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Multi-Part Lesson
1-1
Rational Numbers
PART
A
Main Idea
Express rational
numbers as decimals
and decimals as
fractions.
NGSSS
Preparation for
MA.8.A.6.4 Perform
operations on real numbers
(including integer
exponents, radicals,
percents, scientific notation,
absolute value, rational
numbers, and irrational
numbers) using multi-step
and real world problems.
New Vocabulary
rational number
terminating decimal
repeating decimal
B
C
D
Rational Numbers
MARINE LIFE There are over 360
different species of sharks. The
most common shark species found
around Florida are listed below.
Color
Average
Length (feet)
Sharpnose shark
brown to green-gray
3
Bonnethead shark
gray or gray-brown
3
Blacknose shark
green-gray
5
Blacktip shark
blue-gray
6
Spinner shark
gray-bronze
6
Sandbar shark
brown or gray
6
Nurse shark
yellow-brown
7
Scalloped hammerhead shark
gray-brown
8
Lemon shark
yellow-gray
9
Shark Species
1. What fraction of the shark species have an average length
less than 6 feet?
glencoe.com
2. What fraction of the shark species are a shade of blue?
3. What fraction of the shark species are not a shade of gray?
Rational numbers are numbers that can be written as fractions.
8
-7 _
Since -7 can be written as _
, 2 2 can be written as _
, and 9% can
1
3
3
9
2
, -7, 2_
, and 9% are rational numbers. All integers,
be written as _
100
3
fractions, mixed numbers, and percents are rational numbers.
Rational Numbers
Words
Algebra
Rational numbers are
numbers that can be
written as fractions.
a
_
, where a and b are
b
integers and b ≠ 0
Model
Rational Numbers
-
28
Chapter 1 Rational Numbers and Percent
Integers
Whole -
Numbers
-
Bar Notation Bar notation
is often used to indicate
that a digit or group of
digits repeats. The bar is
placed above the repeating
part. To write 8.636363...
−−,
in bar −notation,
write 8.63
−−
not 8.6 or 8.636. To write
0.3444... in bar notation,
−−.
write 0.34−, not 0.34
Any fraction can be expressed as a decimal by dividing the numerator
by the denominator. A terminating decimal, like 0.625, terminates
because the division ends with a remainder of 0. If the division does
not end, sometimes a pattern of digits repeats. A repeating decimal,
−
like 0.3, has a pattern in its digits that repeats without end.
Write a Fraction as a Decimal
Write each fraction or mixed number as a decimal.
_5
_
-1 2
8
_5 means 5 ÷ 8.
8
3
-5
2
-1_
can be rewritten as _
.
3
3
0.625
8 5.000 Divide 5 by 8.
- 48
−−−−
20
-16
−−−
40
-40
____
0
3
a. _
4
Divide 5 by 3 and add a
negative sign.
1.6...
5.0
3
-3
−−−
2.0
-1.8
−−−−
2
2
The mixed number -1_
−3
can be written as -1.6.
13
b. 4 _
25
2
c. -_
9
1
d. 3 _
11
Repeating decimals often occur in real-world situations. However,
they are usually rounded to a certain place-value position.
BASEBALL In a recent season, Tampa Bay Rays fielder Carl
Crawford had 184 hits in 584 at-bats. To the nearest thousandth,
find his batting average.
To find his batting average, divide the number of hits, 184, by the
number of at-bats, 584.
184 µ 584
0.3150684932
Look at the digit to the right of the thousandths place. Since 0 < 5,
round down.
Carl Crawford’s batting average was 0.315.
Real-World Link
The Tampa Bay Rays
were the thirteenth
expansion team in
Major League history.
They played their first
game on March 31,
1998.
e. AUTO RACING In a recent season, NASCAR driver Jimmie
Johnson won 6 of the 36 total races held. To the nearest
thousandth, find the part of races he won.
Lesson 1-1 Rational Numbers
29
Terminating and repeating decimals are also rational numbers
because you can write them as fractions.
Write Decimals as Fractions
Write 0.45 as a fraction.
45
0.45 is 45 hundredths.
0.45 = _
100
9
= _
20
Simplify.
−
ALGEBRA Write 0.5 as a fraction in simplest form.
−
Assign a variable to the value 0.5. Let N = 0.555... . Then perform
operations on N to determine its fractional value.
N = 0.555...
10(N) = 10(0.555...) Multiply each side by 10 because 1 digit repeats.
10N = 5.555...
- N = 0.555...
−−−−−−−−−−−−
9N = 5
5
N =_
9
Multiplying by 10 moves the decimal point 1 place to the right.
Subtract N = 0.555... to eliminate the repeating part.
Simplify.
Divide each side by 9.
−
5
The decimal 0.5 can be written as _
.
9
Write each decimal as a fraction or mixed number in simplest form.
−
−−
g. 8.75
h. 0.27
i. -0.4
f. -0.14
Examples 1 and 2
(p. 29)
Write each fraction or mixed number as a decimal.
4
1. _
5
5
4. _
9
Example 3
(p. 29)
Examples 4 and 5
(p. 30)
16
5
5. 4_
6
29
3. -1_
40
5
6. -7_
33
7. GOLF In a recent year, Tiger Woods won 7 of the 16 tournaments he
entered. To the nearest thousandth, find his winning average.
Write each decimal as a fraction or mixed number in simplest form.
8. 0.6
−
11. -0.5
30
9
2. _
Chapter 1 Rational Numbers and Percent
9
0.32
−
12. -3.8
10. -1.55
−−
13. 2.15
= Step-by-Step Solutions begin on page R1.
Extra Practice begins on page EP2.
Examples 1 and 2
(p. 29)
Write each fraction or mixed number as a decimal.
