Download Chapter 2 - Lake County Schools

FLORIDA
CHAPTER
2x
Rational Numbers
2A
Rat
Rational
tional Numbers
2-1
Rational Numbers
Rev. MA.6.A.5.1,
Rev. MA.7.A.5.1
2-2
Comparing and Ordering
Rational Numbers
Rev. MA.6.A.5.2
2B
Rational Number
Operations
2-3
Adding and Subtracting
Rational Numbers
Multiplying Rational
Numbers
Dividing Rational
Numbers
Solving Equations with
Rational
Rat
Numbers
2-4
2-5
2-6
MA.8.A.6.4
MA.8.A.6.4
MA.8.A.6.4
MA.8.A.6.4
W
Worktext
pages 5–48
pag
Why Learn This?
Yacht racing speeds, times, and
victory margins can be expressed as
rational numbers, such as fractions and
decimals.
• Add, subtract, multiply and
divide rational numbers.
•U
Use the arithmetic of rational
numbers to solve equations.
58
Chapter 2
Additional instruction, practice, and
activities are available online, including:
• Lesson Tutorial Videos
• Homework Help
• Animated Math
Go to thinkcentral.com
Are You Ready?
Are You Ready?
Go to thinkcentral.com
Vocabulary
V
Choose the best term from the list to complete each sentence.
Choos
1. A number that consists of a whole number and a
fraction is called a(n) ? .
equivalent fraction
2. A(n) ?
a whole.
improper fraction
fraction
is a number that represents a part of
mixed number
3. A fraction whose absolute value is at least 1 is
called a(n) ? , and a fraction whose absolute
value is between 0 and 1 is called a(n) ? .
4. A(n)
?
proper fraction
names the same value.
p
Complete
these exercises to review skills you will need for this chapter.
Model Fractions
M
Write a fraction to represent the shaded portion of each diagram.
5.
6.
7.
8.
Write a Fraction as a Mixed Number
W
Write each improper fraction as a mixed number.
22
9. __
7
18
10. __
5
104
11. ___
25
65
12. __
9
37
13. __
3
1
17. 11__
11
18. 3__56
8
22. __
__
45
15
23. __35 __
75
Write a Mixed Number as a Fraction
W
Write each mixed number as an improper fraction.
14. 7__14
15. 10__37
16. 5__38
Write Equivalent Fractions
W
Supply the missing information.
Suppl
19. __38 __
24
5
20. __
__
13
52
7
21. __
__
12
36
Rational Numbers
59
FLORIDA
CHAPTER
2
Study Guide: Preview
Before
B
Befor
f
Previously, you
•
compared and ordered positive
rational numbers.
•
added, subtracted, multiplied,
and divided integers.
used models to solve equations.
•
Key
Vocabulary/Vocabulario
least common
denominator (LCD)
mínimo común
denominador (mcd)
rational number
número racional
reciprocal
recíproco
relatively prime
primo relativo
Vocabulary Connections
Now
Study Guide: Preview
You will study
•
•
•
•
comparing and ordering
positive and negative fractions
and decimals.
using appropriate operations
to solve problems involving
fractions and decimals.
solving one-step equations.
finding solutions to application
problems using equations.
Why?
You can use the skills
learned in this chapter
60
•
to compare and manipulate
measurements.
•
to find the size of a fraction of a
group or an item.
•
to solve more-complicated
equations in later chapters and
future math courses.
Chapter 2 Rational Numbers
To become familiar with some of the
vocabulary terms in the chapter, consider
the following. You may refer to the chapter,
the glossary, or a dictionary if you like.
1. The word rational has as its root the word
ratio. What do you think a rational number
is in math?
2. The word least means “smallest,” and the
word common means “the same.” What do
you think these words mean in combination
in least common denominator?
3. The word relative means “in relation to each
other.” What do you think relatively prime
numbers are?
FLORIDA
CHAPTER
2
Writing Strategy: Translate Between
Words and Math
LA.8.2.2.3 The student
will organize information
to show...relationships
among facts...(e.g., representing
key points within text through...
paraphrasing...).
When reading a real-world math problem, look for key words
to help you translate between the words and the math.
There are several different ways to indicate a mathematical
operation in words.
• Subtracted from
• Minus
• Difference
• Less than
• Decreased by
• Multiplied by
• Times
• Product
• Groups of
Reading and Writing Math
• Added to
• Plus
• Sum
• More than
• Divided by
• Quotient
• Divided into
In the problem below, use the highlighted terms to translate the
words into math.
The Montez family went to the state fair over the weekend. They spent
$52.50 on rides, food, and drinks, in addition to the $5.50-per-person
price of admission. How much did the Montez family spend at the fair?
They spent $52.50 in addition to $5.50 per person .
Let p represent the number of people.
$52.50
$5.50 p
52.5 5.5p
Try This
Identify the mathematical operation described by the key terms
in each statement. Explain your choice.
1. The male calf weighs 0.55 pounds less than the female calf.
2. Bob has 9 more books than Kerri.
3. The number of treats is divided by the number of students.
4. The rate is $15 plus two times the cost of the paint.
Rational Numbers
61
2-1
Prep for
MA.8.A.6.4
Perform operations
on real numbers (including...
rational numbers...)...
Review MA.6.A.5.1,
Review MA.7.A.5.1
Vocabulary
rational number
relatively prime
Rational
Numbers
I 2007,
In
2007 tthere were 335 NCAA
Division
Di
i i I women’s basketball
teams. At the end of the season,
64 teams were selected for the
women’s NCAA basketball
64
tournament. Only ___
of the
335
teams qualified for the
tournament.
A rational number is any
number that can be written as
a fraction __nd , where n and d are
integers and d 0.
The goal of simplifying fractions
is to make the numerator and
the denominator relatively
prime. Relatively prime
numbers have no common
factors other than 1.
Math
@ thinkcentral.com
The Tennessee Lady Vols women’s basketball team won its
seventh national championship in 2007.
You can often simplify fractions by dividing both the numerator and
denominator by the same nonzero integer. You can simplify the
12
fraction __
to _45 by dividing both the numerator and denominator by 3.
15
12 of the 15
boxes are
shaded.
4
12
3 __
______
5
15 3
4 of the 5
boxes are
shaded.
The same total area is shaded.
EXAMPLE
1
Simplifying Fractions
Simplify.
A
0
__
a 0 for a 0
a
__
a 1 for a 0
7 ___
7 __
7
___
8
8
8
B
9
__
55
933
; there are no common factors.
55 5 11
9
9
__
__
55
55
9 and 55 are relatively prime.
24
____
32
24 2
32 2
8
24 24
____
_______
32
32 8
3
___
, or __3
4
4
62
Chapter 2 Rational Numbers
223
2222
8 is a common factor.
Divide the numerator and denominator by 8.
Lesson Tutorial Videos @ thinkcentral.com
Decimals that terminate or
repeat are rational numbers.
A repeating decimal
can be written with
a bar over the digits
that repeat. So
0.13333 . . . ⫽ 0.13
苶.
EXAMPLE
Rational
Number
Description
Written as
a Fraction
To write a terminating
3.2
Terminating decimal
decimal as a fraction, identify
the place value of the digit
0.13
苶
Repeating decimal
farthest to the right. Then
write all of the digits after the decimal point as the
numerator with the place value as the denominator.
2
2
3_
10
2
15
Writing Decimals as Fractions
Write each decimal as a fraction in simplest form.
A
B
5.59
59
5.59 5___
100
0.5714
5714
0.5714 ______
10,000
2857
____
5000
9 is in the hundredths place.
4 is in the ten-thousandths place.
Simplify by dividing by the common factor 2.
To write a fraction as a decimal, divide the numerator by the
denominator.
EXAMPLE
3
Writing Fractions as Decimals
Write each fraction as a decimal.
A
5
__
4
B
1.25
____
4冄5.00
4
10
8
20 The remainder is 0.
20 This is a terminating
0 decimal.
The fraction _54 is equivalent to
the decimal 1.25.
– __16
0.16
苶
_______
Leave the negative
6冄 1.000
6
40
36
40
sign off while dividing.
The pattern repeats.
The fraction – _16 is equivalent to
the decimal – 0.16
苶.
Think and Discuss
1. Explain how you can be sure that a fraction is simplified.
2. Give the sign of a fraction in which the numerator is negative and
the denominator is negative.
Lesson Tutorial Videos @ thinkcentral.com
2-1 Rational Numbers
63
2-1
Homework Help
Exercises
E
Go to thinkcentral.com
Exercises 1–56, 59, 63
Prep MA.8.A.6.4
GUIDED PRACTICE
See Example 1
See Example 2
See Example 3
Simplify.
11
1. __
22
6
2. __
10
57
6. __
69
7. __68
17
5. __
51
14
4. __
25
16
3. __
24
9
__
8. 28
49
9. ___
112
22
10. __
44
Write each decimal as a fraction in simplest form.
11. 0.75
12. 1.125
13. 0.4
14. 0.35
15. 2.2
16. 0.625
17. 3.21
18. 0.3878
Write each fraction as a decimal.
19. __58
20. __35
5
21. __
12
22. __14
23. __19
18
24. __
9
25. __38
14
26. __
5
27. __54
28. __23
INDEPENDENT PRACTICE
See Example 1
See Example 2
See Example 3
Simplify.
21
29. __
28
25
30. __
65
17
31. __
34
17
32. __
21
25
33. __
30
13
34. __
17
22
35. __
35
64
36. __
76
78
37. ___
126
14
38. __
22
Write each decimal as a fraction in simplest form.
39. 0.6
40. 3.5
41. 0.72
42. 0.183
43. 1.377
44. 1.450
45. 1.4
46. 2.9
Write each fraction as a decimal.
