that sum from 100.
Estimate your answer.
100 - (19 + 41) ≈ 100 - 60 or 40 feet
Solve
3
6
1
1
19_
+ 40_
= 19_
+ 40_
4
8
7
= 59_
8
8
FT
4FSWFS
Explore
8
FT
3ERVER
Rename 40_ with the LCD, 8.
3
4
Simplify.
8
7
7
100 - 59_
= 99_
- 59_
8
Rename 100 with the LCD, 8, as 99_.
8
8
8
1
= 40_
Simplify.
8
1
There is 40_
feet of cable left to connect the third computer.
8
1
Since 40_
is close to 40, the answer is reasonable.
8
8
Check
3
4. INSECTS The speed of a hornet is 13_
miles per hour. The speed of a
10
4
miles per hour. How much faster is the dragonfly than
dragonfly is 17 _
the hornet?
5
Personal Tutor at pre-alg.com
Examples 1, 2
(pp. 263–264)
Find each sum. Write in simplest form.
3
1
+_
1. _
1
1
2. _
+_
5
15
5
1
+ 11_
4. 8_
12
4
Example 3
(p. 264)
3
10
3
5
5. 4_
+ 10_
8
12
1
2
-_
7. _
3
7 - 2
8. -
10 15
3
1
- -5_
10. -9_
2
4
(p. 265)
6
18
3
4
6. 6_
+ -1_
5
4
Find each difference. Write in simplest form.
4
Example 4
7
1
3. -_
+_
5
1
11. 6_
- 2_
6
3
5
7
9. _
-_
8
12
3
1
12. 12_
- 6_
2
8
5
1
13. SEWING Jessica needs 5_
yards of fabric to make a skirt and 14_
yards to
8
make a coat. How much fabric does she need in all?
Extra Examples at pre-alg.com
2
Lesson 5-7 Adding and Subtracting Unlike Fractions
265
HOMEWORK
HELP
For
See
Exercises Examples
14–19
1, 2
20–23
2
24–31
3
32–33
4
Find each sum or difference. Write in simplest form.
3
3
+_
14. _
5
4
8
1
4
+ 3_
20. 8_
2
5
3
7
1
21. _
+ 4_
1
11
22. -4_
+ -7_
5
1
24. _
-_
7
2
25. _
-_
1
2
27. -_
-_
8
2
28. -6_
-_
8
2
7
+ 9_
23. -10_
7
21
3
1
19. -_
+_
7
4
3
8
3
5
5
1
30. 16_ - 12_
6
3
16
7
-_
29. 216
30
15
6
24
8
12
3
7
- -_
26. _
8
5
10
16. _
+ -_
13
3
1
18. -_
+_
8
2
3
5
17. _
+ -_
9
3
15. _
+_
26
10
18
5
3
9
1
1
31. 3_
- -7_
2
3
For Exercises 32–35, select the appropriate operation. Justify your
selection. Then solve.
32. EARTH SCIENCE Did you know that water has a greater
density than ice? Use the information in the table to
find how much more water weighs per cubic foot.
1 Cubic
Foot
Weight
(lb)
water
1
2
9
56 10
33. PUBLISHING The length of a page in a yearbook is
ice
1
inch, and the bottom
10 inches. The top margin is _
2
3
_
margin is inch. What is the length of the page inside
62 4
the margins?
1
of the votes and Sara
34. VOTING In the class election, Murray received _
3
2
of the votes. Makayla received the rest. What fraction of the
received _
5
votes did Makayla receive?
35. ANALYZE TABLES Use the table to find
the sum of precipitation that fell in
Columbia, South Carolina, in
August, September, and October.
EXTRA
PRACTICE
See pages 772, 798.
Self-Check Quiz at
pre-alg.com
36. RESEARCH Use the Internet or another
source to find out the monthly rainfall
totals in your community during the past
year. How much rain fell in August,
September, and October during the past
year?
!MOUNTOF
0RECIPITATIONIN
!UG
??
3EPT
??
/CT
?
Find each difference. Write in simplest form.
3
3
37. -19_
- -4_
8
H.O.T. Problems
-ONTH
5
5
13
39. 8_
- -12_
2
4
38. -3_
- -2_
4
7
12
40. OPEN ENDED Write a real-world problem that you could solve by
3
1
from 15_
.
subtracting 2_
8
4
3
41. CHALLENGE A set of measuring cups has measures of 1 cup, _
cup,
4
1
1
1
1
_
cup, _
cup, and _
cup. How could you get _
cup of milk by using
2
3
these measures?
266 Chapter 5 Rational Numbers
4
6
18
9
7
42. FIND THE ERROR Roberto and Daniel are finding _
+_
. Who is correct so
10
12
far? Explain your reasoning.
Daniel
9 + 7 = 9+7
10 12 10 + 12
Roberto
7
7
9
9 12
_
+_=_·_
+ _ · 10
12
10
43.
10
12
12
10
Writing in Math
Explain how to add and subtract fractions with
different denominators. Illustrate your answer with an example using the
LCM and an explanation of how prime factorization can be used to add
and subtract unlike fractions.
44. For an art project, Halle needs
113 inches of red ribbon and 67 inches
8
9
of white ribbon. Which is the best
estimate for the total amount of
ribbon that she needs?
45. The results of
a grocery store
survey are
listed in the
table. Find the
fraction of
families who
grill out more
than one time
per month.
A 8 in.
B 10 in.
C 18 in.
2
F _
D 26 in.
25
How Often Do You Grill Out?
Times per
Fraction of
Month
People
Less than 1
11
50
1
2
25
2–3
4
25
4 or more
27
50
23
G _
7
H _
100
39
J _
50
10
Find the LCD of each pair of fractions. (Lesson 5-6)
4 _
46. _
, 7
5 _
47. _
, 3
9 12
3 _
48. _
, 2
8 14
1 _
49. _
, 73
15t 5t
3n 6n
Find each sum or difference. Write in simplest form. (Lesson 5-5)
3
3
50. 2_
+ 6_
4
4
3
2
51. 3_
-_
5
5
1
52. 4_
+ 5_
5
6
6
5
1
53. 6_
- -8_
4
15
54. MOVIES A movie is made up of hundreds of thousands of individual
pictures called frames. The frames are shown through a projector at a
rate of 24 frames per second. How many frames would be needed for a
30-minute scene? (Lesson 1-1)
PREREQUISITE SKILL Find each quotient. Round to the nearest tenth,
if necessary. (Page 749)
55. 25.6 ÷ 3
56. 37 ÷ 4.7
57. 30.5 ÷ 11.2
58. 46.8 ÷ 15.6
59. 34.8 ÷ 5.8
60. 63 ÷ 7.5
Lesson 5-7 Adding and Subtracting Unlike Fractions
267
5-8
Solving Equations with
Rational Numbers
Main Idea
• Solve equations
containing rational
numbers.