1
14. _
2
15. _
5
4
7
18. -_
16
4
22. _
33
Example 3
(p. 29)
5
19. -_
32
6
23. -_
11
7
16. _
33
17. _
80
1
20. 2_
8
40
5
21. 5_
16
13
24. -6_
26. FAMILIES The table shows statistics about
the students at Carter Junior High.
Fraction of
Students
Three
_1
15
_1
3
_5
12
_1
Four or more
1
_
None
b. Find the decimal equivalent for the number
of students with three siblings.
c. Write the fraction of students with one sibling
as a decimal. Round to the nearest thousandth.
d. Write the fraction of students with two siblings
as a decimal. Round to the nearest thousandth.
(p. 30)
45
Number of
Siblings
a. Express the fraction of students with no
siblings as a decimal.
Examples 4 and 5
8
25. -7_
15
One
Two
6
60
Write each decimal as a fraction or mixed number in simplest form.
27. -0.4
−
31. 0.2
28. 0.5
−−
32. -0.45
29. 5.55
−−
33. -3.09
WEATHER Write the rainfall amount for each
day as a fraction or mixed number.
30. –7.32
−
34. 2.7
Day
35 Friday
36. Saturday
37. Sunday
Rainfall
(in.)
mL
5
5
4
4
3
Friday
0.08
3
Saturday
2.4
2
2
0.035
1
1
Sunday
0
B
MEASUREMENT Write the length of each insect as a
fraction or mixed number and as a decimal.
38.
39.
0 in.
1
0 in.
40. FROZEN YOGURT The table shows three popular
flavors according to the results of a survey.
What is the decimal value of those who liked
vanilla, chocolate, or strawberry? Round to
the nearest hundredth.
1
Flavor
Vanilla
Chocolate
Strawberry
Fraction
_3
10
_1
11
_1
18
Lesson 1-1 Rational Numbers
31
41. OPEN ENDED Give an example of a repeating decimal where two digits
repeat. Explain why your number is a rational number.
C
42. Which One Doesn’t Belong? Identify the fraction that does not belong with
the other three. Explain your reasoning.
_1
_1
_1
_4
8
4
6
5
43. CHALLENGE Explain why any rational number is either a terminating or
repeating decimal.
−
−−
44.
Compare 0.1 and 0.1, 0.13 and 0.13, and 0.157 and
−−−
0.157 when written as fractions. Make a conjecture about expressing
repeating decimals like these as fractions.
NGSSS Practice
MA.8.A.6.4
45. Which of the following is equivalent to
the fraction below?
13
_
5
46.
A. 2.4
C. 2.55
B. 2.45
D. 2.6
Free Throws
Made
Free Throws
Attempted
Felisa
18
20
Morgan
13
24
Yasmine
15
22
Gail
10
14
Part A Write the fraction of free throws
made in simplest form for each player.
Part B Write each fraction from Part A
as a decimal. Round to the nearest
thousandth if necessary.
Part C Which player has the greatest
fraction of free throws made?
32
-6
F. _
H. 18%
G. 15
I.
11
EXTENDED RESPONSE The table
shows the number of free throws
each player made during the last
basketball season.
Player
47. Which of the following is NOT an
example of a rational number?
Chapter 1 Rational Numbers and Percent
4.23242526. . .
48. While shopping for a new pair of jeans,
Janet notices the sign below.
SALE!
8cca\Xejfek_`jiXZb
(
Xi\f]]k_\
*
fi`^`eXcgi`Z\
(Regularly priced $29.99)
Which of the following expressions can
be used to estimate the total discount on
a pair of jeans?
A. 0.033 × $30
B. 0.33 × $30
C. 1.3 × $30
D. 33.3 × $30
Multi-Part Lesson
1-1
Rational Numbers
PART
Main Idea
Add and subtract
rational numbers.
NGSSS
MA.8.A.6.4 Perform
operations on real
numbers (including integer
exponents, radicals,
percents, scientific notation,
absolute value, rational
numbers, and irrational
numbers) using multi-step
and real world problems.
New Vocabulary
like fractions
unlike fractions
A
B
C
D
Add and Subtract Rational
Numbers
APPLES The amount of apples
Oleta’s family picked is shown.
Person
Amount Picked
(baskets)
1. What is the sum of the
whole-number parts of
the baskets of apples?
Oleta
14
Mr. Davis
2
1
2. How many _
baskets
Mrs. Davis
14
4
are there?
1
3
2
4
Alvin
3. Can you combine all of the
apples into a bushel that
holds five baskets? Explain.
Fractions that have the same denominators are called like fractions.
glencoe.com
Add and Subtract Like Fractions
Words
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Examples
Numbers
Algebra
3
1
4
_
+_=_
a
a+b
b
_
+ _ = _, where c ≠ 0
3
7
4
1
_
- _ = _ or _
a
a-b
b
_
- _ = _, where c ≠ 0
5
8
5
8
c
5
8
c
2
c
c
c
c
You can use the rules for adding integers to determine the sign of the
sum of any two signed numbers.
Add Like Fractions
_ ( _)
8
8
5 + (-7)
_5 + -_7 = _
)
(
8
8
8
Find 5 + - 7 . Write in simplest form.
Add the numerators.
The denominators are the same.
-2
1
=_
or -_
Simplify.
8
5
7
a. _ + _
9
9
4
5
1
b. -_
+_
9
9
( _6 )
-1
c. _ + - 5
6
Lesson 1-1 Rational Numbers
33
Subtract Like Fractions
_ _
Find - 8 - 7 . Write in simplest form.
9
9
8
8
7
7
-_
-_
= -_
+ -_
9
9
9
9
-8
+
(-7)
=_
9
-15
2
= _ or -1_
9
3
( )
-15
2
Rename _ as -1_ or -1_
.
9
6
9
3
3
5
e. _ - _
3
4
d. -_
-_
5
Subtract the numerators by adding the opposite of 7.