47. __38
7
48. __
12
49. __95
13
50. __
20
51. __85
18
52. __
40
23
53. __
5
28
54. __
25
55. __43
56. __74
PRACTICE AND PROBLEM SOLVING
Mental Math Make up a fraction that cannot be simplified and has the
following characteristics.
57. a denominator of 36
58. a denominator of 24
59. Sports The thickness of a surfboard is often matched to the weight of the
rider. For example, a person weighing 170 pounds might need a surfboard
that is 3.375 inches thick. Write 3.375 as a fraction in simplest form.
60. Bondi weighed his cell phone and found it to be approximately
7
__
pound. What is the weight of Bondi’s phone written as a decimal?
25
64
Chapter 2 Rational Numbers
WORKED-OUT SOLUTIONS
on p. WS3
61. a. Simplify each fraction.
8
__
18
8
__
48
5
__
20
18
__
32
21
__
45
45
__
72
24
__
50
36
__
96
b. Write the denominator of each simplified fraction as the product of
prime factors.
c. Write each simplified fraction as a decimal. Label each as a
terminating or repeating decimal.
62. Measurement The ruler is marked at every
1
__
in. Do the labeled measurements convert to
16
terminating or repeating decimals?
63. Critical Thinking The greatest common factor,
GCF, is the largest common factor of two or more
given numbers. Find and remove the GCF of 42
42
and 68 from the fraction __
. Can the resulting
68
fraction be further simplified? Explain.
3
16
7
8
1
2
1
14
64. What’s the Error? A student simplified a
25
fraction in this manner: ___
_56. What error did the student make?
30
65. Write About It Using your answers to Exercise 61, examine the prime
factors in the denominators of the simplified fractions that are equivalent to
terminating decimals. Then examine the prime factors in the denominators
of the simplified fractions that are equivalent to repeating decimals. What
pattern do you see?
66. Challenge A student simplified a fraction to _29 by removing the
common factors, which were 2 and 9. What was the original fraction?
Florida Spiral Review
67. Multiple
Choice
Florida
Spiral Review
Prep for MA.8.A.6.4
If y _39 , which is NOT equal to y ?
1
A. ___
3
1
B. __13
C. ___
3
68. Multiple Choice Which shows the decimal 0.68 as a fraction in
simplest form?
17
F. __
25
( )
34
G. __
50
H. __34
1
D. ___
3
( )
I. __68
119
?
69. Gridded Response What is the decimal equivalent of the fraction ___
4
Evaluate each expression for the given values of the variable. (Lesson 1-1)
70. 3x 5 for x 2 and x 3
71. 4( x 1 ) for x 6 and x 11
Simplify. (Lesson 1-7)
72. 3(6 8)
73. 4(3 2)
74. 5(3 2)
75. 3(1 8)
76. 12(4 9)
77. 15(11 (1))
78. 6(5 (4))
79. 7(1 (17))
2-1 Rational Numbers
65
2-2
Prep for
MA.8.A.6.4
MA
8A64
Perform operations
on real numbers (including...
rational numbers...)...
Review MA.6.A.5.2
Vocabulary
least common
denominator (LCD)
Comparing and Ordering
Rational Numbers
The
h pop
population within the United
States iis constantly changing. The table
S
shows the percent change in populations
from 2000 to 2003 for three states and
the District of Columbia. A negative
percent indicates that the population
declined.
Population Change from 2000–2003
Location
Change (%)
12
___
5
Maine
North Dakota
1.3
Location
Change (%)
Washington
4.0
Washington, D.C.
1
1__
2
To compare or order rational numbers, first write them in the same
form. To compare fractions, find a common denominator. This could
be the least common denominator (LCD), which is the least
common multiple of the denominators.
EXAMPLE
1
Comparing Fractions by Finding a Common Denominator
Compare. Write , , or .
A
5
__
8
7
__
12
Method 1: Multiply to find a common denominator.
8 12 96
Multiply 8 and 12 to find a common
5 __
60
12 5
12 __
__
_____
8 12
8 12
96
7 __
8
78
56
__
_____
__
12 8 96
12 8
60
56
7
__
__
96, so __58 __
96
12
B
The least common
multiple (LCM) of
two numbers is the
smallest number,
other than 0, that is
a multiple of both
numbers. See Skills
Bank page SB5.
66
3
__
4
denominator.
Write the fractions with a common
denominator.
Compare the fractions.
5
__
6
Method 2: Find the least common denominator.
4: 4, 8, 12, … 6: 6, 12, … List multiples of 4 and 6. The LCM is 12.
3 __
9
3 __
__
3 3____
4 3
Write the fractions with a
43
12
5
2
10
2
5 __
common denominator.
__
____
__
6 2 12
6 2
9
10
__
__
, so __34 __56
12
12
Chapter 2 Rational Numbers
Compare the fractions.
Lesson Tutorial Videos @ thinkcentral.com
EXAMPLE
2
Comparing by Using Decimals
Compare. Write , , or .
3__38
A
Math
@ thinkcentral.com
3__35
3__38 3.375 and 3__35 3.6
Write the fractions as decimals.
3.375 3.6, so 3__38 3__35
Compare the decimals.
6
__
10
0.53
B
6
0.6
1
0
6
Write ___
as a decimal.
10
6
0.53 0.6, so 0.53 __
10
Compare the decimals.
9
0.8
11
9
0.8
苶1
苶
11
C
9
Write ___
as a decimal.
11
9
0.8
苶1
苶 0.8, so __
0.8
11
Compare the decimals.
To order fractions and decimals, you can either write them all in the
same form and then compare them, or place them on a number line.
EXAMPLE
3
Social Studies Application
From 2000 to 2003, the percent changes in populations for
three states and the District of Columbia were as follows: 152 for
Maine, 1.3 for North Dakota, 4.0 for Washington, and 121 for
Washington, D.C. List these numbers in order from least
to greatest.
Place the numbers on a number line and read them from left to right.
⫺1
1
2
⫺1.3
2 1.5 1 0.5
12
5
0
0.5
1
1.5
2
2.5
4
3
3.5
4
The percent changes in population from least to greatest are 1_12 ,
12
1.3, __
, and 4.0.
5
Think and Discuss
1. Explain whether you need to find a common denominator to
compare _23 and _12 .
2. Describe the steps you would use to compare 0.235 and 0.239.
Lesson Tutorial Videos
2-2 Comparing and Ordering Rational Numbers
67
2-2
Homework Help
Exercises
E
Go to thinkcentral.com
Exercises 1–26, 27, 29, 41
Prep MA.8.A.6.4
GUIDED PRACTICE
Compare. Write , , or .
See Example 1
1. __38
3
__
7
9
2. __
11
9
__
10
6
3. __
15
See Example 2
5. __78
9
__
11
6. 4.2
See Example 3
9. In Mr. Corsetti’s shop class, students were instructed to measure and cut
boards to a length of 8 inches. In checking four students’ work, Mr. Corsetti
found that one board was 8.25 inches, the second was 8_18 inches, the third was
5
7.5 inches, and the fourth was 7__
inches. List these measurements in order
16
from least to greatest.
7. __37
4__15
2
__
5
0.375
7
4. __
10
__58
8. 1__12
1__79
INDEPENDENT PRACTICE
Compare. Write , , or .
See Example 1
10. __58
14. __23
See Example 2
See Example 3
18. 5__89
22. __97
16
__
21
__57
5__78
10
__
8
13
11. __
11
8
__
7
16
15. __
9
__83
19. __16
23. 1__23
__15
8
1__
12
12. __13
17
16. __
20
20. __47
15
24. __
22
__14
5
__
6
13. __34
17. __29
__25
21. __67
0.681
13
25. __
20
___
9
__
12
__18
0.87
0.65
26. Auto Racing The fastest qualifying speed for a May, 2007 NASCAR
race at the Lowes Motor Speedway in Concord, North Carolina, was
185.312 mi/h. The next five fastest speeds, relative to the fastest speed,
5
2
were approximately 0.68 mi/h, __
mi/h, 1__
mi/h, 1.09 mi/h, and
16
25
1
_
4 mi/h. List these relative speeds from slowest to fastest.
PRACTICE AND PROBLEM SOLVING
Compare. Write , , or .
27. __57
6
__
10
28. 5.00
20
__
29. 7.2
5
7__29
30. 14.7
14.6885
Write a fraction or decimal that has a value between the given numbers.
32. 0.89 and 0.9
33. __23 and 0.5
34. 0.27 and __45
31. __14 and __13
35. Critical Thinking On Tuesday, stock A’s price fell 0.56 and stock B’s
price fell 0.50. Stock C’s price did not fall as much as stock A’s, but it fell
more than stock B’s. What is a reasonable answer for how much stock C’s
price fell? Explain.
36. Multi-Step Alejandro, Becky, Marcus, and Kathy ate lunch at a restaurant.
The total amount of the bill, including tax and tip, was $34.20. Alejandro
paid $10.00, Becky paid 41 of the bill, Marcus paid 0.2 of the bill, and Kathy
paid the rest. Who paid the greatest part of the bill?
68
Chapter 2 Rational Numbers
WORKED-OUT SOLUTIONS
on p. WS3
37. Life Science The lengths of some
butterflies’ wingspans are shown in
the table.
Wingspan (in.)
Great white
a. List the butterflies in order from
smallest to largest wingspan.
Meteorology
NASA uses satellite imagery to
gather information about temperatures on
Earth, such as
the amount of
heat radiated
into space from
Earth’s surface
and atmosphere.
Butterfly
3.75
Large orange
sulphur
b. The pink-spotted swallowtail’s
5
wingspan can measure 3__
inches.
16
Between which two butterflies should
the pink-spotted swallowtail be in
your list from part a?
338
Apricot
sulphur
2.625
White-angled
sulphur
3.5
38. Meteorology One measure of average global temperature shows
how each year varies from a base measure. The table shows results for
several years.