Musical sounds are made by vibrations. If n represents the number of
vibrations for middle C, then the approximate vibrations for the other
notes going up the scale are given below.
Notes
Middle C
D
E
F
G
A
B
C
n
9
n
8
5
n
4
4
n
3
3
n
2
5
n
3
15
n
8
2
n
1
Number of
Vibrations
a. A guitar string vibrates 440 times per second to produce the A
above middle C. Write an equation to find the number of vibrations
per second to produce middle C. If you multiply each side by 3,
what is the result?
b. How would you solve the second equation you wrote in part a?
c. How can you combine the steps in parts a and b into one step?
d. How many vibrations per second are needed to produce middle C?
Solve Addition and Subtraction Equations You can solve equations
with rational numbers using the same properties you used to solve
equations with integers.
EXAMPLE
Look Back
Solve by Using Addition and Subtraction
a. Solve 2.1 = t - 8.5.
To review solving
equations, see
Lessons 3-3 and 3-4.
2.1 = t - 8.5
2.1 + 8.5 = t - 8.5 + 8.5
10.6 = t
3
2
b. Solve x + _
=_
.
5
3
3
2
=_
x+_
3
5
3
3
3
2
x+_
-_
=_
-_
5
5
3
5
9
10
1
_
x=
-_
or _
15
15
15
Solve each equation.
1A. n - 9.7 = -13.9
268 Chapter 5 Rational Numbers
Write the equation.
Add 8.5 to each side.
Simplify.
Write the equation.
Subtract _ from each side.
3
5
Rename the fractions using the LCD and subtract.
7
2
1B. _
+p=_
10
5
Solve Multiplication and Division Equations Use the same process to solve
these equations as those involving integers.
EXAMPLE
Solve by Using Division
Solve -3y = 1.5. Check your solution.
-3y = 1.5
Write the equation.
-3y
1.5
_
=_
-3
-3
Divide each side by -3.
y = -0.5
Simplify. Check the solution.
Solve each equation. Check your solution.
2A. 6a = -8.4
2B. -36 = -5z
1
1
To solve _
x = 3, you can divide each side by _
or multiply each side
2
1
by the multiplicative inverse of _
, which is 2.
2
2
Reciprocals
_1 x = 3
2
Recall that dividing by
a fraction is the same
as multiplying by its
multiplicative inverse.
1
2·_
x =2·3
The product of
any number and
its multiplicative
inverse is 1.
EXAMPLE
2
x =6
Write the equation.
Multiply each side by 2.
Simplify.
Solve by Using Multiplication
-2
Solve _
x = -7. Check your solution.
3
-2
_
x = -7
3
-3 _
-3
-2
_
x =_
(-7)
2
2
3
1
21
x=_
or 10_
2
2
Write the equation.
Multiply each side by -3
.
2
Simplify. Check the solution.
Solve each equation. Check your solution.
5
1
3A. 10 = _
h
3B. _
m = -3
6
8
3
7
PETS Oscar feeds his dog _
cup of dog food in the morning and _
cup
8
4
of dog food in the evening. If a bag of dog food contains 50 cups, how
many days will the bag last?
The amount of dog food that Oscar feeds his dog each day is
_7 + _3 = _7 + _6 or 1_5 cups.
8
4
8
8
8
(continued on the next page)
Extra Examples at pre-alg.com
Lesson 5-8 Solving Equations with Rational Numbers
269
1_ cups per day
5
8
Words
times
d days
equals
50 cups of dog food
d
=
50
Let d the number of days.
Variable
1_
5
8
Equation
·
5
1_
d = 50
8
13
_
d = 50
8
8 _
8
_
· 13 d = 50 · _
13 8
13
400
d=_
≈ 30.8
13
Write the equation.
Rename 1_ as an improper fraction.
5
8
Multiply each side by _.
8
13
Simplify.
The bag of dog food will last approximately 31 days.
4. RETAIL A pair of shoes that normally costs $45 now costs $30. What
fraction of the regular price is the reduced price?
Personal Tutor at pre-alg.com
Solve each equation. Check your solution.
Example 1
(p. 268)
Examples 2, 3
(p. 269)
1. y + 3.5 = 14.9
4. b - 5 = 13.7
5
5
3
_
_
5. c - =
6
5
7. -8.4 = -6f
8. 3.5a = 7
Example 4
HOMEWORK
HELP
For
See
Exercises Examples
14–19
1
20–25
2
26, 27
3
28–31
4
32, 33
5
1
1
3. 4 _
= r + 6_
2
6
1
10. -_
s = 15
4
6. a - 2.7 = 3.2
9. -3.4 = 0.4x
3
11. 9 = _
g
4
6
(pp. 269–270)
3
3
2. _
=w+_
2
12. _
p = -22
3
13. SPACE The weight of an object on the Moon is one-sixth its weight on
Earth. If an object weighs 54 pounds on the Moon, how much does it
weigh on Earth? Write and solve a multiplication equation to determine
the weight of an object on Earth.
Solve each equation. Check your solution.
14. y + 7.2 = 21.9
15. 4.7 = a + 7.1
2
1
16. _
=_
+b
5
7
17. m + _
= -_
3
5
18. 3_
+ n = 6_
1
1
19. y + 1_
= 3_
12
18
8
4
3
8
3
18
20. x - 5.3 = 8.1
21. n - 4.72 = 7.52
8
2
23. x - _
= -_
1
1
24. b - 1_
= 4_
3
1
22. n - _
=_
8
6
1
2
_
25. 7 = r - 5_
26. 4.1p = 16.4
27. -0.4y = 2
2
28. 8 = _
d
1
29. _
t=9
1
30. 4 = -_
q
3
31. -6 = _
a
5
5
270 Chapter 5 Rational Numbers
15
2
8
4
2
3
5
3
-“
œŽ
ÞÊ
33. BUSINESS A store is going out
of business. Items that
normally cost $24.99 now cost
$16.66. What type of discount
is the store offering on those
items?
