CHAPTE R
7
Factors, Fractions,
and Decimals
conn
connectED.mcgraw-hill.com
The
BIG Idea
Investigate
How do prime and
composite numbers,
factors, and
multiples relate to
fractions?
Animations
Vocabulary
Math Songs
Multilingual
eGlossary
Learn
Personal Tutor
Virtual
Manipulatives
Make this Foldable
to help you organize
information about
factors and
multiples.
e
1 Prim
Lesson ization
r
o
t
c
a
ts
F
ponen
and Ex
Audio
Foldables
Practice
Self-Check Practice
eGames
Worksheets
Assessment
Review Vocabulary
fraction fracción A
number that represents
part of a whole or part
of a set.
1
4
1
4
3
4
Key Vocabulary
English
greatest common
factor (GCF)
least common
multiple
composite number
prime number
exponent
290
1
4
Español
máximo común
divisor (MCD)
mínimo común
múltiplo (mcm)
número compuesto
número primo
exponente
When Will I Use This?
Your Turn!
You will solve thhiis teerrr.
problem in the chap
Factors, Fractions, and Decimals 291
Are You Ready
You have two options for checking
Prerequisite Skills for this chapter.
for the Chapter?
Text Option
Take the Quick Check below.
Write all of the factors of each number.
1. 8
2. 11
3. 6
4. 15
5. 32
6. 24
List the first four multiples of each number.
7. 4
8. 8
9. 3
10. 12
11. 5
12. 10
Find a fraction that is equivalent to each fraction.
2
13. _
3
14. _
6
15. _
2
16. _
1
17. _
5
18. _
5
4
3
8
19.
6
20.
Online Option
292
10
Take the Online Readiness Quiz.
Factors, Fractions, and Decimals
Multi-Part
Lesson
1
Prime Factorization and Exponents
PART
A
Main Idea
I will explore using
models and divisibility
rules to identify prime
and composite
numbers.
Vocabulary
V
prime number
composite number
Materials
color tiles
Get ConnectED
GLE 0506.2.2
Write natural numbers (to 50)
as a product of prime factors
and understand that this is
unique (apart from order).
B
C
D
E
Prime and Composite
Numbers
Three bass drums are
stored on shelves in these
two arrangements.
1
3
1
3
These rectangular
arrangements show that the
only factors of 3 are 1 and 3.
1×3
3×1
When a number, like 3, has exactly two factors,
the number is prime .
1
You can store 4 drums
in any of the three
ways shown at the
right. What are the
factors of 4?
4
1
4
1×4
2
4×1
When a number, like 4,
has more than two
factors, the number is composite .
2
2×2
The numbers 0 and 1 are neither prime nor composite.
Use models to determine whether 6 is prime or composite.
U
6
1
1
1×6
2
3
6
3
2
2×3
3×2
6×1
You can arrange the 6 color tiles in four different ways.
So, 6 is a composite number.
Lesson 1A Prime Factorization and Exponents 293
Use models to determine whether 5 is prime or composite.
U
5
1
1
1×5
5
5×1
You can arrange the 5 tiles in only 2 ways: 5 × 1 and 1 × 5.
So, 5 is a prime number.
You can also use divisibility rules to determine if a number is prime or
composite. If a number is divisible by another number (besides 1 and
itself), then it is composite.
Divisibility Rules
Examples
Even numbers are divisible by 2.
4, 8, 12, 16, 18, 182 are all
divisible by 2.
A number is divisible by 3 if the
sum of the digits is divisible by 3.
342
3+4+2=9
9
9÷3=3
342 is divisible by 3.
Numbers divisible by 5 will end in
a 0 or a 5.
10, 15, 85, and 375 are all
divisible by 5.
There is no common rule for
numbers divisible by 7. Check
by dividing.
14, 21, 70, 140 are all
divisible by 7.
Use divisibility rules to determine if 117 is prime or composite.
U
Try
Is it a factor?
2
No, 117 is not even.
3
Yes, because 1 + 1 + 7 = 9
and 9 ÷ 3 = 3.
So, 117 is a composite number because it is divisible by 3.
Check 117 ÷ 3 = 39. 294
Factors, Fractions, and Decimals
Use divisibility rules to determine if 61 is prime or composite.
U
Try
Is it a factor?
2
No, 61 is not even.
3
No, because 6 + 1 = 7
and 7 is not divisible by 3.
5
No, 61 does not end in 0 or 5.