Year
Difference from Base
1958
1964
1965
1978
2002
0.10 °C
0.17 °C
0.10 °C
1 °C
50
0.54 °C
a. Order the five years from coldest to warmest.
b. In 1946, the average temperature varied by 0.03°C from the base
measure. Between which two years should 1946 fall when the years
are ordered from coldest to warmest?
39. What’s the Error? A student compared _14 and 0.3. He changed _14 to
the decimal 0.25 and wrote, “Since 0.3 is greater than 0.25, 0.3 is
greater than 0.25.” What was the student’s error?
40. Write About It Describe two methods to compare 1137 and 0.82. Which
do you think is easier? Why?
41. Challenge Write ⏐_23 ⏐, ⏐0.75⏐, ⏐0.62⏐, and ⏐_56 ⏐ in order from least to
greatest.
Florida Spiral Review
Prep for MA.8.A.6.4
42. Multiple Choice Which pair of numbers does _37 NOT come between?
9
and __12
C. 0.2 and __13
D. __25 and 0.65
B. __
25
43. Multiple Choice Which list of numbers is in order from least to greatest?
A. 0.3 and 0.45
F. 0.3, __45 , __14 , 0
G. __45 , 0.3, 0, __14
H. __14 , __45 , 0, 0.3
I. 0, __14, 0.3, __45
Simplify. (Lesson 1-6)
44. 5 ( 4 )
45. 8 ( 2 )
46. 19 13
47. 72 119
48. 24 37
9
52. __
15
19
53. __
20
Write each fraction as a decimal. (Lesson 2-1)
49. __34
50. __18
10
51. __
4
2-2 Comparing and Ordering Rational Numbers
69
Ready To Go On?
Ready To Go On?
SECTION 2A
Go to thinkcentral.com
Quiz for Lessons 2-1 Through 2-2
Rational Numbers
2-1 R
Simplify.
S
Si
i
lif
12
1. __
36
15
2. __
48
33
3. __
88
55
4. ___
122
12
5. __
96
6
6. __
16
10
7. ____
15
14
8. __
42
Write each number as a fraction in simplest form.
9. 0.4
13. 0.45
10. 0.05
11. 0.12
12. 0.625
14. 8.3
15. 1.019
16. 3.425
Ready to Go On?
Write each fraction as a decimal.
17. __38
18. __14
19. __94
7
20. __
12
12
21. __
25
22. __94
23. __83
11
24. __
12
14 hours on his math homework
25. John realized that he had worked __
5
last night. What is this time written as a decimal?
Comparing and Ordering Rational Numbers
2-2 C
Compare. Write ⬎, ⬍, or .
C
26. __67
30. 1__58
11
27. __
15
4
__
5
1__23
31. 2__18
9
__
10
2__15
28. __13
12
32. __
17
5
__
6
0.75
29. __45
1
__
8
33. 5__78
5.9
Write the numbers in order from least to greatest.
34. 1.2, __23 , 0.5, __34
35. 3__57, 0.1, __78 , 0.275
36. 2.3, __32 , 3, 3__89
10
33
37. 2__
, 1.3, __
, 2.99
13
8
38. Peggy, Will, Pat, and Eric have been collecting donations for a class
gift to the school library. The total collected so far is $90.00. Peggy
has collected about __29 of the total so far, Will collected __15 of the total
so far, Pat collected 0.3 of the total so far, and Eric collected the rest,
or about $25.00. Who has collected the most donations so far?
70
Chapter 2 Rational Numbers
Look Back
• Is your answer reasonable?
After you solve a word problem, ask yourself if your answer makes
sense. You can round the numbers in the problem and estimate to
find a reasonable answer. It may also help to write your answer in
sentence form.
Read the problems below and tell which answer is most reasonable.
1 Tonia’s rectangular vegetable garden is
8__12 feet long and 3__12 feet wide. What is the
approximate area of the garden?
A. about 5 ft2
B. about 15 ft2
C. about 20 ft2
D. about 30 ft2
2 The Qin Dynasty in China began about 2170
years before the People’s Republic of China
was formed in 1949. When did the Qin
Dynasty begin?
F.
G.
H.
I.
before 200 B.C.E.
between 200 B.C.E. and 200 C.E.
between 200 C.E. and 1949 C.E.
after 1949 C.E.
3 On Mercury, the lowest temperature
is about 600 °C below the highest
temperature of 430 °C. What is the
lowest temperature on the planet?
A. about 1030 °C
B. about ⴚ1030 °C
C. about ⴚ170 °C
D. about 170 °C
4 Julie walked from her home to school in
0.24 hour. Mark walked from his home
5
to school in __
hour. Anna’s trip to school
12
2
__
takes hour. Which statement is true?
9
F.
G.
H.
I.
Julie’s trip takes longest.
Julie’s time is greater than Anna’s time.
It takes Mark less time than Anna.
Anna’s time is closest to __12 hour.
Focus on Problem Solving
71
2-3
MA.8.A.6.4
Perform operations
on real numbers
(including...absolute value
[and] rational numbers...)...
using...real-world problems.
EXAMPLE
Adding and Subtracting
Rational Numbers
Olympic swimming events are measured
in hundredths of a second. In the Athens
2004 Summer Olympic Games, the
difference in times between the gold
and silver medal winners in the men’s
100-meter backstroke was 0.29 second.
1
Sports Application
In the Athens 2004 Olympic Games, Aaron Peirsol of the
United States won the gold medal in the 100-meter
backstroke with a time of 54.06 seconds. The eighth place
finisher, Marco di Carli, completed the race in 55.27 seconds.
What was the difference in times between the first- and
eighth-place finishers?
Write the numbers so that the
55.27
54.06
decimals line up.
1.21
The difference between the first- and eighth-place finishers was
1.21 seconds.
You can add and subtract fractions with like or unlike denominators.
ADDING AND SUBTRACTING WITH LIKE DENOMINATORS
Words
To add or subtract rational
numbers with the same
denominator, add or
subtract the numerators
and keep the denominator.
EXAMPLE
2
Numbers
( )
1 __
4 1
__
( 4 )
_______
5
5
5
3
___
, or __35
5
Algebra
a
b
b
__
__
a_____
d
d
d
Adding and Subtracting Fractions with Like Denominators
Add or subtract. Write each answer in simplest form.
A
72
7
11
__
___
13
13
7
11
11 7
__
______
__
Add numerators. Keep the denominator.
13
13
13
18
5
__
, or 1__
13
13
Chapter 2 Rational Numbers
Lesson Tutorial Videos @ thinkcentral.com
Add or subtract. Write each answer in simplest form.
B
__38 __58
3
5
__38 __58 ___
___
8
8
Subtracting a
number is the
same as adding its
opposite.
5
5
To subtract __
, add the opposite, __
.
8
8
5
8
_________
3
___
1
8
8
(
)
To add or subtract fractions with different denominators, rewrite the
fractions with a common denominator.
TWO WAYS TO FIND A COMMON DENOMINATOR
EXAMPLE
3
Method 1: Multiply the
denominators to write fractions
with the same denominator.
Method 2: Find the LCM (least
common multiple) of the
denominators.
6
1 __
11
4 Multiply the
__
____
1____
4
6
46
6 4 denominators.
6
4
__
__
24
24
2 __
1
__
24
12
3
1 __
11
2
__
____
1____
4
6
43
62
3
2
__
__
12
12
1
__
12
The LCM
of the
denominators
is 12.
Adding and Subtracting Fractions with Unlike Denominators
Add or subtract. Write each answer in simplest form.
A
4 __
1
__
5
6
Method 1: __45 __16
4
6
5
____
1____
56
65
5
24 __
__
30
30
29
__
30
B
Find a common denominator: 5(6) 30.
Multiply the denominators.
Rewrite with a common denominator.
Simplify.
2_16 2__29
Method 2: 2_16 2_29
You may also use
prime factorization
to find the LCD. See
Skills Bank p. SB4.
Lesson Tutorial Videos
13
20
__
__
6
9
Multiples of 6: 6, 12, 18,...
Multiples of 9: 9, 18, 27,...
Write as improper fractions.
List the multiples of each denominator
and find the LCD.
13
3
2
_____
_____
20
63
92
Write equivalent fractions with a
common denominator.
39
40
__
__
18
18
1
__
18
Rewrite with the LCD.
Simplify.
2-3 Adding and Subtracting Rational Numbers
73
EXAMPLE
4
Recreation Application
Two hikers on the Appalachian Trail are 5_34 miles from the trail
head. The hikers cover 2_18 miles before taking a break. They
then hike another 1_12 miles before taking a second break. How
many more miles do the hikers have to go before reaching the
trail head?
Recreation
Clingman’s Dome, on
the Tennessee-North
Carolina border, is the
Appalachian Trail’s
highest point at
6643 feet.
2_18 1_12
Add to find the distance hiked.
17
__
__32
8
Write as improper fractions.
17
12
__
__
8
8
The LCD is 8.
29
__
, or 3_58
8
Simplify.
The hikers have hiked 3_58 miles. Now find the number of miles
remaining.
5__34 3__58
Subtract the distance hiked from the total distance.
23
29
__
__
4
8
Write as improper fractions.
46
29
__
__
8
8
The LCD is 8.
17
__
, or 2__18
8
Simplify.
The hikers have 2_18 miles to go before reaching the trail head.
EXAMPLE
5
Simplifying Absolute-Value Expressions
1 __
1 __
1.
Simplify __
3
2
6
1 __
1
__
3
2
1
__
6
3
__
6
2
__
6
1
__
6
The absolute value
of a number is its
distance from 0 on
a number line. See
Lesson 1-4 p. 19.
1
__
6
1
__
6
Rewrite with a common
denominator: the LCD is 6.