Î
Ó
£
ä
œÕ ˜Ì À i >
>ˆ Ì
˜Ã
À
>˜
`Ê
>
˜Þ
œ˜
"
ޓ
«ˆ
V
9œ
Ãi
,œ
“
ˆÌi
VŽ
ÞÊ
œÕ
˜Ì
>ˆ
˜Ã
9i
œ
ÜÃ
̜
˜i
.UMBEROF6ISITORSMILLIONS
For Exercises 32 and 33, write an equation and solve the problem.
32. ANALYZE GRAPHS The graph
)''*DfjkM`j`k\[
shows the most visited national
L%J%EXk`feXcGXibj
£ä
parks in the United States in
™
2003. The Great Smoky
n
Mountains had 5.25 million
Ç
more visitors than the Grand
È
Canyon. How many people
x
{
visited the Grand Canyon?
53.ATIONAL0ARKS
-œÕÀVi\4OP4ENOF%VERYTHING
Solve each equation. Check your
solution.
1
2
n=_
34. _
3
5
1
35. _
= -_
r
9
8
28
7
36. -_
t = -_
2
9
36
For Exercises 37–40, write an equation and solve the problem.
37. METEOROLOGY When a storm struck, the barometric pressure was
28.79 inches. Meteorologists said that the storm caused a 0.36-inch drop in
pressure. What was the pressure before the storm?
38. MUSIC Carla downloaded some songs onto her digital music player and
5
1
full. If the player was _
full before the download, what
now the player is _
6
5
fraction of the space on the player do the new songs occupy?
1
batches of cookies for a bake sale and used
39. COOKING Gabriel made 2_
2
3
cups of sugar. How much sugar is needed for one batch of cookies?
3_
4
1
inches wide and 22 inches long. This is
40. PUBLISHING A newspaper is 12_
4
1
inches narrower and one-half inch longer than the old edition. What
1_
4
were the previous dimensions of the newspaper?
EXTRA
PRACTICE
See pages 772, 798.
Self-Check Quiz at
pre-alg.com
H.O.T. Problems
3
square inches.
41. GEOMETRY The area A of the triangle is 33_
4
1
bh to find the height h of the
Use the formula A = _
2
triangle with the given base b.
h
42. FIND THE DATA Refer to the United States Data File on
9 in.
pages 18–21. Choose some data and write a real-life
problem in which you would solve equations with rational numbers.
43. FIND THE ERROR Grace and Ling are solving 0.3x = 4.5. Who is correct?
Explain your reasoning.
Grace
0.3x = 4.5
0.3x _
_
= 4.5
3
3
x = 1.5
Ling
0.3x = 4.5
0.3x _
_
= 4.5
0.3
0.3
x = 15
Lesson 5-8 Solving Equations with Rational Numbers
271
44. Which One Doesn’t Belong? Identify the equation that does not belong with
the other three. Explain your reasoning.
_5 j = 20
c + 3.17 = -3.17
3
2
4_
= w - 2_
9
4
-0.8k = 10
8
45. CHALLENGE The denominator of a fraction is 4 more than the numerator. If
both the numerator and denominator are increased by 1, the resulting
1
fraction equals _
. Find the original fraction.
2
46.
Writing in Math
Explain how fractions are used to compare musical
notes. How are reciprocals useful in finding the number of vibrations per
second needed to produce certain notes? Give an example.
47. GRIDDABLE A
hamburger is formed
into the shape of a
circle with a radius of
3
1_
inches. If a grill is
4
28 inches wide, how
many hamburgers can
fit across the grill?
48. Mary and Tabitha ran in a race.
Mary’s time was 12 minutes, which
3
was _
of Tabitha’s time. Using t for
4
Tabitha’s time, which equation
represents the situation?
Î
£Ê{ ˆ˜°
3
A _
t = 12
3
C t-_
= 12
3
= 12
B t+_
3
D 12t = _
4
4
4
4
Find each sum or difference. Write in simplest form. (Lesson 5-7)
3
1
49. _
+_
5
5
1
50. _
-_
3
6
5
1
+ -_
52. _
9
1
55. _
2
1
1
51. -4_
-_
4
3
1
53. -3_
+ -2_
4
12
Replace each
8
6
9
1
54. 8_
- 1_
8
10
6
with <, >, or = to make a true statement. (Lesson 5-6)
5
_
12
56.
16 _
9
_
50 30
4
57. _
5
48
_
60
58.
7
_3 _
8
12
1
. (Lesson 5-5)
59. ALGEBRA Evaluate a - b if a = 9_ and b = 1_
5
6
6
60. GEOMETRY Express the area of the rectangle as a monomial. (Lesson 4-5)
61. HEALTH According to the National Sleep Foundation, teens should
get approximately 9 hours of sleep each day. What fraction of the day
is this? Write in simplest form. (Lesson 4-4)
PREREQUISITE SKILL Find each sum. (Lesson 2-2)
62. 24 + (-12) + 15
63. (-2) + 5 + (-3)
64. 4 + (-9) + (-9) + 5
65. -10 + (-9) + (-11) + (-8)
272 Chapter 5 Rational Numbers
{X ÓY
™XY Ó
EXPLORE
5-9
Algebra Lab
Analyzing Data
Often, it is useful to describe or
represent a set of data by using a
single number. The table shows
the daily maximum temperatures
for twenty days during a recent
September in Phoenix, Arizona.
0HOENIX!RIZONA
-AXIMUM4EMPERATURES²&
3EPTEMBERn
One number to describe this data
set might be 96. Some reasons for
choosing this number are listed
below.
• It occurs four times, more
often than any other number.
• If the numbers are arranged in order from least to greatest, 96 falls in
the center of the data set.
There is an equal number of
data above and below 96.
80 89 90 94 94 94 95 96 96 96 96 97 97 98 98 98 98 98 99 99
So, if you wanted to describe a typical high temperature for Austin during
August, you could say 96°F.
COLLECT THE DATA
Collect a group of data. Use one of
the suggestions below, or use your
own method.
• Research data about the weather in
your city or in another city, such as
temperatures, precipitation, or wind
speeds.
• Find a graph or table of data in the
newspaper or a magazine. Some
examples include financial data,
population data, and so on.
• Conduct a survey to gather some
data about your classmates.
• Count the number of raisins in
a number of small boxes.
ANALYZE THE RESULTS
1. Choose a number that best
describes all of the data in the set.