8
5
5
4
f. _ - -_
( 7)
7
8
Fractions with unlike denominators are called unlike fractions. To
add or subtract unlike fractions, rename the fractions using prime
factors to find the least common denominator. Then add or subtract
as with like fractions.
Add and Subtract Unlike Fractions
Add or subtract. Write in simplest form.
_1 + (-_2 )
3
4
_1 + -_2 = _1 · _3 + -_2 · _4
4
4
4 3
3
3
3
8
= _ + -_
12
12
3 + (-8)
=_
12
5
= -_
12
( )
Least Common
Denominator (LCD)
To find the LCD of two
fractions, write the prime
factorization of each
denominator. Identify
common prime factors.
9=3×3
15 = 3 × 5
The LCD is the product of
each common prime
factor and any remaining
factors.
LCD = 3 × 3 × 5 or 45
( )
( )
Rename using the LCD.
Add the numerators.
Simplify.
_ ( _)
9
15
-8 - - 7
8
7
8
7
-_
- -_
= -_
+_
( 15 )
9
6
( 2)
Chapter 1 Rational Numbers and Percent
To subtract -_, add _.
7
15
15
9
5
8
7 _
= -_ · _ + _
·3
9 5
15 3
40
21
= -_
+_
45
45
-40
+
21
=_
45
19
= -_
45
5
1
g. -_
+ -_
34
The LCD is 3 · 4 or 12.
The LCD is 9 · 5 or 45.
Rename using the LCD.
Add the numerators.
Simplify.
1
3
h. _ + _
14
7
15
4
3
5
i. -_
+_
6
10
Add and Subtract Mixed Numbers
Add or subtract. Write in simplest form.
_ _
57 + 84
9
9
7
4
7
4
5_
+ 8_
= (5 + 8) + _
+_
9
9
9
9
7
+
4
= 13 + _
9
11
2
= 13_
or 14_
9
9
(
)
Add the whole numbers
and fractions separately.
Add the numerators.
11
_
= 1_2
9
9
_ _
23 - 31
4
3
3
10
1
11
2_
- 3_
=_
-_
4
3
Write as improper fractions.
4
3
33
40
=_
-_
12
12
33 - 40
-7
=_
or _
12
12
5
4
j. 6_
+ 3_
11 _
33
10 4
40
_
· 3 =_
and _ · _ = _
4
3
3
12
12
Subtract the numerators. Then simplify.
5
3
k. 9_
- 3_
7
7
5
1
m. 3_
+ 2_
6
4
4
5
2
l. -8_
+ -6_
(
(
8
8
7
2
n. 5_
- 4_
9
3
)
)
9
9
3
2
o. -3_
+ -8_
5
3
Sometimes you need to regroup before you can subtract.
Real-World Link
Dressage is an
Olympic sport where
a horse and rider
must go through a
series of tests. These
tests require very
controlled movements
for the horse and
require years of
training.
ANIMALS Horses are measured by a unit called a handbreadth or
1
hand. How much taller is a horse that is 14_
hands tall than one
4
3
that is 12_
hands tall?
4
1
14_
→
5
13_
4
4
3
_
_
-12 → -12 3
4
4
2
1
1_
or 1_
2
4
14_ = 13 + 1 + _ or 13_
1
4
5
4
1
4
Subtract the whole numbers
and fractions separately.
1
The first horse is 1_
hands taller.
2
3
p. BAKING A recipe for chocolate cookies calls for 2_
cups of flour.
4
1
If Alexis has 1_
cups of flour, how much more will she need?
4
Lesson 1-1 Rational Numbers
35
Examples 1–6
(pp. 33–35)
Add or subtract. Write in simplest form.
2
4
1. _
+ -_
( 5)
3
1
5. _
+ (-_
4
6)
5
9
7
3. -_
-_
5
1
6. -_
+_
3
7
7. _
-_
4
4
10
2
8
4
2
9. 5_
- 2_
9
3
1
2. -_
+_
8
3
2
10. -1_
+ -2_
9
(
7
10
7
)
4
5
2
11 -3_ + 1_
5
6
3
7
4. _ - _
8
8
7
2
8. _ - _
13
9
5
1
12. 3_
- 1_
8
3
Example 7
13. HOMEWORK Venus wrote a report for her middle school history class in
(p. 35)
1
2_
hours. Her sister Tia is in high school, and she wrote a history paper
4
3
in 4_
hours. How much longer did it take Tia to write her paper?
4
= Step-by-Step Solutions begin on page R1.
Extra Practice begins on page EP2.
Examples 1–6
(pp. 33–35)
Add or subtract. Write in simplest form.
4
1
14. -_
+_
3
2
15. -_
+ -_
5
7
16. -_
+_
8
5
17. _ + -_
3
4
18. -_
-_
15
9
19. _
-_
1
7
20. _
-_
2
8
21. _ - _
1
7
22. _
+ -_
5
3
23. -_
+_
6
1
24. -_
+ -_
5
3
25. _ + -_
1
7
26. _
-_
4
2
27. _
- -_
9
9
5
7
16
5
(
4
3
12
)
12
12
16
9
12
7
( 15 )
30. ELECTIONS The table shows the fraction of
students who voted for Josh or Chuan in the
election for class president. What fraction of
the students voted for either Josh or Chuan?
9
( 8)
7
12
29. -_
- (-_
25 )
15
9
Class President
3
31. DOGS Omar feeds his dog _
cup dog food in the
4
3
2
morning, _
cup in the afternoon, and _
cup in
3
( 9)
9
12
( 2)
3
2
28. -_
- (-_
9
11 )
6
8
5
8
( 7)
Candidate
Fraction of
Students
Josh
_4
9
_2
5
Chuan
4
the evening. How many cups does he feed the
dog altogether?
Add or subtract. Write in simplest form.