Subtract.
__16 __16
__16 is __16 unit from 0.
__26 __13
Simplify.
Think and Discuss
1. Give an example of two denominators with no common factors.
(
)
3
is positive or negative. Explain.
2. Tell if 2_15 2__
16
3. Explain how to add 2_25 9_13 without first writing them as improper
fractions.
74
Chapter 2 Rational Numbers
Lesson Tutorial Videos @ thinkcentral.com
2-3
Homework Help
E
Exercises
Go to thinkcentral.com
Exercises 1–27, 29, 31
MA.8.A.6.4
GUIDED PRACTICE
See Example 1
1. Sports In the Beijing 2008 Olympic Games, Natalie Coughlin of the
United States won the gold medal in the 100-meter backstroke swim
with a time of 58.96 seconds. The silver medal winner, Kirsty Coventry of
Zimbabwe, completed the race in 59.19 seconds. What was the difference
between the two times?
Add or subtract. Write each answer in simplest form.
(
See Example 2
2. __16 __56
3
9
3. __
__
10
10
3
7
4. __
__
12
12
9
4
5. __
__
25
25
See Example 3
6. __47 __13
7. __12 __78
8. 3__12 7__45
(
)
(
7
9. 3__
2__45
12
See Example 4
10. Gavin cut 2 pieces of fabric, each measuring 2_58 yards, from a bolt
containing 9_14 yards of fabric. How much fabric remains on the bolt?
See Example 5
Simplify each expression.
11. __12 __25
12. 3__13 __23
)
13. __49 __13
)
14. 8__14 3__25
INDEPENDENT PRACTICE
See Example 1
15. Sports Reaction time measures how quickly a runner reacts to the starter
pistol. In a race, a runner had a reaction time of 0.214 second and a total
race time of 11.03 seconds. How long did it take to run the actual distance?
Add or subtract. Write each answer in simplest form.
( )
See Example 2
7
5
__
16. __
13
13
13
1 __
17. __
17
17
9
16
18. __
__
17
17
See Example 3
11 __
4
20. __
5
12
14
21. __25 __
15
22. 5__45 3__27
See Example 4
24. A tank contained 212_23 liters of water before 27_13 liters were used. If the
tank can hold 240_38 liters, how much space in the tank is unused?
See Example 5
Simplify each expression.
9
3
__
25. __45 __
10
10
19
11 __
19. __
33
33
(
26. 1__34 __13 __56
)
11
23. __59 __
14
7
27. 4__
__35 5__13
15
PRACTICE AND PROBLEM SOLVING
28. Multi-Step Five basketball players are 78_18 in., 74 in., 71_58 in.,
70_34 in., and 69_12 in. tall. Find their average height.
29. Measurement A water pipe has an outside diameter
5
of 1_14 inches and a wall thickness of __
inch. What is the
16
inside diameter of the pipe?
WORKED-OUT SOLUTIONS
on p. WS3
1 14 in.
5
in.
16
2-3 Adding and Subtracting Rational Numbers
75
Social Studies
Niagara Falls, on the border of Canada
and the United States, has two major
falls, Horseshoe Falls on the Canadian
side and American Falls on the U.S. side.
Surveys of the erosion of the falls began
in 1842. From 1842 to 1905, Horseshoe
Falls eroded 239_25 feet.
30. In 1986, Thomas Martin noted that American
Falls eroded 7_12 inches and Horseshoe Falls
4
eroded 2__
feet. What is the difference between
25
the two measurements?
31. From 1842 to 1875, the yearly erosion of
61
Horseshoe Falls varied from a minimum of ___
100
17
__
meter to a maximum of 150 meters. By how much
did these rates of erosion differ?
32. The shoreline of the Niagara River is also eroding.
3
1
Along the upper river, ___
of the shoreline is eroding, while __
of the
100
25
shoreline of the tributaries is eroding. The lower river’s shoreline is less
7
protected and about __
is eroding. What part of the total shoreline of the
50
Niagara River and its tributaries is actively eroding?
Challenge Rates of erosion of American Falls have been recorded
C
33.
23
9
as ___
meter per year for 33 years, __
meter per year for 48 years, and
100
40
1
_
meter per year for 4 years. What is the total amount of erosion
5
during these three time spans?
Florida Spiral Review
MA.8.A.6.4
34. Multiple Choice A 4_58 ft section of wood was cut from a 7_12 ft board. How much
of the original board remained?
A. 3__58 ft
9
B. 3__
ft
C. 2__78 ft
D. 2__38 ft
16
35. Extended Response A rectangular swimming pool measured 75_12 feet by
25_14 feet. Schmidt Pool Supply computed the perimeter of the pool to be 200_13 feet.
Explain what the company did incorrectly when computing the perimeter. What
is the correct perimeter?
Evaluate each expression for the given value of the variable. (Lesson 1-5)
36. c 4 for c 8
37. m 2 for m 13
38. 5 d for d 10
Compare. Write , , or . (Lesson 2-2)
39. 0.25
76
1
__
3
__
40. 0.53
Chapter 2 Rational Numbers
0.5
41. __47
0.57
9
42. __
11
___
0.81
WORKED-OUT SOLUTIONS
on p. WS3
2-4
Multiplying Rational
Numbers
MA.8.A.6.4
Perform operations
on real numbers
(including...rational numbers...)
using...real-world problems.
Andrew walks his dog each day.
His route is _18 mile. What is the
total distance that Andrew walks
his dog in a 5-day week?
Recall that multiplication is
repeated addition.
()
3 __14 __14 __14 __14
11
1________
4
3
__
4
Notice that multiplying a fraction
by a whole number is the same
as multiplying the whole number
by just the numerator of the
fraction and keeping the same
denominator.
RULES FOR MULTIPLYING TWO RATIONAL NUMBERS
If the signs of the factors are the same, the product is positive.
() () () or () () ()
If the signs of the factors are different, the product is negative.
() () () or () () ()
EXAMPLE
1
Multiplying a Fraction and an Integer
Multiply. Write each answer in simplest form.
A
32
To write __
as a
5
mixed number,
divide:
32
___
6 R2
5
6_25
()
6( __23 )
6 __23
B
6
2
____
3
12
__
3
Multiply.
4
Simplify.
Lesson Tutorial Videos @ thinkcentral.com
( )
2( 3__15 )
16
2( __
5)
1 ______ 3__
5
5
5
32
__
Multiply ()
2 3__15
5
6__25
3(5) 1
16
() ().
Simplify.
2-4 Multiplying Rational Numbers
77
EXAMPLE
2
Multiplying Fractions
Multiply. Write each answer in simplest form.
A
A fraction is in
lowest terms, or
simplest form, when
the numerator and
denominator have
no common factors.
B
( 1)
3
3 1
1
___
__5 ( __4 ) ___
5 ( 4 )
3
__5 __4
(
(
( 3 )( 1 )
________
5( 4 )
Multiply numerators.
Multiply denominators.
3
__
20
Simplify.
)
)
12
5
__
__
5
12
5
5
12
12
__
__
5 __
__
5
12
12
(
)
( ⫺12 )1
5______
12( 5 )
Look for common
factors: 12, 5.
___ 1
1
1
Simplify.
1
1
EXAMPLE
3
1
Multiplying Decimals
Multiply.
A
Math
@ thinkcentral.com
EXAMPLE
4
5.2( 5 )
Product is
5.2 (5 ) 26.0 positive with 1
decimal place.
26
You can drop the zero after
the decimal point.
B
0.07( 4.6 )
0.07 4.6 0.322
Product is negative
with 3 decimal
places.
Recreation Application
Andrew walks his dog __18 mile each day. What is the total distance
that Andrew walks his dog in a 5-day week?
5
1( 5 ) 1
__
____
8
8
__58
Multiply.
Andrew walks his dog _58 mile in a 5-day week.
Think and Discuss
1. Give an example of a multiplication problem with two factors
where the product is a. greater than the factors. b. between the
factors. c. less than the factors.
2. Give an example of two fractions whose product is an integer
due to common factors.
78
Chapter 2 Rational Numbers
Lesson Tutorial Videos @ thinkcentral.com
2-4
Homework Help
E
Exercises
Go to thinkcentral.com
Exercises 1–38, 39, 41, 45, 47, 49, 51, 53
MA.8.A.6.4
GUIDED PRACTICE
Multiply. Write each answer in simplest form.
See Example 1
See Example 2
See Example 3
()
5. __14 ( __58 )
1. 5 __12
()
6__37 ( __78 )
( )
__35 ( __59 )
3. 3 __58
4. 4 5__23
6.
7.
8.
Multiply.
9. 2.1( 7 )
See Example 4
( )
3 __
__
7
8 ( 10 )
2. 7 1__34
11. 4.8( 2 )
10. 0.03( 5.4 )
12. 0.15( 2.8 )
13. Tran jogs _34 mile each day. How far does Tran jog in 6 days?
INDEPENDENT PRACTICE
See Example 1
Multiply. Write each answer in simplest form.
15. 3 1__56
( )
19. 3( 6__79 )
23. __29 ( __78 )
7
27. 2__17 ( __
10 )
4
16. 9 __
21
( )
20. 8( __34 )
5
24. 5__78 ( __
11 )
28. __23 ( __19 )
( )
21. 7( 3__15 )
25. __13 ( __78 )
29. __78( __35 )
30. 1.7( 4 )
31. 0.05( 4.7 )
32. 6.2( 7 )
33. 0.75( 5.5 )
34. 6.2( 9 )
35. 0.08( 6.2 )
36. 2.4( 9 )
37. 0.04( 9.2 )
()
14
18. 9( __
15 )
22. __23 ( __56 )
26. __37 ( __56 )
14. 5 __17
See Example 2
See Example 3
See Example 4
17. 7 1__23
Multiply.