2. Explain what your number means, and explain which method you used
to choose your number.
3. Describe how your number might be useful in real life.
Explore 5-9 Algebra Lab: Analyzing Data
Geoff Butler
273
5-9
Measures of Central
Tendency
Main Ideas
• Use the mean,
median, and mode as
measures of central
tendency.
• Choose an
appropriate measure
of central tendency
and recognize
measures of statistics.
New Vocabulary
measures of central
tendency
mean
median
mode
The Iditarod is a 1150-mile dogsled
race across Alaska. The winning
times for 1977–2004 are shown.
Winning Times (days)
a. Which number appears most
often?
b. If you list the data in order from
least to greatest, which number
is in the middle?
17
15
15
14
12
16
13
13
18
12
11
11
11
11
13
11
11
11
9
9
9
9
10
9
9
8
9
9
Source: Anchorage Daily News
c. What is the sum of all the
numbers divided by 28?
d. If you had to give one number that best represents the winning
times, which would you choose? Explain.
Mean, Median, and Mode When you have a list of numerical data, it is
often helpful to use one or more numbers to represent the whole set.
These numbers are called measures of central tendency.
Measures of Central Tendency
mean
Mean, Median,
Mode
• The mean and
median do not have
to be part of the
data set.
• If there is a mode, it
is always a member
of the data set.
the sum of the data divided by the number of items in the data set
median the middle number of the ordered data, or the mean of the middle
two numbers
mode
the number or numbers that occur most often
a. SPORTS The heights of the players
on the girls’ basketball team are shown.
Find the mean, median, and mode.
sum of heights
number of players
63 + 61 + . . . + 59
= __
12
732
_
=
or 61 The mean height is 61 inches.
12
mean = __
Height of Players (in.)
63 58 61 60
61 59 68 55
63 59 66 59
To find the median, order the numbers from least to greatest.
55, 58, 59, 59, 59, 60, 61, 61, 63, 63, 66, 68
60 + 61
_
= 60.5
2
There is an even number of items. Find
the mean of the two middle numbers.
The median height is 60.5 inches.
The height 59 inches appears three times so 59 is the mode.
274 Chapter 5 Rational Numbers
b. HURRICANES The line plot shows the number of Atlantic hurricanes
that occurred each year from 1974 to 2004. Find the mean, median,
and mode.
1
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
2
3
4
5
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
6
7
8
9
10
11
12
Source: National Weather Service
2 + 3(4) + 4(6) + 5(5) + 6(2) + 7(4) + 8(3) + 9(4) + 10 + 11
31
mean = _____ ≈ 5.9
Real-World Link
A hurricane can be up
to 600 miles in diameter
and can reach 8 miles
in the air.
Source: sptimes.com
There are 31 numbers. So the median is the 16th number, or 5.
You can see from the graph that 4 occurs most often. So 4 is the mode.
1 , 11, 5,
1. SHOES The shoe sizes of students in Ms. Alberti’s classroom are 10 _
1 , 6, 6 _
1 , 11, 7, 7 _
1 , 8, 9, 5 _
1 , 10 _
1 , 4, 10 _
1 , 10, 5, 14, and 12 _
1 . Find2the
6, 10 _
2
2
2
2
2
2
2
mean, median, and mode.
Choose Appropriate Measures Different circumstances determine which of
the measures of central tendency are most useful.
Using Mean, Median, and Mode
mean
• the data set has no extreme values (values that are much greater or
much less than the rest of the data)
median • the data set has extreme values
• there are no big gaps in the middle of the data
mode
• the data set has many repeated numbers
Choose an Appropriate Measure
WEATHER The table shows daytime high
temperatures for a week. Which measure of
central tendency best represents the data?
Then find the measure of central tendency.
Since the set of data has no extreme values or
numbers that are identical, the mean would
best represent the data.
Day
Temperature
Sun.
84°F
Mon.
83°F
Tues.
89°F
Wed.
90°F
Thurs.
91°F
84 + 83 + . . . + 80
602
mean: __ = _
or 86
Fri.
85°F
Sat.
80°F
7
7
The temperature 86°F best represents the data.
2. EXERCISE The following set of data shows the number of sit-ups Pablo
had done in one minute for the past 6 days: 40, 37, 45, 49, 50, 56.
Which measure of central tendency best represents the data? Justify
your selection and then find the measure of central tendency.
Extra Examples at pre-alg.com
NASA
Lesson 5-9 Measures of Central Tendency
275
Using
measures of
central
tendency can help you
analyze the data from
fast-food restaurants.
Visit pre-alg.com to
continue work on your
project.
NUTRITION The table shows
the number of Calories per
serving of each vegetable.
Tell which measure of central
tendency best represents the
data. Then find the measure
of central tendency.
Vegetable
There is one value that is much
greater than the rest of the data,
66. Also, there does not appear
to be a big gap in the middle of
the data. There is only one set of
identical numbers. So, the median
would best represent the data.
Calories
Vegetable
Calories
asparagus
14
cauliflower
10
beans
30
celery
17
bell pepper
20
corn
66
broccoli
25
lettuce
9
cabbage
17
spinach
9
carrots
28
zucchini
17
9, 9, 10, 14, 17, 17, 17, 20, 25, 28, 30, 66
The median is 17 Calories.
CHECK You can check whether the median best represents the data by
finding mean with and without the extreme value.
mean with extreme value
262
sum of values
__
=_
12
number of values
Interpreting
Data
You need to interpret
information carefully
so that you do not
give a false
impression for a set of
data. As you have
seen, extreme values
affect how a set of
data is perceived.
≈ 21.8
mean without extreme value
sum of values
196
__
=_
11
number of values
≈ 17.8
The mean without the extreme value is closer to the median. The
extreme value increases the mean by about 4. Therefore, the median
best represents the data.
3. RETAIL An electronics store recorded the number of customers it had each
hour during the day. 86, 71, 79, 86, 79, 32, 88, 86, 82, 69, 71, 70
Which measure of central tendency best represents the data? Justify your
selection and then find the measure of central tendency.
Measures of central tendency can be used to show different points of view.
The average wait times for 10 different rides at an amusement park are
65, 21, 17, 52, 25, 17, 11, 22, 60, and 44 minutes. Which measure of data
would the park advertise to show the wait times for its rides are short?