7
No, 61 ÷ 7 has a remainder.
The divisibility rules show that 61 is a prime number.
About It
1. Are all even numbers composite? Use a drawing in your
explanation.
2. Are all odd numbers prime? Support your explanation with
a drawing.
and Apply It
Use color tiles or divisibility rules to determine whether each
number is prime or composite.
3. 13
4. 27
5. 11
6. 63
7. 71
8. 51
9. Bruce made 12 dinner rolls. He placed the rolls in
3 rows of 4 on a table. In what other ways could
he have arranged the rolls in equal rows?
10. Write a number between 20 and 30. Then use
objects or pictures to show whether the number
is prime or composite.
11.
E
WRITE MATH Is there a connection between the number
of rectangular arrangements that are possible when modeling a
number and the number of factors the number has? Explain your
reasoning.
Lesson 1A Prime Factorization and Exponents 295
Multi-Part
Lesson
1
Prime Factorization and Exponents
PART
A
Main Idea
I will identify prime
and composite
numbers.
Vocabulary
V
even number
odd number
Get ConnectED
GLE 0506.2.2 Write
natural numbers (to 50) as a
product of prime factors and
understand that this is
unique (apart from order).
B
C
D
E
Prime and Composite
Numbers
In Lesson 1A, you learned that a composite number has more
than two factors. So, 12 is a composite number because its
factors are 1, 2, 3, 4, 6, and 12.
The number 3 has only two factors:
1 and 3. So, 3 is a prime number.
The numbers 1 and 0 are neither prime
nor composite.
• 1 has only one factor: 1
• 0 has a never-ending number of factors: 0 × 1, 0 × 2, . . .
Tell whether the number 10
is prime or composite.
The model shows 2 rows of 5 squares.
The squares could also be arranged in 5 rows
of 2 squares, 10 rows of 1 square, or 1 row of 10 squares.
The number 10 is a composite number because it has more
than 2 factors.
Use Divisibility Rules
Tell whether 91 is prime or composite. Use divisibility rules.
Try
Is it a factor?
2
No, 91 is not even.
3
No, because 9 + 1 = 10
and 10 is not divisible by 3.
5
No, 91 does not end in 0 or 5.
7
Yes, 91 ÷ 7 = 13
So, 91 is a composite number because it has more than
two factors.
296
Factors, Fractions, and Decimals
D
DINING
A banquet hall has
24 square tables that are to be
2
placed together to form a
rectangle. Is 24 prime or
composite? What does this
mean in the problem? What
would happen if the banquet
hall had only 23 tables?
factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
You can use models
to identify 24 as
prime or composite.
Twenty-four counters
can be arranged in
equal rows in more
than two ways. So,
24 is composite.
Since 24 has more than two factors, it is a composite number.
This means that there are more than two ways to arrange the
24 tables. Some of the ways are listed below.
• 1 row of 24 tables
• 2 rows of 12 tables
• 3 rows of 8 tables
• 4 rows of 6 tables
If the banquet hall had only 23 tables, there could be only
two possible arrangements, since 23 has only two factors.
This is because 23 is a prime number.
• 1 row of 23 tables
• 23 rows of 1 table each
Tell whether the number represented by each model is prime or
composite. See Example 1
1. 5
2. 6
Tell whether each number is prime or composite. Use objects or
models to justify your answer. See Examples 1 and 3
3. 28
4. 44
5. 61
6. 31
Tell whether each number is prime or composite.
Use divisibility rules. See Example 2
7. 135
8. 51
9. 19
11. Is there more than one way for Mark
to display 21 model cars if each row
has the same number of cars? Explain.
12.
E
10. 119
TALK MATH Is 33 prime or
composite? Explain how you know.
Lesson 1B Prime Factorization and Exponents 297
EXTRA
%
)# E
# T4 IC
!C
2A
0R
P
Begins on page EP2.
Tell
or
T
ll whether
h th the
th number
b represented
t d by
b each
h model
d l is
i prime
i
composite. See Example 1
13. 2
14. 8
15. 7
16. 4
Tell whether each number is prime or composite. Use objects or
models to justify your answer. See Examples 1 and 3
17. 18
18. 29
19. 15
20. 26
21. 13
22. 16
23. 11
24. 53
Tell whether each number is prime or composite.
Use divisibility rules. See Example 2
25. 58
26. 3
27. 87
28. 150
29. 37
30. 752
31. 4,002
32. 2,433
33. A mountain range has 90 mountains
that are one mile or more in height.