38. There was _34 of a pizza left over from a family gathering. The next day, Tina
ate _12 of what was left. How much of the whole pizza did Tina eat?
PRACTICE AND PROBLEM SOLVING
39. Consumer Economics At a bookstore, the ticketed price of a book is _14
off the original price. Kayla has a discount coupon for _12 off the ticketed
price. What fraction of the original price is the additional discount?
Multiply.
()
4 __
4
44. __
11 ( 7 )
40. 6 __37
( )
45. 3__56( __79 )
8
41. 5 1__
11
()
46. __89( __35 )
42. 7 __45
( )
5
11
47. __
__
12 ( 16 )
43. 5 3__19
Estimate each product.
48. 1.499 3.998
WORKED-OUT SOLUTIONS
on p. WS3
49. 0.95 5.03
( )( )
8 __
12
50. __
15 25
( )
10
51. 4 __
19
2-4 Multiplying Rational Numbers
79
52. Health The directions for a pain reliever recommend that children
96 pounds and over take 4 tablets every 4 hours as needed, and children
who weigh between 60 and 71 pounds take only 2_12 tablets every 4 hours
4
as needed. Each tablet is __
gram.
25
Animals
a. If a 105-pound child takes 4 tablets, how many grams of pain reliever is
he or she receiving?
b. How many grams of pain reliever is the recommended dose for a child
weighing 65 pounds?
There are fewer
than 30 veterinary
colleges in the
United States.
53. Animals The label on a bottle
of pet vitamins lists dosage
guidelines. What dosage would
you give to each of these
animals?
Do-Good Pet Vitamins
Adult dogs:
1 tsp per 20 lb body weight
2
Puppies, pregnant dogs, or nursing dogs:
a. a 50 lb adult dog
1 tsp per 10 lb body weight
2
b. a 12 lb cat
c. a 40 lb pregnant dog
Cats:
1 tsp per 2 lb body weight
4
54. What’s the Error? A student
multiplied two mixed numbers
in the following fashion: 2_47 ⴢ 3_14 ⴝ 6__17. What’s the error?
55. Write About It In the pattern _13 ⴙ _14 ⴙ _15 ⴙ . . ., which fraction makes the
sum greater than 1? Explain.
56. Challenge Of the 42 presidents who preceded George W. Bush, _13 were
elected to a second term. Of those elected to a second term, _17 were former
vice presidents of the United States. What fraction of the first 42 presidents
were elected to a second term and were former vice presidents?
Florida Spiral Review
MA.8.A.6.4
57. Multiple Choice Lindsay walked _34 mile on Monday. She walked 1_58 that
distance on Tuesday. How far did she walk on Tuesday?
7
A. 1__
miles
32
15
B. 1__
miles
C. 2__38 miles
32
58. Multiple Choice What is the product of ⴚ5__13 and 3__34?
15
D. 2__
miles
32
G. ⴚ15__14
59. Multiple Choice Multiply: ⴚ0.98 ⴛ ⴚ8.4.
H. 15__14
I. 20
C. 8.232
D. 82.83
F. ⴚ20
A. ⴚ82.83
B. ⴚ8.232
Compare. Write ⬍, ⬎, or ⴝ. (Lesson 1-4)
60. |ⴚ9|
ⴚ9
61. ⴚ13
ⴚ22
62. |5|
|ⴚ5|
63. |ⴚ17|
Find each sum. (Lesson 2-3)
64. ⴚ1.7 ⴙ 2.3
80
( )
65. ⴚ__56 ⴙ ⴚ __1
6
Chapter 2 Rational Numbers
66. 23.75 ⴙ ( ⴚ25.15 )
67. ⴚ__49 ⴙ __23
|ⴚ13|
2-5
Dividing Rational Numbers
MA.8.A.6.4
Perform operations
on real numbers
(including...rational numbers...)
using...real-world problems.
A number and its reciprocal have a product of 1. To find the reciprocal
of a fraction, exchange the numerator and the denominator. Remember
that an integer can be written as a fraction with a denominator of 1.
Number
Reciprocal
Product
Vocabulary
3
__
4
__
reciprocal
4
3
5
___
12
___
6
1
__
()
5 ___
12 1
___
12 ( 5 )
1 1
6( __
6)
3 __
4 1
__
4 3
5
12
6
Multiplication and division are inverse operations. They undo each other.
( ) 152
1 __
2
__
3 5
2
___
___
15
2 __
1
__
5 3
Notice that multiplying by the reciprocal gives the same result
as dividing.
1
( 152 )( 52 ) 152 52 10
30 3
___ __
______
___
__
DIVIDING RATIONAL NUMBERS IN FRACTION FORM
Numbers
Words
To divide by a
fraction, multiply
by the reciprocal.
EXAMPLE
1
1
__
7
Algebra
5 ___
5
4 __
1 __
__
5
7 4
28
a
__
b
a __
d ____
ad
__c __
c
d
b
bc
Dividing Fractions
Divide. Write each answer in simplest form.
A
7
__
__45
15
7 __
7
__
__45 __
5
15 4
15
Multiply by the reciprocal.
7 51
_____
15 4
Divide out common factors.
7
__
12
Simplest form
3
Lesson Tutorial Videos @ thinkcentral.com
2-5 Dividing Rational Numbers
81
Divide. Write each answer in simplest form.
B
1
( 7 )
5__
3
1 ( 7 ) __
16
5__
__71
3
3
( )
16 __
__
1
3 ( 7)
Write as improper fractions.
Multiply by the reciprocal.
1
________
16
37
No common factors
16
__
21
Simplest form
(
)
When dividing a decimal by a decimal, multiply both numbers by a
power of 10 so you can divide by a whole number. To decide which
power of 10 to multiply by, look at the denominator. The number of
decimal places is the number of zeros to write after the 1.
( )
1.32
10
1.32 __
13.2
____
____
____
0.4
4
0.4 10
1 decimal place
EXAMPLE
2
1 zero
Dividing Decimals
Find 7.48 0.4.
( )
10
7.48 __
74.8
____
7.48 0.4 ____
4
0.4 10
18.7
EXAMPLE
3
10
0.4 has 1 decimal place, so use __
.
10
Divide.
Evaluating Expressions with Fractions and Decimals
Evaluate each expression for the given value of the variable.
A
7.2
___
n for n 0.24
( 100 )
7.2
7.2 ___
100
____
____
0.24
0.24
720
___
24
100
0.24 has 2 decimal places, so use ___
.
100
Divide.
30
7.2
When n 0.24, ___
n 30.
B
5
for m 3__34
m __
24
Rewrite 3_34 as an improper fraction and
5
multiply by the reciprocal of ___
.
3 __
5 __
15 __
24
3__
4 5
4
24
3
24
______
15
45
24
6
Divide out common factors.
1
1
18
__
18
1
3 , m __
5 18.
When m 3__
4
82
Chapter 2 Rational Numbers
24
Lesson Tutorial Videos @ thinkcentral.com
EXAMPLE
4
PROBLEM SOLVING APPLICATION
Ella ate _23 cup of lowfat yogurt. The serving
size listed on the container is 6 ounces, or
_3 cup. How many servings did Ella eat?
4
How many Calories did Ella eat?
1
Understand the Problem
The number of Calories Ella ate is the
number of Calories in the fraction of
a serving.
List the important information:
• Ella ate _23 cup.
• A full serving is _34 cup.
• There are 100 Calories in one serving.
22 Make a Plan
Set up an equation to find the number of servings Ella ate.
amount Ella ate serving size number of servings
Using the number of servings, find the number of Calories Ella ate.
number of servings Calories per serving total Calories
33 Solve
Let n number of servings. Let c total Calories.
Servings: __23 __34 n
2 __
4n
__
33
8
__
n
9
Calories: __89 100 c
8
100 c
______
9
800
___
⬇ 88.9
9
Ella ate __89 of a serving, which is about 88.9 Calories.
44
Look Back
Ella did not eat a full serving, so _89 of a serving is a reasonable
answer. Since _89 is less than 1 and 88.9 calories is less than 100, the
Calories in a full serving, 88.9 Calories is a reasonable answer.
Think and Discuss
1. Explain how to write a division problem where the quotient is
greater than both the dividend and divisor.
2. Explain how to find the quotient _23 4 by modeling the product
of _23 and _14.
Lesson Tutorial Videos @ thinkcentral.com
2-5 Dividing Rational Numbers
83
2-5
Homework Help
Exercises
E
Go to thinkcentral.com
Exercises 1–46, 47, 49
MA.8.A.6.4
GUIDED PRACTICE
See Example 1
See Example 2
See Example 3
See Example 4
Divide. Write each answer in simplest form.
3
1. _12 __
4
2. 4__15 5__23
3. __67 3
4. __56 __38
1 4__
4
5. 5__
18
9
6. __58 12
14 __
2
7. __
15
3
3
8. 4__
__35
10
Find each quotient.
9. 3.72 0.3
10. 2.1 0.07
11. 10.71 0.7
12. 1.72 0.2
13. 2.54 0.6
14. 11.04 0.4
15. 2.45 0.005
16. 4.41 0.7
Evaluate each expression for the given value of the variable.
9.7
17. ___
x for x 0.5
6.2
18. ___
x for x 0.2
40.5
19. ____
x for x 0.9
9.2
20. ___
x for x 2.3
32.4
21. ____
x for x 1.8
14.7
22. ____
x for x 0.07
23. You eat _14 ounce of cheddar cheese. One serving of cheddar cheese is
1_12 ounces. How much of a serving did you eat?
INDEPENDENT PRACTICE
See Example 1
See Example 2
See Example 3
See Example 4
Divide. Write each answer in simplest form.