A Mode
B Median
C Mean
D Cannot be determined
Read the Test Item
To find which measure of central tendency to use, find the mean, median,
and mode of the data and select the least measure.
276 Chapter 5 Rational Numbers
Solve the Test Item
Analyzing Data Use
these clues to help
you analyze data.
• Extremely high or
low values affect the
mean.
• A value with a high
frequency affects
the mode.
• Data that is
clustered affect the
median.
65 + 21 + … + 44
10
334
Mean: __ = _
or 33.4
Mode: 17
10
Median: 11, 17, 17, 21, 22, 25, 44, 52, 60, 65
22 + 25
_
or 23.5
2
The mode is the least measure. So the answer is A.
4. Serena received the following scores on her first six math tests: 90, 68, 89,
94, 60, and 93. Which measure of data might she want to use when
describing how she is doing in math class?
F Mode
G Median
H Mean
J Cannot be determined
Personal Tutor at pre-alg.com
Example 1
(pp. 274–275)
Find the mean, median, and mode for each set of data. If necessary, round
to the nearest tenth.
1. 4, 5, 7, 3, 9, 11, 23, 37
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
1
2
3
4
3.
Example 2
(p. 275)
Example 3
(p. 276)
2. 7.2, 3.6, 9.0, 5.2, 7.2, 6.5, 3.6
5
⫻
⫻
⫻
⫻
6
7
8
4. VACATIONS The table shows the number
of annual vacation days for nine
countries. Which measure of central
tendency best represents the data? Justify
your selection and then find the measure
of central tendency.
5. BOOKS The number of books sold during
the past week is shown below. Which
measure of central tendency best
represents the data? Justify your
selection and then find the measure of
central tendency.
53, 61, 46, 59, 61, 55, 49
Example 4
(pp. 276–277)
Annual Vacation Days
Country
Number of Days
Brazil
34
Canada
26
France
37
Germany
35
Italy
42
Japan
25
Korea
25
United Kingdom
28
United States
13
Source: World Tourism Organization
6. MULTIPLE CHOICE Suppose 83 books sold on the eighth day in Exercise 5.
Which measure of central tendency would change the most?
A The mean
B The median
C The mode
D All measures were affected equally.
Lesson 5-9 Measures of Central Tendency
277
HOMEWORK
HELP
For
See
Exercises Examples
7–12
1
13–14
2, 3
15–17
4
Find the mean, median, and mode for each set of data. Round to the
nearest tenth, if necessary.
7. 41, 37, 43, 43, 36
8. 2, 8, 16, 21, 3, 8, 9, 7, 6
9. 14, 6, 8, 10, 9, 5, 7, 13
11.
15
⫻
⫻
⫻
16
10. 7.5, 7.1, 7.4, 7.6, 7.4, 9.0, 7.9, 7.1
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
17
18
19
20
21
12.
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
4.1
4.2
4.3
4.4
⫻
22
4.5
4.6
4.7
4.8
For Exercises 13–14, which measure of central tendency best represents the
data? Justify your selection and then find the measure of central tendency.
13.
University
Michigan
All-Time
Football Wins
833
Notre Dame
796
Nebraska
781
Texas
776
Alabama
758
Source: The World Almanac
14.
2003 Corn Production
State
Bushels (millions)
CA
27.2
GA
36.8
MD
50.4
MS
71.6
TX
194.7
Source: U.S. Dept. of Agriculture
15. BASKETBALL Refer to the cartoon at the right.
Which measure of central tendency would
make opponents believe that the height of
the team is much taller than it really is?
Explain.
EXTRA
PRACTICE
16. TESTS Which measure of central tendency
best summarizes the test scores shown below?
Explain. 97, 99, 95, 89, 99, 100, 87, 85, 89, 92,
96, 95, 60, 97, 85
See pages 772, 798.
Self-Check Quiz at
pre-alg.com
H.O.T. Problems
17. ICE SKATING Sasha needs to average 5.8 points from 14 judges to win the
competition. The mean score of 13 judges was 5.9. What is the lowest score
Sasha can have from the 14th judge and still win?
18. OPEN ENDED Write a set of data with at least four numbers that has a mean
of 8 and a median that is not 8.
19. CHALLENGE A real estate guide lists the “average” home prices for counties
in your state. Do you think the mean, median, or mode would be the most
useful average for homebuyers? Explain.
20.
Writing in Math
Explain how measures of central tendency are used in
the real world. Include in your answer examples of real-world data from
home or school that can be described using the mean, median, or mode.
278 Chapter 5 Rational Numbers
21. The graph shows the number of
siblings that Ms. Cantor’s students
have. Which measure of data best
represents the data?
22. If 18 were added to the data set
below, which statement is true?
16, 14, 22, 16, 16, 18, 15, 25
F The mode increases.
ÀœÌ…iÀÃÊ>˜`Ê-ˆÃÌiÀÃ
G The mean increases.
ՓLiÀʜvÊ-ÌÕ`i˜ÌÃ
£ä
H The mode decreases.
n
J The median increases.
È
23. The hourly salaries of
employees in a small store
are shown. Which measure
of data would the store are
shown use to attract people
to work there?
{
Ó
ä
<iÀœ "˜i /ܜ /…Àii œÕÀ
ՓLiÀʜvÊ-ˆLˆ˜}Ã
A Mode
ˆÛi
C Mean
B Median D Cannot be determined
$17
$24
$7
$8
$10
$6
$8
$8
A Mode
C Mean
B Median
D Cannot be determined
ALGEBRA Solve each equation. Check your solution. (Lesson 5-8)
3
1
25. 1_
+ p = -6_
24. k - 4.1 = -9.38
4
c
26. _
= 0.845
2
10
Find each sum or difference. Write in simplest form. (Lesson 5-7)
3
7
27. _
-_
8
10
5
1
28. -_
+_
5
12
2
1
29. 9_
+_
3
3
1
30. -2_
- 1_
6
4
31. SEWING Stacey is making a linen suit. The portion of the pattern
envelope that shows the yards of fabric needed for different
sizes is shown at the right. If Stacey is making a size 6 jacket
and skirt from 45-inch fabric, how much fabric should she
buy? (Lesson 5-5)
Express each number in scientific notation. (Lesson 4-7)
32. The flow rate of some Antarctic glaciers is 0.00031 mile
per hour.