5
5
32. 3_
+ 7_
8
9
1
33. 8_
+ -2_
8
5
5
- 3_
36. -1_
6
(
10
6
2
37. 7 - 5_
5
10
)
1
1
34. 3_
+ -8_
( 2)
3
1
38. 8_
- (-6_
7
2)
5
5
2
35. -15_
+ 11_
8
3
5
1
39. -8_
- 4_
3
6
Example 7
1
40. HOME IMPROVEMENT Andrew has 42_
feet of molding to use as borders
(p. 35)
2
around the windows of his house. If he uses 23_
feet of the molding on
3
3
the front windows, how much remains for the back windows?
6
41 WEATHER One year, Brady’s hometown of Powell received about 42_
10
3
inches of snow. The following year only 14 _
inches of snow fell. What is
10
the difference in the amount of snow between the two years?
36
Chapter 1 Rational Numbers and Percent
B
Simplify each expression.
3
4
1
42. -7_
+ 3_
- 2_
5
5
3
1
43. -8_
- -3 _
+ 6_
( 5)
5
(
8
6
)
4
ALGEBRA Evaluate each expression for the given values.
1
1
and b = -2_
44. a - b if a = 5 _
3
3
3
7
_
46. c - d if c = - and d = -12_
8
4
5
1
45. x + y if x = -_
and y = -_
12
12
5
5
_
47. r - s if r = -_ and s = 2
6
8
1
48. ACTIVITIES Tamara played a computer game for 1_
hours, studied for
4
1
_
2.25 hours, and did some chores for hour. How long did it take Tamara
2
Adding Rational
Numbers It is helpful
to write all rational
numbers in the same
form before adding.
In Exercise 48, write
all the numbers as
fractions or decimals.
Then add.
to do these things?
49. HOMEWORK Rob recorded the amount of time he spent on
homework last week. Express his total time for the week
in terms of hours and minutes.
5
50. PLUMBING A plumber has a pipe that is 64_
inches
8
7
long. The plumber cuts 2_
inches off the end of the
8
3
pipe, then cuts off an additional 1_
inches. How long is
8
the remaining pipe after the last cut is made?
MEASUREMENT Find the missing measure for each figure.
51.
52.
3
6 4 ft
1
1
Time
Mon
2_ h
Tue
2_ h
Wed
1_ h
Thu
2_ h
Fri
1_ h
1
6
1
2
3
4
5
12
1
4
x in.
1
4 3 ft
Day
7
9 2 in.
11 8 in.
4 3 ft
1
13 4 in.
x ft
11
perimeter = 17 12 ft
3
perimeter = 40 4 in.
53. FIND THE DATA Refer to the Data File on pages 2–5. Choose some data
and write a real-world problem in which you would add or subtract
fractions or mixed numbers.
C
54. OPEN ENDED Write a subtraction problem using unlike fractions with a
least common denominator of 12. Find the difference.
55. NUMBER SENSE Without doing the computation, determine whether
_4 + _5 is greater than, less than, or equal to 1. Explain.
7
9
56. CHALLENGE Suppose a bucket is placed under two faucets. If one faucet is
turned on alone, the bucket will be filled in 5 minutes. If the other faucet
is turned on alone, the bucket will be filled in 3 minutes. Write the
fraction of the bucket that will be filled in 1 minute if both faucets are
turned on.
57.
Write a real-world situation that can be solved by adding
or subtracting mixed numbers. Then solve the problem.
Lesson 1-1 Rational Numbers
37
NGSSS Practice
MA.6.A.1.2, MA.8.A.6.4
58. Use the figure shown below.
1
cups
59. A recipe for snack mix contains 2_
3
1
of mixed nuts, 3_
cups of granola, and
2
1
13
2
23
_3 cup of raisins. What is the total
4
1
13
amount of snack mix?
2
F. 5_
c
2
H. 6_
c
3
7
G. 5_
c
12
What is the length of the segment
connecting the centers of the two
smaller circles?
3 _
denominator to simplify _
– 2?
3
2
B. 5_
units
3
1
C. 5_
units
3
5
D. 4_
units
3
A.
B.
C.
D.
Write each fraction or mixed number as a decimal.
5
62. _
20
1
65. 2_
4
I.
60. Which of the following shows the next
step using the least common
1
A. 6_
units
14
61. _
7
66. -3_
9
4
_3 × _5 - _2 × _5
5
3
5
4
( ) ( )
(_34 × _66 ) - (_23 × _55 )
(_34 × _33 ) - (_23 × _44 )
(_34 × _44 ) -( _23 × _33 )
(Lesson 1-1A)
3
64. _
9
63. _
6
3
7
6_
c
12
8
11
7
67. 4_
12
8
68. 1_
10
69. HOCKEY The sheet of ice that covers a hockey rink is created in two layers.
1
First, an _
-inch layer of ice is made for the lines to be painted on. Then,
8
6
-inch layer of ice is added on top of the painted layer for a total thickness
a_
8
7
-inch. Write the total thickness of the ice as a decimal. (Lesson 1-1A)
of _
8
Write each decimal as a fraction or mixed number in simplest form.
(Lesson 1-1A)
70. 0.25
71. 1.6
72. -4.35
73. 0.94
−
74. 1.6
−
75. -2.2
−
76. 0.7
−−
77. 4.65
78. ELEVATORS In one hour, an elevator traveled up 5 floors, down 2 floors,
up 8 floors, down 6 floors, up 11 floors, and down 14 floors. If the elevator
started on the seventh floor, on which floor is it now? (Lessons 0-3 and 0-4)
38
Chapter 1 Rational Numbers and Percent
3
Multi-Part Lesson
1-1
Rational Numbers
PART
Main Idea
Multiply rational
numbers.
NGSSS
MA.8.A.6.4 Perform
operations on real
numbers (including integer
exponents, radicals,
percents, scientific notation,
absolute value, rational
numbers, and irrational
numbers) using multi-step
and real world problems.