24. __16 __34
25. 4__25 3__12
5
26. __
__23
12
27. __45 __12
28. 1__23 2__16
7
29. __29 __
12
3
30. __23 __
10
31. 2__38 1__16
32. 12.11 0.7
33. 2.49 0.03
34. 6.64 0.4
35. 4.85 0.5
36. 5.49 0.003
37. 32.44 0.8
38. 9.36 0.03
39. 12.24 0.9
Find each quotient.
Evaluate each expression for the given value of the variable.
7.2
40. ___
x for x 0.4
9.6
41. ___
x for x 0.8
15
42. __
x for x 0.05
15.4
43. ____
x for x 1.4
4.24
44. ____
x for x 0.8
22.2
45. ____
x for x 0.06
5
feet wide.
46. The platform on the school stage is 8_34 feet wide. Each chair is 1__
12
How many chairs will fit across the platform?
PRACTICE AND PROBLEM SOLVING
47. Maya is drinking her favorite juice. There are 2_34 servings remaining in the
bottle. Maya pours only _14 of a serving into her glass at a time. How many
glasses can Maya have before the bottle is empty?
84
Chapter 2 Rational Numbers
WORKED-OUT SOLUTIONS
on p. WS3
48. Estimation The width of a DVD case is about _13 inch. About how many
DVD cases are in a box set if the set is about 1_34 inches thick?
49. Social Studies Nesting dolls called
matrushkas are a well-known type
of Russian folk art. Use the
information in the picture
to find the height of
the largest doll.
6
x
25
7
8
x in.
in.
50. Estimation A cereal box holds 9_15 servings. Leo’s bowl holds 1_58 servings.
Approximately how many times can Leo fill his bowl?
51.
5 Choose a Strategy Before 2000, the prices of all stocks traded on the New
York Stock Exchange were given in fractions. When a stock is split 2-for-1,
the price of the stock is halved and the number of shares doubles. A stock
trading at $20_14 was split 2-for-1. What was the price of the stock after the split?
5 Write About It What are the effects of multiplying and dividing a
52.
rational number by a rational number greater than 1? by a rational
number between 0 and 1?
27
53.
5 Challenge In 2006, the U.S. Census Bureau estimated that about _____
of
10,000
the U.S. population resided in Mecklenburg County, North Carolina, which
3
represented ___
of North Carolina residents. What fraction of U.S. residents
100
lived in North Carolina?
Florida Spiral Review
MA.8.A.6.4
7.92
54. Multiple Choice Evaluate the expression ____
x for x 3.3.
A. 2.4
B. 4.62
C. 11.22
D. 26.136
55. Multiple Choice A recipe calls for 2_12 cups of sugar to make a batch of
cookies. To make one-third of a batch, Betty needs to divide the amount of
each ingredient in the recipe by 3. How many cups of sugar will she use?
F. __34 cup
G. __56 cup
H. 1_15 cups
I. 7_12 cups
56. Gridded Response Frank bought 12.5 gallons of gasoline for $26.75.
How much, in dollars, was the cost per gallon of gasoline?
Evaluate each expression for the given values of the variables. (Lesson 1-1)
57. 7x 4y for x 5 and y 6
58. 6.5p 9.1q for p 2.5 and q 0
Add or subtract. Write each answer in simplest form. (Lesson 2-3)
59. __78 __16
60. 4__23 5__34
1
61. 6__58 2__
20
62. 2_89 __45
2-5 Dividing Rational Numbers
85
2-6
MA.8.A.6.4
Perform operations
on real numbers
(including...rational numbers...)
using...real-world problems.
Solving Equations with
Rational Numbers
Painting
i i a house can be a difficult task. In
order
d to have a good surface for the new
paint, the old paint must be cleaned, and
sometimes even scraped off completely.
Sully runs his own house-painting business.
When he plans a job, he estimates that he
Math
2
@ thinkcentral.com can paint _
of a house in one work day. You
5
can write and solve an equation to find how
long it would take Sully to paint 3 houses.
EXAMPLE
1
Solving Equations with Decimals
Solve.
A
Once you have
solved an equation,
it is a good idea to
check your answer.
To check your
answer, substitute
your answer for the
variable in the
original equation.
B
y 17.5 11
y 17.5 11
17.5 17.5
y 28.5
4.2p 4.2p 4.2p
______
4.2
12.6
12.6
12.6
_____
4.2
Use the Division Property of
Equality: Divide both sides by 4.2.
p 3
C
t
___
4
7.5
t
___
4
7.5
t
7.5 ___
4 7.5
7.5
t 30
EXAMPLE
Use the Addition Property of
Equality: Add 17.5 to both sides.
2
Use the Multiplication Property of
Equality: Multiply both sides by 7.5.
Solving Equations with Fractions
Solve.
A
x __19 __49
x __19 __49
x __19 __19 __49 __19
Subtract _19 from both sides.
x __59
86
Chapter 2 Rational Numbers
Lesson Tutorial Videos @ thinkcentral.com
Solve.
B
9
x __18 __
16
9
x __18 __
16
9
x __18 __18 __
__18
16
9
2
x __
__
16
16
11
x __
16
C
EXAMPLE
3
3
3
__
w __
5
16
3
3
__
w __
5
16
3
3
__
w __35 __
__35
5
16
1
3
5 1 1__
__
__
w
3 __5
3 1 16 31
15
5
w __
16
1 to both sides.
Add __
8
Find a common denominator, 16.
Divide both sides by _35 .
Multiply by the reciprocal. Simplify.
Solving Word Problems Using Equations
Sully has agreed to paint 3 houses. If he knows that he can
paint _25 of a house in one day, how many days will it take him to
paint all 3 houses?
Write an equation:
number of days
houses per day
number of houses
d
2
__
5
3
d __25 3
d __25 __25 3 __25
d __25 __52 3 __52
15
d __
, or 7__12
2
Divide both sides by _25 .
Multiply by the reciprocal.
Simplify.
Sully can paint 3 houses in 7_12 days.
Think and Discuss
1. Explain the first step in solving an addition equation with
fractions having like denominators.
2. Explain the first step in solving an addition equation with
fractions having unlike denominators.
Lesson Tutorial Videos
2-6 Solving Equations with Rational Numbers
87
2-6
Homework Help
Exercises
E
Go to thinkcentral.com
Exercises 1–28, 29, 31, 33, 35, 39, 43
MA.8.A.6.4
GUIDED PRACTICE
See Example 1
See Example 2
Solve.
1. y 17.3 65
2. 5.2f 36.4
m
6
3. ___
3.2
4. r 15.8 24.6
s
5. _____
6.3
15.42
6. 0.06g 0.474
7. x __19 __49
10. m __43 __43
See Example 3
8. __38 k __78
7
56
11. __
y __
17
17
7
9. __56w __
18
4 __
12
12. t __
13
39
13. Alonso runs a company called Speedy House Painters. His workers can
paint _34 of a house in one day. How many days would it take them to paint
6 houses?
INDEPENDENT PRACTICE
See Example 1
See Example 2
See Example 3
Solve.
l
14. y 16.7 49
15. 4.7m 32.9
h 2
16. ___
7.8
17. k 3.2 6.8
z
18. ____
6
11.4
19. c 5.98 9.1
20. j __13 __34
3
21. __56d __
15
14
22. 7h __
33
23. __23 x __58
7
1 __
24. x __
16
16
25. r __47 __17
7
26. __56 c __
24
11
27. __78 d __
12
28. A professional lawn care service can mow 2_34 acres of lawn in one hour.
How many hours would it take them to mow a lawn that is 6_78 acres?
PRACTICE AND PROBLEM SOLVING
Earth Science The largest of all known diamonds, the Cullinan diamond,
weighed 3106 carats before it was cut into 105 gems. The largest cut, Cullinan I,
or the Great Star of Africa, weighs 530_13 carats. Another cut, Cullinan II, weighs
317_25 carats. Cullinan III weighs 94_25 carats, and Cullinan IV weighs 63_35 carats.
29. How many carats of the original Cullinan diamond were left after the
Great Star of Africa and Cullinan II were cut?
30. How much more does Cullinan II weigh than Cullinan IV?
31. Which diamond weighs 223 carats less than Cullinan II?
32. Nutrition An entire can of chicken noodle soup has 6.25 grams of total
fat. There are 2.5 servings per can. How many grams of total fat are in a
single serving of chicken noodle soup?
88
Chapter 2 Rational Numbers
WORKED-OUT SOLUTIONS
on p. WS3
Solve.
33. z __29 __19
34. 5f 1.5
j
35. ___
3
7.2
36. __25 x 0.25
37. t __34 6__14
x
38. ___
__78
0.5
39. __67 d __37
40. 4.7g 28.2
v
41. ___
5.5
5.5
42. r __56 3__16
43. y 2.8 1.4
45. y 57 2.8
m
46. ___
7
0.8
Solve. Justify each step.
3
44. 3c __
20
47. Multi-Step Jack is tiling along the walls of the
rectangular kitchen with the tile shown. The kitchen
has a length of 243_34 inches and a width of 146_14 inches.
a. How many tiles will fit along the length of the room?
b. How many tiles will fit along its width?
c. If Jack needs 48 tiles to tile around all four walls of
the kitchen, how many boxes of ten tiles must he
buy? (Hint: He must buy whole boxes of tile.)
4
48. What’s the Error? Janice is thinking about buying a
DVD writer that burns 4.8 megabytes of data per
second. She figures that it would take 16 minutes to
burn 200 megabytes of data. What was her error?
49.
4 Write About It If a is _13 of b, is it correct to say _13a b? Explain.
5 Challenge A 200-carat diamond was cut into two equal pieces to
50.
form two diamonds. One of the diamonds was cut again, reducing
it by _15 its weight. In a final cut, it was reduced by _14 its new weight.
How many carats remained?