33. A human blinks about 6.25 million times a year.
8
Size
6
8
10
JACKET
45”
2_
2_
2_
60”
2
2
2
SKIRT
45”
_7
1_
1_
60”
5
8
8
_7
8
3
4
3
4
1
8
7
_
8
1
8
7
_
8
Math and Science
You Are What You Eat It is time to complete your project. Use the information and data you have
gathered to prepare a brochure or Web page about the nutritional value of fast-food meals. Include the
total Calories, grams of fat, and amount of sodium for five meals that a typical student would order from
at least three fast-food restaurants.
Cross-Curricular Project at pre-alg.com
Lesson 5-9 Measures of Central Tendency
279
EXTEND
5-9
Graphing Calculator Lab
Mean and Median
A graphing calculator is able to perform operations on large data sets efficiently. You can
use a TI-83/84 Plus graphing calculator to find the mean and median of a set of data.
ACTIVITY
SURVEYS Fifteen eighth graders were surveyed and
asked what was their weekly allowance (in dollars).
The results of the survey are shown at the right.
Find the mean and median allowance.
15
15
5
10
5
5
5
5
10
5
5
5
10
10
10
Step 1 Enter the data.
• Clear any existing lists.
KEYSTROKES:
STAT
ENTER
CLEAR ENTER
• Enter the allowances as L1.
KEYSTROKES:
20 ENTER 10 ENTER … 5 ENTER
Step 2 Find the mean and median.
• Display a list of statistics for the data.
KEYSTROKES:
STAT
The first value,
x, is the mean.
ENTER ENTER
Use the down arrow key to locate “Med.” The
median allowance is $5 and the mean allowance is $8.
EXERCISES
Clear list L1 and find the mean and median of each data set. Round decimal
answers to the nearest hundredth.
1. 6.4, 5.6, 7.3, 1.2, 5.7, 8.9
2. -23, -13, -16, -21, -15, -34, -22
3. 123, 423, 190, 289, 99, 178, 156, 217, 217
4. 8.4, 2.2, -7.3, -5.3, 6.7, -4.3, 5.1, 1.3, -1.1, -3.2, 2.2, 2.9, 1.4, 68
ANALYZE THE RESULTS
5. Look back at the medians found. When is the median a member of the data set?
6. Refer to Exercise 4.
a. Which statistic better represents the data, the mean or median? Explain.
b. Suppose the number 68 should have been 6.8. Recalculate the mean and median. Is
there a significant difference between the first pair of values and the second pair?
c. When there is an error in one of the data values, which statistic is less
likely to be affected? Why?
280 Chapter 5 Rational Numbers
Other calculator keystrokes at pre-alg.com
CH
APTER
5
Study Guide
and Review
Download Vocabulary
Review from pre-alg.com
Key Vocabulary
Be sure the following
Key Concepts are noted
in your Foldable.
…>«ÌiÀÊx\
,>̈œ˜>
ՓLiÀÃ
Key Concepts
Fractions and Decimals
(Lessons 5-1 and 5-2)
• Any number that can be written as a fraction is a
rational number.
• Decimals that are terminating or repeating are
rational numbers.
Multiplying and Dividing Rational
Numbers (Lessons 5-3 and 5-4)
• To multiply fractions, multiply the numerators and
then multiply the denominators.
bar notation (p. 229)
common multiples (p. 257)
dimensional analysis (p. 241)
least common denominator (LCD) (p. 258)
least common multiple (LCM) (pp. 257)
mean (p. 274)
measures of central tendency (p. 274)
median (p. 274)
mixed number (p. 228)
mode (p. 274)
multiple (p. 257)
multiplicative inverse (p. 245)
rational number (p. 234)
reciprocals (p. 245)
repeating decimal (p. 229)
terminating decimal (p. 228)
• To divide by a fraction, multiply by its
multiplicative inverse.
Adding and Subtracting Rational
Numbers (Lessons 5-5 to 5-7)
• To add or subtract fractions with like denominators,
add or subtract the numerators and write the sum
or difference over the denominator.
• To add or subtract fractions with unlike
denominators, rename the fractions with the least
common denominator. Then add or subtract.
Comparing and Applying Rational
Numbers (Lessons 5-7 to 5-9)
• To compare fractions with unlike denominators,
write the fractions using the least common
denominator and compare the numerators.
• To solve an equation, use inverse operations to
isolate the variable.
• The mean, median, and mode can be used to
describe a set of data.
Vocabulary Review at pre-alg.com
Vocabulary Check
Choose the correct term to complete each
sentence.
1. A mixed number is an example of a
(whole, rational) number.
2. The decimal 0.900 is a (terminating,
repeating) decimal.
x is an example of a(n) (algebraic
3. 15
fraction, mixed number).
4. The numbers 12, 15, and 18 are (factors,
multiples) of 3.
5. To add unlike fractions, rename the
fractions using the (LCD, GCF).
6. The product of a number and its
multiplicative inverse is (0, 1).
7. The (mean, median) is the most useful
measure of central tendency when there
are extreme values in the data.
Chapter 5 Study Guide and Review
281
CH
A PT ER
5
Study Guide and Review
Lesson-by-Lesson Review
5–1
Writing Fractions as Decimals
(pp. 228–233)
Write each fraction or mixed number as
a decimal. Use a bar to show a repeating
decimal.
3
8. _
7
9. _
10
15
10
7
= 2 + 0.7 Write _
as a decimal.
10
= 2.7
Add.
20
3
_
13. 8
25
7
12. - _
10
Write as the sum of an integer and
a fraction.
7
7
2_
=2+_
9
10. -_
8
10
2
11. 3 _
3
7
Example 1 Write 2_
as a decimal.
14. CARPENTRY Antoine is cutting a
5
5_
-inch board for a project. Write
16
this length as a decimal.
5–2
Rational Numbers
(pp. 234–238)
Write each decimal as a fraction or
mixed number in simplest form.
Example 2 Write 0.16 as a fraction in
simplest form.
15. 0.23
16. -0.05
16
0.16 = _
18. 4.44
21. -1.7
19. -8.002
22. 0.72
17. 0.125
3
20. 0.
100
4
=_
25
23. 3.
36
0.16 is 16 hundredths.
Simplify. The GCF of 16 and 100 is 4.
24. MUSIC Suzanne practiced playing the
piano for 1.
6 hours. Write the time
Suzanne practiced playing the piano
as a mixed number.
5–3
Multiplying Rational Numbers
(pp. 239–244)
Find each product. Write in simplest
form.