New Vocabulary
dimensional analysis
A
B
C
D
Multiply Rational Numbers
You can use an area model to find _ of _. The model
1
2
also represents the product of _ and _.
1
2
3
4
3
4
Step 1 Draw a rectangle with four columns.
3
4
Step 2 Divide the rectangle into two rows.
3
1
Step 3 Shade a rectangle that is _ unit by _ unit
4
2
1
2
blue.
The shaded area represents _ of _.
3
4
3
1
_
_
1. What is the product of and ?
4
2
1
2
2. Use an area model to find each product.
glencoe.com
3 1
a. _ · _
2 _
b. _
·2
1 3
c. _ · _
2 4
d. _ · _
4
4
5
2
3
3
5
5
3. What is the relationship between the numerators of the factors
and the numerator of the product?
4. What is the relationship between the denominators of the factors
and the denominator of the product?
The area model suggests the following rule for multiplying fractions.
Multiply Fractions
Words
Examples
To multiply fractions, multiply the numerators and multiply
the denominators.
Numbers
Algebra
8
2 _
4
_
· =_
c
ac
a _
_
· = _, where b and d ≠ 0.
3
5
15
b
d
bd
You can use the rules for multiplying integers to determine the sign of
the product of any two signed numbers.
Lesson 1-1 Rational Numbers
39
Multiply Fractions and Mixed Numbers
Multiply. Write in simplest form.
Negative Fractions
5
-65 , -5
, and -6
are all
6
equivalent fractions.
__
_
_ _
-5 · 3
6
8
1
3
-5 _
5 _
-_
· 3 =_
·
6
8
6/
2
Divide 6 and 3 by their GCF, 3.
8
-5 · 1
5
=_
or -_
2·8
16
41 · 22
2
3
9 _
1
2
4_ · 2_
=_
·8
2
3
2 3
_ _
3
2 _
Rename 4_ as _ and 2_
as .
1
2
9
2
8
3
3
4
9 8
=_·_
2
3
1
1
Divide out common factors.
3·4
12
=_
or _
1·1
Multiply. Then simplify.
1
1 2
a. _ · _
4
Multiply. Then simplify. The fractions have different
signs, so the product is negative.
1
1
c. 2_
· 1_
1
b. -_
( 2 ) (-_67 )
3
6
5
Recall that probability can be expressed as a fraction. Independent
events are two or more events in which the outcome of one does not
affect the outcome of the other. For example, rolling a number cube
and spinning a spinner have no affect on each other. To find the
probability of independent events, multiply the probability of the first
event by the probability of the second event.
Independent Events
The spinner at the right is spun,
and a coin is tossed. What is the
probability of spinning an odd
number and tossing tails?
Probability of a
Simple Event
The probability of rolling
an even number on a
number cube is
P(even)
8
1
P(tossing tails) = _
2
6
4
1
P(spinning an odd number) = _
or _
8
1
7
3
5
4
2
2
1 _
1
· 1 or _
P(odd and tails) = _
2
2
4
even numbers
= __
total possible numbers
3
1
=_
or _
6
2
Refer to the situation above to find each probability.
d. P(less than 4 and heads)
40
Chapter 1 Rational Numbers and Percent
e. P(prime and tails)
Dimensional analysis is the process of including units of
measurement when you compute.
AIRCRAFT Refer to the information at the left. Suppose a VH-71
helicopter is traveling at its cruising speed. How far will it
_
travel in 1 3 hours?
4
Real-World Link
Marine One is the
helicopter used to
transport the President
and Vice President.
The latest model is
the VH-71, which has
a cruising speed of
172 miles per hour.
Words
Distance equals the rate multiplied by the time.
Variable
Let d represent the distance.
Equation
d = 172 miles per hour · 1 hours
_3
4
172 miles
3
d=_
· 1_
hours
Write the equation.
4
1 hour
172
miles
hours
7
d=_·_·_
4
1
1 hour
1_ = _
3
4
7
4
43
172 miles _
hours
d=_
· 7 ·_
1 hour
4
1
Divide by common factors and units.
1
d = 301 miles
3
hours.
A VH-71 will travel 301 miles in 1_
4
f. AIRCRAFT The VH-71 has 200 square feet of cabin space. What is
the size of its cabin space in square yards? (Hint: 1 square yard =
9 square feet)
Examples 1 and 2
(p. 40)
Multiply. Write in simplest form.
3 _
1. _
·5
5 7
-12
4. _
13
4 _
2. _
·3
5
2
-_
3
( )( )
Example 3
(p. 40)
1
2
5. 2_
· 1_
2
5
9
(p. 41)
1 _
3. -_
·4
9
3
7
6. -6_ · 1_
9
4
8
A number cube is rolled and a marble is selected from
the bag shown. Find each probability.
7. P(even and blue)
Example 4
8
P(greater than 2 and yellow)
8. P(3 and red)
10. P(odd and green)
5
11. FRUIT Terrence bought 2_
pounds of grapes that cost $2
8
per pound. What was the total cost of the grapes? Use
dimensional analysis to check the reasonableness of the
answer.
Lesson 1-1 Rational Numbers
41
= Step-by-Step Solutions begin on page R1.
Extra Practice begins on page EP2.
Examples 1 and 2
(p. 40)
Multiply. Write in simplest form.
1 _
12. _
·4
12
7
9 _
16. -_
·2
10 3
1 _
20. 3_
·1
3 4
Example 3
(p. 40)
3 _
13. _
·1
9
15
12 _
17. -_
25 32
1
1
21. 4_
· 3_
3
4
16
( )
9 2
15. _ · _
5 _
14. _
·4
10
8
5
3
1
18. -_
-_
5
3
3
2
22. -3_ · -_
8
3
4
1
19. -_
-_
( )( )
( )
The spinner at the right is spun once and
a coin is tossed. Find each probability.