Florida Spiral Review
MA.8.A.6.4
12
51. Multiple Choice If __
2w, what is the value of w?
36
24
A. __
36
24
B. __
72
C. __13
1
D. ᎏ6ᎏ
52. Short Response The performance of a musical arrangement lasted 6_14 minutes.
The song consisted of 3 verses that each lasted the same number of minutes.
Write and solve an equation to find the length of each verse.
Write an algebraic expression for each word phrase. (Lesson 1-2)
53. 15 less than a number p
54. half of the sum of m and 19
Divide. Write each answer in simplest form. (Lesson 2-5)
4 __2
55. __
11
7
56. __49 8
7
14
57. __
__
15
25
58. 3__13 __79
2-6 Solving Equations with Rational Numbers
89
Ready To Go On?
Ready To Go On?
SECTION 2B
Go to thinkcentral.com
Quiz for Lessons 2-3 Through 2-6
Adding and Subtracting Rational Numbers
2-3 A
Add
A
Ad
dd or subtract. Write each answer in simplest form.
(
1. 65.8 24.24
2. __37 2__47
3. __56 2__16
4. __27 __14
5. 1__23 3__59
6. 6__47 3__15
)
7. Darius and Jamal ride their bicycles home from school every day. Each
day this week, they have timed themselves to see how long the ride
takes. On Monday, they made it home in about 0.25 hour. Today, it took
3
them __
hour. How much longer did it take today?
10
Multiplying Rational Numbers
2-4 M
2-4
Ready to Go On?
Multiply.
M
l i l Write each answer in simplest form.
( )
8. 2 43_2
( )
7
9. 2__25 __
36
10. 3.8(4)
3
1 ___
11. ___
7 4
( )
12. Robert has a piece of twine that is _34 yard long. He needs a piece of twine
that is _23 of this length. What length of twine does Robert need?
Dividing Rational Numbers
2-5 D
2-5
Divide.
D
i id Write
W
each answer in simplest form.
4
13. __35 __
15
14. 2.7 3
15. __23 1
16. 4__67 2__56
17. To make pasta, you need to boil water. Your pan holds 6 cups of water, and
your measuring glass holds 1_12 cups. How many glassfuls do you need?
Solving Equations with Rational Numbers
2-6 S
2-5
Solve.
S
l
18. p 1.2 5
19. 9w 13.5
m
8
20. ___
3.7
21. x __19 __47
22. m __34 __4
7
56
23. __
y __
33
3
y
24. ____
3.2
2.6
25. s 0.45 10.07
26. p 2.7 4.5
3
27. From start to finish, Ellen took 1532 days to write a research paper for
her literature class. This was 190 the time it took Rebecca to
write her paper. How long did it take Rebecca to write her
research paper?
90
Chapter 2 Rational Numbers
FLORIDA
The Florida Trail
The Florida Trail is a continuous footpath
that makes it possible to hike from one end of the Sunshine State
to the other. The entire system consists of 1400 miles of trails that
wind through prairies, forests, and marshes.
F LO R I D A
Osceola 38.8 mi
Jacksonville
10
Olustee 28 25 mi
95
75
Camp Blanding 24.1 mi
Real-World Connections
Florida Trail
er
ns Riv
a. On the first day, she hikes _14 of
the segment. How many miles
does she hike the first day?
15 mi
Joh
3. A hiker plans to cover the entire
Olustee segment of the trail.
0
St.
2. The map gives information on
four segments of the trail in
north Florida. What is the total
length of these four segments?
N
GEORGIA
Atlantic Ocean
1. The Florida Trail is divided into
more than 40 segments. The
Highlands segment is 32.3 miles
long. The Three Lakes segment is
32_35 miles long. Which segment is
longer? How much longer is the
longest segment?
Gainesville
Goldhead/Etoniah 41103 mi
b. How many more miles must
she hike to complete the segment?
c. Assuming she hikes the same number of miles each day, how
many additional days will it take her to complete the segment?
4. The Florida Trail Association sells a guidebook for the trail.
The book costs $19.95. A hiking club places an order that totals
$119.70. How many books has the hiking club ordered?
Real-World Connections
91
Egyptian Fractions
If you were to divide 9 loaves of bread among 10 people, you
9
would give each person __
of a loaf. The answer was different on
10
the ancient Egyptian Ahmes papyrus, because ancient Egyptians
used only unit fractions, which have a numerator of 1. All other
fractions were written as sums of different unit fractions.
So _56 could be written as _12 _13, but not as _16 _61 _16 _16 _16 .
Method
Example
Suppose you want to write a fraction
as a sum of different unit fractions.
9
___
10
Step 1. Choose the largest fraction of
the form __n1 that is less than the
fraction you want.
0
Step 2. Subtract __n1 from the fraction
you want.
9
1 __
2 remaining
___
__
5
10
2
Step 3. Repeat steps 1 and 2 using the
difference of the fractions
until the result is a unit fraction.
0
Step 4. Write the fraction you want
as the sum of the unit fractions.
1 1 1
5 4 3
1 1 1 2
5 4 3 5
1
2
1
2
9 1
10 1
1
1
2 __
1 ___
1 remaining
__
5
15
3
9
1 __
1 ___
1
___
__
15
10
2
3
Write each fraction as a sum of different unit fractions.
3
1. __
4
5
2. __
8
11
3. __
12
Egg Fractions
This game is played with an empty egg carton.
Each compartment represents a fraction with
a denominator of 12. The goal is to place
tokens in compartments with a given sum.
A complete copy of
the rules is available
online.
92
Games
Go to thinkcentral.com
Chapter 2 Rational Numbers
3
4. __
7
7
5. __
5
Materials
• film canister
with lid
• adding-machine
tape
• ruler
• markers
• self-stick label
PROJECT
Canister Carry-All
A
Turn a film canister into a handy carrying
case for a number line and notes about rational
numbers.
Directions
1 If necessary, cut off a strip along the bottom
edge of the adding-machine
tape so that the tape will fit into the film
canister when it is rolled up. When you’re done,
the tape should be about 1_34 in. wide.
B
Figure A
2 Use a ruler to make a long number line
on one side of the adding-machine tape.
Figure B
3 Write the number and title of the chapter on a
self-stick label. Then peel the backing off the
label and place the label on the outside of the
canister.
Taking Note of the Math
Place examples of rational numbers
on the number line. Choose examples
that will help you remember how to
compare and order rational numbers.
Then turn the adding-machine
tape over, and use the other side
to write notes and sample
problems from the chapter.
It’s in the Bag!
93
FLORIDA
CHAPTER
2
Study Guide: Review
Multi–Language
Glossary
Go to thinkcentral.com
Vocabulary
Voca
abulary
a
bulary
least common denominator (LCD) . . . . . .66
reciprocal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
rational number . . . . . . . . . . . . . . . . . . . . . . .62
relatively prime . . . . . . . . . . . . . . . . . . . . . . . . 62
Complete the sentences below with vocabulary words from the
list above.
1. Any number that can be written as a fraction __nd (where n and d
are integers and d 0) is called a ___?___.
2. Integers that have no common factors other than 1 are ___?___.
Study Guide: Review
3. The product of a number and its ___?___ is 1.
EXAMPLES
2-1
■
2-2
■
94
EXERCISES
Rational Numbers (pp. 62–65)
Write 0.8 as a fraction.
8
8 is in the tenths place.
0.8 ___
10
82
______
Divide numerator and
10 2
denominator by 2.
4
__
5
Prep MA.8.A.6.4
Write each decimal as a fraction.
4. 0.6
5. 0.25
Write each fraction as a decimal.
7. __74
Chapter 2 Rational Numbers
9. __79
4
8. __
15
Simplify.
14
10. __
21
22
11. __
33
Comparing and Ordering Rational Numbers (pp. 66–69)
5
__
Compare __23
. Write ⬍, ⬎, or .
8
5
2
__
__
3
8
2
8
16
____ __
24 is the LCD.
3 8 24
5
3 __
____
15
8 3 24
16 __
5
2 ⬎ __
__
⬎ 15 , so __
24 24
3 8
6. 0.525
75
12. ___
100
Prep MA.8.A.6.4
Compare. Write ⬍, ⬎, or .
13. __57
9
__
10
14. __78
28
__
32
Write the numbers from least to greatest.
15. __23 , 0.25, __12 , 0.9
9
16. 0.67, __
, 0, 0.11
10
17. Paul ran these distances with respect to
his daily target: _23, 0.75, 1_12, 0.25, and
_35 mi. Order these distances from least
to greatest.
EXAMPLES
EXERCISES
Adding and Subtracting Rational Numbers (pp. 72–76)
2-3 A
Add or subtract.
3
■ __
7
Add or subtract. Write each answer in
simplest form.
__47
4
__77 1
3_____
7
8
2
■ __ __
11
11
(
)
( 2 )
2
10
8_______
8_____
__
11
11
11
3
2
__
__
■
5
4
18. 0.36 1.097
19. 1.7 0.76
8
2
__
20. ___
13
13
4
21. __35 ___
5
7
2 __
22. ___
9
9
( )
5
7
23. ___
( ___
12
12 )
9
10
__
24. ___
11
11
( 7 )
5
25. __
____
13
13
24
8 Multiply denominators,
__
20 4 5 20.
54
Add and
15
8
15 8
23
3
__
__
______
__
1__
simplify.
20
20
20
20
20
26. __56 __13
27. __56 __59
28. 3__12 7 __45
3
1 2__
29. 7__
4
10
Simplify.
19
__73
30. __
20
31. 1__59 7__34
35
15
__
20
45
■
MA.8.A.6.4
1
2
__
__
5
3
1
2
__
__
3 5
15
3
____
2____
35
53
6
1
5_____
__
15
15
32. 14.2 3.6 2.9 ()
33. __34 __23 __12
2-4 M
Multiplying Rational Numbers (pp. 77–80)
Multiply. Write the answer in simplest form.