2 _
·1
25. _
3
8
7 _
26. -_
·5
4
27. -8 · -_
5
15 9
1
4
28. 2_
· -_
4
3
29. BACKPACKING A liter of water weighs
1
approximately 2_
pounds. Enrique
5
carries 3.5 liters of water with him.
How heavy is his backpack?
282 Chapter 5 Rational Numbers
1
4
Example 3 Find _
· 3_
. Write in simplest
5
3
form.
1
Rename 3_ as an improper
10
_4 · 3_1 = _4 · _
3
5
3
5
fraction.
3
2
4 _
=_
· 10
5
1
Divide by the GCF, 5.
3
8
2
=_
or 2_
3
3
Multiply. Then simplify.
Mixed Problem Solving
For mixed problem-solving practice,
see page 798.
5–4
Dividing Rational Numbers
(pp. 245–249)
Find each quotient. Write in simplest
form.
6
3
Example 4 Find _
÷_
. Write in simplest
7
4
form.
2
1
÷ -_
30. _
_6 ÷ _3 = _6 · _4
3
6
31. -3_
÷_
5
15
5
2
32. _
÷_
8
3
5
7
11
1
33. 1_
÷ 4_
2
18
7
4
2
Adding and Subtracting Like Fractions
4 3
Divide by the GCF, 3.
1
1
= _ or 1_
8
7
7
Multiply. Then simplify.
(pp. 250–254)
Find each sum or difference. Write in
simplest form.
3
7
Example 5 Find 3_
+ 9_
. Write in
8
8
simplest form.
3
8
+ -_
35. _
3
7
3_
+ 9_
= (3 + 9) +
11
11
1
1
_
37. - + -3_
8
8
19
17
36. _
-_
20
20
3
7
_
38. 8 - 5_
10
10
8
8
39. EXERCISE Samantha is going to walk
5
3
3_
miles today and 2 _
miles
16
16
tomorrow. What is the total distance
she will walk?
5–6
3
6 _
·4
=_
7 3
3
glasses of milk
34. FOOD Pilar drinks 1_
4
each day. At this rate, in how many
days would it take her to drink a total
of 14 glasses?
5–5
7
Multiply by the reciprocal
3 4
of _ , _.
Least Common Multiple
_78 + _38 Add the whole numbers
and fractions separately.
7+3
= 12 + _
The denominators are the
same. Add the numerators.
10
= 12_
Simplify.
8
8
2
1
= 13_
or 13_
8
4
Simplify.
(pp. 257–261)
Find the least common multiple (LCM)
of each pair of numbers or monomials.
Example 6 Replace with <, >, or = to
7
_5 a true statement.
make _
40. 4, 18
The LCD is 32 · 5 or 45. Rewrite the
fractions using the LCD.
41. 24, 20
42. 4a, 6a
Replace each with <, >, or = to make
a true statement.
3
43. _
8
5
_
12
2
44. _
9
4
_
15
5
45. _
20
_1
4
15
9
7
21
_
=_
15
25
_5 = _
9
45
45
25
5
21
7
<_
. So _
<_
.
Since 21 < 25, _
45
45
15
9
46. RADIO A radio station is giving away a
free CD to every 30th caller and a pair
of concert tickets to every 45th caller.
Which caller will be the first to win
both the CD and the concert tickets?
Chapter 5 Study Guide and Review
283
CH
A PT ER
5
5–7
Study Guide and Review
Adding and Subtracting Unlike Fractions
(pp. 263–267)
Find each sum or difference. Write in
simplest form.
5
7
Example 7 Find _
-_
. Write in
9
12
simplest form.
1 _
+5
47. _
3
5
5 _
7 _
_7 - _
=_
· 4 -_
·
3
5
3
1
1
_
50. -2 + 5_
6
3
1
2
49. 5_
- 2_
2
2
1
48. 1_
- -_
6
3
9
12
3
51. COOKING Monica needs 2_
cups of
4
1
flour for a batch of cookies and 3_
cups
3
of flour for a dozen muffins. How many
cups of flour does Monica need?
5–8
Solving Equations with Rational Numbers
2
8
54. 0.2t = 6
The LCD is 32 · 22 or 36.
Rename the fractions
using the LCD.
Subtract like fractions.
Example 8 Solve 1.6x = 8. Check your
solution.
53. x - 1.5 = 1.75
1.6x = 8
Write the equation.
4
55. 2 = -_
n
8
1.6x
_
=_
1.6
1.6
Divide each side by 1.6.
5
1
gallons of
56. PICNIC Luther is mixing 5_
2
orange drink for his class picnic. Every
_3 gallon requires 1 packet of orange
4
drink mix. Write and solve an equation
to determine how many packets Luther
will need.
5–9
3
(pp. 268–272)
Solve each equation. Check your
solution.
3
1
=a+_
52. _
9 4
12
28
15
=_
-_
36
36
13
_
=
36
Measures of Central Tendency
x=5
Simplify.
Check
1.6(x) = 8
Write the equation.
1.6(5) = 8
Replace x with 5.
8 = 8 Simplify.
(pp. 274–279)
Find the mean, median, and mode for
each set of data. Round to the nearest
tenth, if necessary.
Example 9 Find the mean, median, and
mode of 8, 9, 5, 2, 4, and 2. Round to the
nearest tenth.
57. 11, 5, 3, 37, 9, 23, 4, 7
8+9+5+2+4+2
30
mean: __ = _
or 5
58. 3.6, 7.2, 9.0, 5.2, 7.2, 6.5, 3.6
59. MOVIES At the movie theater, six movies
are playing and their lengths are 138,
117, 158, 145, 135, and 120 minutes.
Which measure of central tendency
best represents the data? Justify your
selection and then find the measure of
central tendency.
284 Chapter 5 Rational Numbers
6
median:
2, 2, 4, 5, 8, 9
4+5
_
= 4.5
2
mode:
2
6
Arrange the numbers from least to
greatest.
Find the middle number or the median
of the two middle numbers.
Two is the number that occurs most often.
CH
A PT ER
5
Practice Test
Write each fraction or mixed number as a
decimal. Use a bar to show a repeating decimal.
9
1. _
3. 4_2
2. -_7
20
9
8
17. MULTIPLE CHOICE Use the table to find the
fraction of people who voted for Collins,
Johnson, and Shaw in the election for
freshman class president.
Write each decimal as a fraction or mixed
number in simplest form.