24. P(even and heads)
25. P(less than 8 and heads)
9
8
7
( 7 )( 20 )
5
4
23. -_
· -1_
5)
6 (
10 1
26. P(composite and tails)
3
6 5
2
3
4
27. P(factor of 12 and tails)
Example 4
(p. 41)
Solve each problem. Use dimensional analysis to check the reasonableness
of the answer.
3
28. BAKING A recipe calls for _
cup of sugar per batch of cookies. If Gabe
4
wants to make 6 batches of cookies, how many cups of sugar does
he need?
29. POPULATION Population density measures how many people live
within a certain area. In a certain city, there are about 150,000 people per
1
square mile. How many people live in an area of 2_
square miles?
_
4
_
_
_
8
1
2
2
B ALGEBRA Evaluate each expression if r = , s = , t = , and v = - .
4
31. rt
30. rs
5
9
33 rtv
32. stv
34. GEOGRAPHY There are about 57 million
square miles of land on Earth covering
seven continents.
a. What is the approximate land area
of Europe?
Approximate Fraction
of Earth’s Landmass
Continent
_1
Africa
5
Antarctica
9
_
b. What is the approximate land area
of Asia?
Asia
3
_
3
c. Only about _
of Australia’s land area
Australia
11
_
is able to support agriculture. What
fraction of Earth’s land is this?
Europe
7
_
North America
33
_
South America
3
_
10
100
10
200
100
200
25
Find each product. Write in simplest form.
3 _
1
35. _
· -_
·4
3
( 8)
5
1
_
38. 10 · 3.78 ·
5
42
3
Chapter 1 Rational Numbers and Percent
5
2 _
36. -_
· 1 · -_
5
2
6
−
2
_
39. - · 0.3
( )
9
( )
1
1
37. 3_
· 1_
·5
3
2
7
40. -_
· (-2.375)
16
C
1
1
41. FIND THE ERROR Danielle is finding 2_
· 3_
.
2
4
Find her mistake and correct it.
1
1
1 _
2_
• 3_
=2•3+_
•1
2
4
2 4
1
=6+_
8
1
= 6_
8
1
and less
42. OPEN ENDED Select two fractions with a product greater than _
2
than 1. Use a number line to justify your answer.
3
43. CHALLENGE Find the missing fraction. _
·
4
14
1
7
1
Explain why the product of _
and _
is less than _
.
2
8
2
44.
NGSSS Practice
9
=_
MA.6.A.1.2, MA.8.A.6.4
46. Find the area of the parallelogram. Use
the formula A = bh.
45. A whole number greater than one is
multiplied by a positive fraction less
than one. The product is always
3
in.
4
A. greater than the whole number.
B. between the fraction and the whole
number.
3
2 5 in.
C. less than the fraction.
5 2
F. _
in
19 2
H. 1_
in
D. all of the above.
3
G. 2_
in2
I.
6
7
20
10
Add or subtract. Write in simplest form.
1
1
47. _
+_
9
7
1
48. _
-_
8
6
_4 in2
5
(Lesson 1-1B)
1
4
49. -5_
- 6_
2
5
1
2
50. 2_
+ 5_
2
3
BIOLOGY Write the weight of each animal as a fraction
or mixed number. (Lesson 1-1B)
51. queen bee
52. hummingbird
Weight
(ounces)
0.004
Queen Bee
Hummingbird 0.11
3.5
Hamster
Animal
53. hamster
Divide.
(Lesson 0-5)
54. 51 ÷ (-17)
55. -81 ÷ (-3)
56. -92 ÷ 4
Lesson 1-1 Rational Numbers
43
Multi-Part Lesson
1-1
Rational Numbers
PART
Main Idea
Divide rational numbers.
NGSSS
MA.8.A.6.4 Perform
operations on real
numbers (including integer
exponents, radicals,
percents, scientific notation,
absolute value, rational
numbers, and irrational
numbers) using multi-step
and real world problems.
New Vocabulary
multiplicative inverses
reciprocals
A
B
D
C
Divide Rational Numbers
ANIMALS An antelope is one of the
fastest animals on Earth. It can run
about 60 miles per hour. A squirrel
runs one fifth of that speed.
1. Find the value of 60 ÷ 5.
1
2. Find the value of 60 × _
.
5
3. Compare the values of 60 ÷ 5
1
and 60 × _
.
5
4. What can you conclude about
the relationship between dividing
1
by 5 and multiplying by _
?
5
glencoe.com
Two numbers with a product of 1 are multiplicative inverses, or
1
reciprocals, of each other. For example, 5 and _
are multiplicative
5
1
inverses because 5 · _
= 1.
5
Inverse Property of Multiplication
Words
The product of a number and its multiplicative inverse is 1.
Examples
Numbers
Algebra
3 _
4
_
· =1
a _
b
_
· = 1, where a and b ≠ 0
4
3
b
a
Find a Multiplicative Inverse
_
Write the multiplicative inverse of -5 2 .
3
2
17
2
_
_
_
Write -5 as an improper fraction.
-5 = 3
3
3
3
3
2
17
Since -_
-_
= 1, the multiplicative inverse of -5_
is -_
.
3
3
17
17
(
)
Write the multiplicative inverse of each number.
1
a. -2_
3
44
Chapter 1 Rational Numbers and Percent
5
b. -_
8
c. 7
Complex Fractions
Recall that a fraction
bar represents division.
So,
_a
_a ÷ _c = __bc .
b
d
d
Multiplicative inverses are used in division. Consider _a ÷ _c ,
b
d
which can be written as a fraction.
_a
_a · _d
_c
_c · _d
d c
_a · _d
b c
_
Multiply the numerator and denominator
d
c
by _c , the multiplicative inverse of _.
b c
_b = _
d
=
d
_c · _d = 1
c
d
1
d
= _a · _
c
b
d
.