Is 90 a prime or composite number?
35. FIND THE ERROR Rico is
determining whether 119 is
prime or composite using the
divisibility rules. Help find and
correct his mistake.
34. Brian’s birthday is February 29. Is 29 a
prime or composite number?
Since 119 is
not divisible by
2, 3, or 5, it must
be prime.
36. CHALLENGE Two prime numbers that have a difference of 2 are
called twin primes. For example, 5 and 7 are twin primes. Find all
pairs of twin primes less than 50.
37.
E
WRITE MATH Explain how you can use objects or models to
tell if a number is prime or composite.
298
Factors, Fractions, and Decimals
Test Practice
38. The table shows how many Calories
you can burn in 10 minutes for
certain activities.
Activity
Number of Calories
Basketball
64
Dancing
35
Hiking
47
Roller skating
57
40. Which group of numbers below are
all composite?
A. 6, 15, 28, 100
B. 3, 14, 37, 115
C. 13, 40, 52, 63
D. 13, 18, 77, 210
For which activity is the number of
Calories a prime number?
A. basketball
41. The table shows the number of
students participating in each activity.
Activity
B. dancing
C. hiking
D. roller skating
39. Rachel’s birthday is a date in
November that is a prime number.
Which of the following could be her
birthday?
Number of Students
Baseball
24
Science Club
19
Student Council
17
Ski Club
23
For which activity is the number of
students a composite number?
F. baseball
G. science club
F. November 21
H. November 15
H. student council
G. November 19
I. November 10
I. ski club
Odd and Even Numbers
An even number is a whole number that is divisible by 2.
An odd number is not divisible by 2. An even number of
objects can be divided into two equal sets; an odd number
of objects cannot.
Classify each number as even or odd.
42. 25
43. 120
10 is an even number.
11 is an odd number.
44. 256
45. 1,001
46. MAKE A CONJECTURE Is the sum of two odd numbers always,
sometimes, or never an odd number? Use a model to explain
your reasoning.
Lesson 1B Prime Factorization and Exponents 299
Multi-Part
Lesson
1
Prime Factorization and Exponents
PART
A
Main Idea
I will find the prime
factorization of
numbers.
Vocabulary
V
prime factorization
square numbers
B
C
D
E
Prime Factorization
You can write every composite number as a product of prime
factors. This is called the prime factorization of a number.
A factor tree is a diagram that shows the prime factorization of
a composite number.
square root
Get ConnectED
GLE 0506.2.2 Write
natural numbers (to 50) as a
product of prime factors and
understand that this is unique
(apart from order). SPI 0506.2.2
Write the prime factorization of
numbers through 50 using
both exponential and standard
notation. Also addresses SPI
0506.2.7.
A
AGE
Mr. Dempsey tells
his
h class that he is
36 years old. Find the
prime factorization of 36.
One Way
36
Another Way
Write the number to be
factored at the top.
36
2 × 18
Choose any pair of whole
number factors of 36.
3 × 12
2×2 × 9
Continue to factor any
number that is not prime.
3×3 × 4
2×2×3×3
Except for the order, the
prime factors are the same.
3×3×2×2
The prime factorization of 36 is 2 × 2 × 3 × 3.
Check
Check your answer by working backward. Multiply all the
prime factors. Then compare your product with the
composite number.
2 × 2 = 4, 4 × 3 = 12, 12 × 3 = 36 300
Factors, Fractions, and Decimals
Prime Factorization
Find the prime factorization of 24.
24
You can choose any
pair of whole number
factors, such as 6 × 4
or 12 × 2. Except for
the order, the prime
factors of the number
are the same.
Choose any pair of whole
number factors of 24.
3 × 8
Continue to factor any
number that is not prime.
3×2×4
3×2×2×2
The prime factorization of 24 is 2 × 2 × 2 × 3.
Prime Factorization
Find the prime factorization of 37.
Use divisibility rules.
Neither 2, 3, 5, or 7 are factors of 37.
So, 37 is a prime number.
The prime factorization is 37.
Find the prime factorization of each number.
number See Examples 11–33
1. 16
2. 22
3. 30
4. 42
5. 50
6. 81
7. 65
8. 19
9. The state of Pennsylvania has 67 counties.
Write the prime factorization of 67.
10. There are 45 students in the gymnasium.
Find the prime factorization of 45.
11.
E
Pennsylvania
TALK MATH What are the first ten
prime numbers?