4. 0.24
5. 5.06
6. -2.3
Candidate
2
7. SEWING Allie needs 4 _
yards of lace to finish
3
3
sewing a blanket. She only has _
of that
4
amount. How much lace does Allie have?
Find each product or quotient. Write in
simplest form.
5
6
8. _
· -_
8
3
F _
11
4
1
_
_
9. 4 · 9
7
3
7
4
_
_
10. ÷
9
15
4
1
11. -8_
÷ 2_
9
9
3
Collins
20
Johnson
21
50
Murphy
10
Shaw
100
3
3
G _2
3
H _
5
10
Fraction of
People
7
J _
5
10
Solve each equation. Check your solution.
12. MULTIPLE CHOICE The front bicycle gear has
52 teeth, and the back gear has 20 teeth.
How many revolutions must each gear
make for them to align again as shown?
18. x + 4.3 = 9.8
19. 12 = 0.75x
20. 3.1m = 12.4
4
2
21. p - _
=_
5
3
=_
a
22. -_
5
1
23. y - 2 _
= 1_
6
4
8
5
3
3
24. WEATHER The low temperatures on
March 24 for twelve different cities are
recorded below. Find the mean, median,
and mode. Round to the nearest tenth,
if necessary.
Low Temperatures ( °F )
back gear
front gear
31
29
30
22
A front gear: 10; back gear: 26
45
35
29
30
B front gear: 5; back gear: 13
40
29
31
38
C front gear: 26; back gear: 10
D front gear: 13; back gear: 5
Find each sum or difference. Write in
simplest form.
3
1
+_
13. _
4
15.
3
5
14. - _
+_
16
1
1_
5
7
-_
15
7
16.
14
A mode
C mean
6
B median
D none
3
1
6_
- 2_
4
Chapter Test at pre-alg.com
25. MULTIPLE CHOICE Tanya scored the
following number of points in her last six
basketball games: 8, 6, 11, 9, 20, and 10.
Which measure of data would make
Tanya’s average the greatest?
Chapter 5 Practice Test
285
CH
A PT ER
Standardized Test Practice
5
Cumulative, Chapters 1–5
Read each question. Then fill in the correct
answer on the answer document provided by
your teacher or on a sheet of paper.
1. Anthony has an average of 85 on nine math
quizzes this year. If his teacher drops his
lowest quiz score, a 71, which equation can
be used to find n, Anthony’s new quiz
average?
(85 × 9) - 71
9
(85
×
9) - 71
B n = __
8
(85 - 71) × 9
C n = __
8
(71 × 9) - 85
D n= 8
A n = __
Question 1 Check that your answer satisfies the conditions of
the original problem.
4. The low temperature overnight was -1°F.
Each night for the next four nights, the low
temperature was 7° lower than the previous
night. What was the low temperature during
the last night?
A -29°F
C -22°F
B -27°F
D -8°F
5. Express 0.00000035 in scientific notation.
F 35 × 10-8
G 3.5 × 10-7
H 0.35 × 10-6
J
-3.5 × 107
6. The number of strike-outs the Bronco’s
softball team had in each of the last 10 games
is shown in the table. Which measure of
central tendency would the team use to
convince fans that they had a lot of
strikeouts?
Number of Strikeouts
2. A punch recipe calls for 1.5 quarts of apple
1
cups of lemon-lime soda, 96 ounces
juice, 3 _
7
3
11
9
4
6
8
5
3
2
2
of pineapple juice, 2 quarts of water,
1
cup of lemon juice. Which is the
and _
4
smallest container that will hold all of the
ingredients?
F 6 qt
A mode
B median
C mean
D cannot be determined
G 7 qt
H 8 qt
J
9 qt
7. The expression below describes a pattern of
numbers.
n(n + 2) - 3
1 off the
3. GRIDDABLE Paula has a coupon for 4
total amount of her order when she spends
at least $50 at a department store. By what
decimal can she multiply the cost of her
order to find the amount of discount?
286 Chapter 5 Rational Numbers
If n represents a number’s position in the
sequence, which pattern of numbers was
created by the expression?
F 0, 5, 10, 15, 20, …
H 0, 5, 12, 21, 32, …
G 0, 4, 10, 18, 28, …
J 1, 6, 13, 22, 33, …
Standardized Test Practice at pre-alg.com
Preparing for
Standardized Tests
For test-taking strategies and more practice,
see pages 809–826.
8. Fourteen inches is what part of 1 yard?
11. A certain bacterium measures approximately
0.00000071 centimeter in length. How is this
length expressed in scientific notation?
1
A _
3
5
B _
18
7
C _
18
9
D_
25
A 7.1 × 10-7 cm
C 71 × 10-6 cm
B 71 × 10-7 cm
D 7.1 × 10 cm
12. A contractor has 20 boxes of square patio
tiles. Each box contains 10 square tiles. He
wants to use the tiles to build a patio that is
15 tiles long and 13 tiles wide. Which
procedure can he use to see if he has enough
tiles?
9. There are 4 children in the Gentry family.
2 times Sam’s age. Jenny is
Tim’s age is 2
3
2 years younger than Tim. Beth is 9 years old,
which is 6 years older than Sam. How old is
Jenny?
F Multiply 10 by 15.
G Add 15 and 13 and then multiply by 10.
F 5 years old
H Subtract 13 from 15 and then multiply by 10.
G 6 years old
J Multiply 20 by 10 and then compare to the
product of 15 and 13.
H 7 years old
J 8 years old
Pre-AP
10. GRIDDABLE A survey of 110 people asked
which country they would most like to visit.
The bar graph shows the data. How many
people chose Canada, England, or Australia
as the country they would most like to visit?
Record your answers on a sheet of paper.
Show your work.
13. During seven regular season games, the
Hawks basketball team scored the points
shown in the table below.
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œÕ˜ÌÀÞÊ̜Ê6ˆÃˆÌ¶
a. Find the mean, median,
and mode of these seven
scores.
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Óx
Points
1
10
2
20
3
30
4
40
5
50
6
78
b. During the first playoff
game after the regular
season, the Hawks scored
only 62 points. Find the
mean, median, and mode
of all eight scores.
Óä
£x
£ä
7
73
c. Which of these three
measures of central
tendency—mean, median, or
mode—changed the most as a result of the
playoff score? Explain your answer.
x
À
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>˜
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ä
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Game
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Chapter 5 Standardized Test Practice
287