Therefore, _a ÷ _c = _a · _
b
d
c
b
Divide Fractions
Words
To divide by a fraction, multiply by its multiplicative inverse.
Symbols
Numbers
Algebra
3
2
2 4
_
÷_=_·_
5
4
5 3
c
a
a d
_
÷ _ = _ · _, where b, c and d ≠ 0
d
b
b c
Divide Fractions and Mixed Numbers
Divide. Write in simplest form.
_ _
-4 ÷ 6
7
5
6
4 _
4
-_
÷_
=-_
·7
7
5 6
5
The multiplicative inverse of _ is _.
6
7
7
6
2
4 7
= -_ · _
5
Divide -4 and 6 by their GCF, 2.
/
6
3
14
= -_
Multiply.
15
_ ( _)
3
2
4 2 ÷ -3 1
2
1
14
7
4_
÷ -3 _
=_
÷ -_
3
2
( )
( )
3
2
14
2
=_
· -_
3
7
2
14
2
= -_
· -_
3
( 7)
2
4_
= _, -3_ = -_
3
14
3
7
2
1
2
The multiplication inverse of -_ is -_
.
7
7
2
2
Divide 14 and 7 by their GCF, 7.
1
1
4
= -_
or -1_
Multiply.
Dividing By a Whole Number
When dividing by a whole
number, rename it as an
improper fraction first.
Then multiply by its
reciprocal.
3
3
3
1
d. _ ÷ _
4
2
7
1
e. -_
÷_
4
8
3
1
f. 2_
÷ -2_
4
(
5
)
1
g. -1_
÷ 12
2
Lesson 1-1 Rational Numbers
45
1
13
7
13
CRAFTS Lina’s class is making flags for the
school’s International Day celebration. She
1
1
feet of paper for the blue portion on
needs 1_
1 6 ft
6
each flag. If the class has a 21-foot roll of blue
paper, how many flags can she make?
1
Divide 21 by 1_
.
6
1
21
7
=_
÷_
21 ÷ 1_
6
1
6
3
21
1
1
6
7
6
Multiply by the multiplicative inverse of _, which is _.
7
6
Divide 7 and 21 by their GCF, 7.
7
6
21 · _
=_
1
Write 21 as _. Write 1_ as _.
7
6
1
Real-World Link
The width of the blue
portion of the Stars and
7
Stripes is _ of the
18
=_
or 18
1
Simplify.
Lina’s class can make 18 flags using the 21-foot roll of paper.
13
width of the entire flag.
The width of each
1
stripe is _ of the
1
h. LUMBER How many 1_
-inch-thick boards are in a stack that is
2
36 inches tall?
13
width of the entire flag.
Example 1
(p. 44)
Write the multiplicative inverse of each number.
5
1. _
Examples 2 and 3
(p. 45)
(p. 46)
4
Divide. Write in simplest form.
3
2
4. _
÷_
3
4
4
8. _ ÷ 8
5
Example 4
3
3. -2_
2. -12
7
5
1
5. _
÷_
8
2
9
9. _ ÷ 3
10
3
9
6. _
÷ -_
( 10 )
5
2
10. -5_
÷ (-4_
6
3)
8
7
7
7. -_
÷ -_
( )
8
16
5
7
11 -3_ ÷ 6_
6
12
12. BIRDS The smallest owl found in the United
1
States is the Elf Owl, which weighs 1_
ounces.
2
One of the largest owls is the Eurasian Eagle
Owl, which weighs nearly 10 pounds or
156 ounces.
The Eurasian Eagle Owl
is how many times as
heavy as the Elf Owl?
Elf Owl
46
Chapter 1 Rational Numbers and Percent
Eurasian Eagle Owl
= Step-by-Step Solutions begin on page R1.
Extra Practice begins on page EP2.
Example 1
(p. 44)
Write the multiplicative inverse of each number.
7
13. -_
5
14. -_
9
16. 18
Example 2 and 3
(p. 45)
3
2
÷_
19. _
3
2
20. _
÷_
8
2
1
22. _ ÷ _
5
2
21. _
÷_
8
3
3
2
24. _ ÷ -_
10
3
9
28. _
÷3
16
1
1
32. 7_
÷ 2_
2
10
4
3
4
23. -_
÷_
5
4
2
27. _
÷4
5
3
1
31. 3_
÷ 2_
2
4
(p. 46)
1
18. 4_
Divide. Write in simplest form.
5
Example 4
15. 15
8
2
_
17. 3
5
3
( )
6
10
5
7
26. -_ ÷ -_
6
12
6
30. _ ÷ 4
7
3
1
34. 10_
÷ -_
5
15
5
5
2
25. -_
÷ -_
9
3
4
29. _
÷6
5
1
2
33. -12_
÷ 4_
3
4
( )
( )
35. HUMAN BODY The table shows the
composition of a healthy adult male’s
body. Examples of body cell mass are
muscle, body organs, and blood.
Examples of supporting tissue are
blood, plasma, and bones.
( )
Composition of Human Body
Component
Body Cell Mass
Supporting Tissue
a. How many times more of a healthy
adult male’s body weight is made up
of body cell mass than body fat?
Body Fat
Fraction of
Body Weight
11
_
20
_3
10
_3
20
b. How many times more of a healthy adult male’s body weight is made
up of body cell mass than supporting tissue?
1
36. PAINTING It took 3 people 2_
hours to paint a large room. How long
2
would it take 5 people to paint a similar room?
B
Real-World Link
99% of the mass of
the human body is
made up of six
elements: oxygen,
carbon, hydrogen,
nitrogen, calcium, and
phosphorus.
37 BIOLOGY How many of the
small hummingbirds need
to be placed end-to-end to
have the same length as the
large hummingbird?