■
3(4) 1
5 3__14 __51 _______
4
( ) ( )(
13
( __51 )( __
4)
65
16__14
__
4
■
)
Write as improper
fractions.
Multiply and simplify.
3.2(2.3) 3.2 2.3
7.36
Study Guide: Review
Simplify.
Product is negative
with 3 decimal places.
MA.8.A.6.4
Multiply. Write each answer in simplest
form.
( )
2 ___
4
36. ___
3 ( 5 )
38. 5__14 ( __37 )
40. 4__78 ( 2__23 )
( )
8 ____
22
37. __
11 ( 4 )
3
39. 2__12 ( 1__
10 )
42. 2.4(0.5)
43. 0.36(0.4)
34. 3 __25
35. 2 3__45
41. ( 1.75 )( 4 )
44. A file transferred 2_34 GB of data each
minute for 13_12 minutes. How many
GB of data were transferred?
Lesson Tutorial Videos @ thinkcentral.com
Study Guide: Review
95
EXAMPLES
EXERCISES
Dividing Rational Numbers (pp. 81–85)
2-5 D
■
Divide. Write the answer in simplest form.
Divide. Write each answer in simplest form.
3
7 __
__
__78 __43
4
8
Multiply by the reciprocal.
45. 3.4 0.2
46. 0.1 80
Write as one fraction.
47. __34 __18
3
48. __
__45
10
Remove common factors.
49. __23 3
1
50. 4 ___
4
51. 3__34 3
52. 1__13 __23
74
8 3
1
74
71
83
23
2
1
7 1__
__
6
6
■
Find the quotient.
5.8 0.2
Evaluate each expression for the given
value of the variable.
5.8 __
10
___
0.2 10
0.2 has 1 decimal place,
10
so use __
.
10
4.6
53. ____
n for n 9.2
58
__
29
2
Divide.
55. Waylon spent $84.13 buying mixed nuts
for a party. The nuts cost $8.95 per pound.
How many pounds of mixed nuts did
he buy?
( )
■
Evaluate the expression.
Evaluate a 2__13 for a 7.
Study Guide: Review
MA.8.A.6.4
a 2__13 7 __73
__71 __37
1
3
7_____
17
1
3
10
54. x __
for x 3__37
21
Rewrite ⴚ2_13 as an
improper fraction.
Multiply by the
reciprocal.
Divide out
common factors.
MA.8.A.6.4
Solving Equations with Rational Numbers (pp. 86–89)
2-6 S
Solve.
Solve.
■
x 13.7 22
13.7 13.7
x 8.3
Add 13.7 to each side.
7
■ __x
__25
9
9 __
7
__
x __97 __25
79
9
Multiply both sides by __
.
7
18
x __
35
96
Chapter 2 Rational Numbers
56. y 7.8 14
57. 2.9z 52.2
58. w __34 __18
59. __38 p __34
2
60. x __79 __
11
61. 7.2x 14.4
62. y 18.7 25.9
19
38
t __
63. __
7
21
64. Freda paid $126 for groceries for her
family. This was 116 as much as she paid
the previous time she shopped. How
much did Freda pay on her previous
shopping trip?
FLORIDA
Chapter Test
CHAPTER
2
Simplify.
36
1. __
72
16
3. __
88
21
2. __
35
18
4. __
4
25
Write each decimal as a fraction in simplest form.
5. 0.225
7. 0.101
6. 0.04
8. 0.875
Write each fraction as a decimal.
9. __78
5
11. __
12
13
10. __
25
4
12. __
33
Write the numbers in order from least to greatest.
2
13. 3, 0.36, 0.2, __14
14. 0.55, __78 , 0.8, __56
9
15. __
, 0.7, 1.6, __75
10
Add or subtract. Write each answer in simplest form.
3
4
___
16. ___
11
11
( )
17. 4.5 5.875
18. 8__15 __23
Chapter Test
19. Simplify the expression 7__15 __13 1__23 .
20. Justin worked 2_23 hours on Thursday and 6_34 hours on Friday. How many hours
did he work both days?
Multiply or divide. Write each answer in simplest form.
21. 9( 0.63 )
( )
5
24. 3__37 1__
16
5
22. __78 __
24
9
23. __23 ___
20
25. 34 3.4
26. 4__23 1__16
( )
27. Kory is making Thai food for several friends. She needs to triple her recipe. The
recipe calls for _34 teaspoon of curry. How much curry does she need?
28. Lucie drank _34 pint of bottled water. One serving of the water is _78 pint. How
much of a serving did Lucie drink?
Solve.
29. x __1 __38
30. 3.14y 53.38
x
31. __
11
12
1
32. 2k __
4
33. h 3.24 1.1
34. __47y 24
4
35. Rachel walked to a friend’s house, then to the store, and then back home. The
distance from Rachel’s house to her friend’s house is 1_56 miles. This is twice
the distance from Rachel’s house to the store.How far does Rachel live from
the store?
36. Tickets to an orchestra concert cost $25.50 apiece. If Jamal spent $102, how
many tickets did he purchase?
Chapter 2 Test
97
FLORIDA
CHAPTER
2
Mastering the
Standards
Florida Test
Practice
Go to thinkcentral.com
Cum
Cumulative
mulative A
Asses
Assessment, Chapters 1–2
M
Multiple
ltii l Ch
Choice
i
1. What is the value of the expression
12 k if k 3?
A. 15
C. 9
B. 9
D. 15
7. According to the graph, what fraction
of games resulted in something other
than a tie?
Football Season Results
Ties
Mastering the Standards
2. Which expression is equivalent to
2x 5 if x 4?
F. 13
H. 3
G. 3
I. 13
3. Which value of x is the solution of the
equation __3x 12?
A. x 36
C. x 4
B. x 15
D. x 9
4. Which is the best estimate of the value
of ⏐2.3 k⏐ for k 1.05?
F. 3
H. 1
G. 1
I. 3
5. If a pitcher contains _34 gallon of juice
and each glass will hold _18 gallon of
juice, how many glasses can be filled?
3
A. ___
glass
32
C. 6 glasses
B. _34 glass
D. 8 glasses
6. Skip drove 55.6 miles on Tuesday.
Then he drove another 42_15 miles on
Wednesday. How many more miles
did he drive on Tuesday than on
Wednesday?
98
F. 97.8 miles
H. 13.4 miles
G. 13.5 miles
I. 13.1 miles
Chapter 2 Rational Numbers
Wins
1
10
3
10
3
5
Losses
9
A. ___
10
3
C. ___
10
6
B. ___
15
9
D. ___
50
8. A bag of dried fruit contains 1_23 lb of
fruit. William pours dried fruit from
6_12 bags in a box. The box itself weighs
1_12 lb. What is the total weight of the
fruit and the box?
F. 9_23 lb
H. 12__13 lb
G. 10__56 lb
I. 17__13 lb
9. Which value of x makes the equation
2
_
x _56 true?
3
A. x 1__14
C. x __16
B. x __59
D. x 1__14
10. The solution of the equation 9s =
is
1
__
15 . What number is missing from the
equation?
F. __53
5
H. ___
27
G. __35
I. 15
11. Marie jogs the same distance each day.
After 9 days, the total distance she has
jogged is 40.5 kilometers. Jean-Claude
jogs 2.3 km less than Marie each day.
How far does Jean-Claude jog each day?
A. 2.2 km
C. 6.7 km
B. 4.5 km
D. 38.2 km
H OT !
Tip
Make
M
ake sure you look at all the answer
choices before making your decision. Try
cchoic
substituting each answer choice into the
ssubst
problem if you are unsure of the answer.
prob
p
12. Oscar bought a bag of almonds. He ate
3
ᎏᎏ of the bag on Sunday. On Monday,
8
he ate ᎏ23ᎏ of the almonds left. What
fraction of the entire bag did he eat
on Monday?
H. __14
5
G. ___
12
1
I. ___
12
Gridded Response
13. The diameter of a standard CD is 4ᎏ34ᎏ in.
The diameter of the circular hole in the
middle is ᎏ12ᎏ in. What is the distance, in
inches, from the edge of the hole to
the outer edge of the CD? Write your
answer as a decimal.
S1. A health club charges $39 per month
for membership. Let m represent the
number of months, and let C represent
the total amount of money spent on
the health club membership.
a. Write an equation that relates m
and C.
b. If Jillian has spent $702 on her
membership, how many months has
she been a member of the club?
S2. The sum of 7 and the absolute value of
a number is the same as 12.
a. Write an equation that can be used
to solve for the number.
b. Describe the first step of solving the
equation.
c. Determine how many numbers
make the equation true. Explain
your reasoning.
S3. Brigid has a 21_14 in. long ribbon. For a
project she is cutting it into _34 in. pieces.
Into how many _34 in. pieces can she cut
the ribbon? Show or explain how you
found your answer.
Extended Response
1
2
in. x
E1. Use a diagram to model the expression
4
_
_43.
5
3
4 4 in.
a. Draw a diagram to model the
fraction _45.
14. What is the value of the
expression 2 ⏐3 8⏐?
15. Alana has three times as many pairs of
shoes as Électra. If Alana has 18 pairs
of shoes, how many pairs of shoes does
Électra have?
5
16. Gregory filled a bucket with 4__
gallons
12
11
__
of water. Linda used 3 gallons of the
b. What fraction do you multiply by
that is equivalent to dividing by _43?
c. Use your answer from part b and
shade that fraction of the _45 that is
already shaded. What does this
shaded area represent?
d. Use your diagram to write the
quotient in simplest form.
12
water. How many gallons of water are
left in the bucket?
Cumulative Assessment, Chapters 1-2
99
Mastering the Standards
9
F. ___
16
Short Response