1
5 2 cm
22 cm
38. GEOMETRY The circumference C, or distance around a
44
circle, can be approximated using the formula C = _
r,
7
where r is the radius of the circle. What is the radius of
the circle at the right? Round to the nearest tenth.
r
C = 53.2 m
39. BAKING Emily is baking chocolate cupcakes. Each batch of 20 cupcakes
2
1
requires _
cup of cocoa. If Emily has 3_
cups of cocoa, how many full
3
4
batches of cupcakes will she be able to make and how much cocoa will
she have left over?
Lesson 1-1 Rational Numbers
47
C
40. CHALLENGE Give a counterexample to the statement The quotient of two
fractions between 0 and 1 is never a whole number.
3
3
41. NUMBER SENSE Which is greater: 30 · _
or 30 ÷ _
? Explain.
4
4
CHALLENGE Use mental math to find each value.
43 _
641
· 641 ÷ _
42. _
76
594
44.
NGSSS Practice
783 _
72
43. _
· 241 ÷ _
594
241
783
53
Write a real-world problem that can be solved by
dividing fractions or mixed numbers. Solve the problem.
MA.8.A.6.4
45. Some of the ingredients required for one
batch of muffins are shown below.
46. Mr. Jones is doing a science experiment
with his class of 20 students. Each
3
student needs _
cup of vinegar. If he
4
currently has 15 cups of vinegar, which
equation could Mr. Jones use to
determine if he has enough vinegar for
his entire class?
Strawberry Muffins
flour
strawberries
2
3 cup
3
4 cup
2
Claudio’s father used 1_
cups of flour
3
7
and 1_
cups of strawberries. How many
8
batches of muffins did he make?
A. 3
C. 2
1
B. 2_
2
3
D. 1_
4
6
8
I.
x = 15(20)
GRIDDED RESPONSE Lucas is storing
a set of art books on a shelf that has
1
11_
inches of space. If each book is
4
_3 inch wide, how many books can be
stored on the shelf?
(Lesson1-1C)
5 _
49. _
·4
6
H. x = 20 – (15)
3
G. x = 15 ÷ _
4
4
Multiply. Write in simplest form.
1 _
48. _
·3
47.
F. x = 15 ÷ 20
5
_
2
50. 1_
· 41
3
5
52. VOLUNTEERING The table shows the number of hours four students
volunteered to work at an animal shelter after school. How much
time did the students volunteer in all? (Lesson 1-1B)
3
53. HEALTH A newborn baby weighs 6_
pounds. Write this weight as a
4
decimal. (Lesson 1-1A)
2
1
51. _ · 3_
3
4
Student
Anabel
Damon
Jeremiah
Meghan
48
Chapter 1 Rational Numbers and Percent
Time (h)
2_
1
3
7
_
1
8
5
1_
6
1
2_
4
CHAPTE R
1
Mid-Chapter
Check
7
1. Write 1_
as a decimal.
16
Multiply. Write in simplest form.
(Lesson 1-1A)
−
2. Write 0.4 as a fraction in simplest form.
(Lesson 1-1A)
1 _
9. -_
·7
3
3
1
10. -2_
· -_
(
) ( 5)
4
2
12. -2_
· -3_
7 (
3)
8
5 3
11. _ · _
6
3. NGSSS PRACTICE The table gives the
Year
Duration (h)
Challenger
(41–B)
1984
191_
Discovery
(51–A)
1984
191_
Endeavour
(STS–57)
1992
190 _
1999
1
191_
Discovery
(STS–103)
5
4
13. WEATHER The table shows the approximate
number of sunny days each year for certain
3
cities. Oklahoma City has about _
as many
5
sunny days as Phoenix. About how many
sunny days each year are there in
Oklahoma City? (Lesson 1-1C)
durations of spaceflights.
Mission
4
15
3
4
Sunny Days Per Year
1
2
6
Which of the following correctly orders
these durations from least to greatest?
(Lesson 1-1A)
City
Days
Austin, TX
120
Denver, CO
115
Phoenix, AZ
215
Sacramento, CA
195
Santa Fe, NM
175
3
1
1
4
A. 190 _
, 191_
, 191_
, 191_
Divide. Write in simplest form.
3
1
4
1
, 191_
, 191_
, 190 _
B. 191_
1
3
14. _ ÷ -_
2
4
6
6
4
15
15
( 4)
1
2
16. 6_
÷ (-1_
6
3)
2
2
3
1
1
4
, 191_
, 191_
, 191_
C. 190 _
2
6
15
(Lesson 1-1C)
4
(Lesson 1-1D)
1
1
15. -1_
÷ -_
(
3
) ( 4)
1
7
17. 8_
÷ 1_
2
10
3
1
4
1
, 191_
, 190 _
, 191_
D. 191_
6
15
2
4
1
18. NGSSS PRACTICE A board that is 25_
feet
Add or subtract. Write in simplest form.
2
1
long is cut into pieces that are each 1_
feet
(Lesson 1-1B)
1
4
4. _ + -_
6
4
5. -3_
- 3_
5
7
6. _ + -_
3
1
7. 5_
- 12_
5
12
( 5)
( 15 )
7
5
2
long. Which of the steps below would give
the number of pieces into which the board
is cut? (Lesson 1-1D)
7
2
1
1
F. Multiply 1_
by 25_
.
8. PIZZA A pizza has 3 toppings with no
1
of
toppings overlapping. Pepperoni tops _
3
2
. The rest is
the pizza and mushrooms top _
5
topped with sausage. What fraction is
topped with sausage?
2
2
1
1
G. Divide 25_ by 1_.
2
2
1
1
H. Add 25_
to 1_
.
2
2
1
1
I. Subtract 1_
from 25_
.
2
2
Chapter 1 Mid-Chapter Check
49