Lesson 1C Prime Factorization and Exponents 301
EXTRA
%
)# E
# T4 IC
!C
2A
0R
P
Begins on page EP2.
Find
Fi
d th
the prime
i
ffactorization
t i ti
off each
h number.
b
See Examples
l 1–3
12. 63
13. 18
14. 40
15. 75
16. 27
17. 32
18. 49
19. 25
20. 44
21. 104
22. 55
23. 77
Use the table that shows the average
weights of popular dog breeds.
24. Which weight(s) have a prime
factorization of exactly three factors?
25. Which weight(s) have a prime
factorization with factors that are
all the same number?
26. Which dog breeds have weights
that are prime numbers?
Breed
Weight (lb)
Cocker Spaniel
20
German Shepherd
81
Labrador Retriever
67
Beagle
25
Golden Retriever
70
Siberian Husky
50
Boxer
60
Rottweiler
112
Dalmatian
55
Poodle
57
27. Of the Beagle, Golden Retriever, Siberian Husky,
Rottweiler, and Dalmatian breeds, which have
weights that are composite numbers?
28. CHALLENGE Find the prime factorization of 2,800.
29. WHICH ONE DOESN’T BELONG? Which of the numbers below is
not a prime factor of 70?
2
7
3
30. REASONING Explain why the prime factorization 3 × 3 × 5 × 7
is for the same number as the prime factorization 5 × 3 × 3 × 7.
31.
E
WRITE MATH Explain how tree diagrams help you find
the prime factorization of a number.
302
Factors, Fractions, and Decimals
5
Factors
Numbers that have two identical factors are called square numbers .
For example, 9 is a square number.
3×3=9
A square root of a number is one of two identical factors of a number.
The square root of 9 is 3.
The table shows other examples of square numbers and square roots.
Model
Multiplication Fact
Square Number
Square Root
2×2=4
4
2
4 × 4 = 16
16
4
Name the square number and square root shown in each model.
32.
33.
34. Use the multiplication fact 7 × 7 = 49 to name a square number
and its square root.
35. What is the largest square number less than 200?
36. MAKE A CONJECTURE The prime factorization of 100 is
2 × 2 × 5 × 5. Explain how to find a square root of 100 using
the prime factorization.
Lesson 1C Prime Factorization and Exponents 303
Multi-Part
Lesson
1
PART
Prime Factorization and Exponents
A
B
D
C
E
Exponents
Main Idea
I will explore using
exponents.
Materials
hole punch
construction paper
Any number can be written
as a product of prime factors.
Step 1
Fold a piece of
paper in half
and make one
hole punch.
Open the paper
and count the number of holes. Copy the
table below and record the results.
Get ConnectED
GLE 0506.1.1
Use mathematical language,
symbols, and definitions
while developing mathematical
reasoning. SPI 0506.2.2 Write
the prime factorization of
numbers through 50 using
both exponential and standard
notation. Also addresses GLE
0506.1.3.
Number
of Holes
Prime
Factorization
…
Number
of Folds
1
5
Step 2
Find the prime factorization of the number of
holes and record the results in the table.
Step 3
Fold another piece of paper in half twice. Then
make one hole punch. Complete the table for
two folds.
Step 4
Complete the table for three, four, and five folds.
About It
1. What prime factors did you record?
2. How does the number of folds relate to the number of
factors in the prime factorization of the number of holes?
3. Write the prime factorization of the number of holes made if
you folded it eight times.
304
Factors, Fractions, and Decimals
Multi-Part
Lesson
1
PART
Prime Factorization and Exponents
A
Main Idea
I will use powers
and exponents in
expressions.
Vocabulary
V
exponent
base
power
B
C
E
D
Powers and Exponents
A product of identical factors can be written using an exponent
and a base. The base is the number used as a factor. The
exponent indicates how many times the base is used
as a factor.
2 × 2 × 2 × 2 × 2 = 25
5 factors
squared
cubed
Get ConnectED
GLE 0506.1.1
Use mathematical language,
symbols, and definitions
while developing mathematical
reasoning. SPI 0506.2.7
Recognize equivalent
representations for the same
number. Also addresses
SPI 0506.2.2.
exponent
base
Numbers expressed using exponents are called powers .
Numbers raised to the second or third power have special names.
Powers
Words
25
2 to the fifth power
32
3 to the second power or 3 squared
103
10 to the third power or 10 cubed
Use Exponents
Write 3 × 3 × 3 × 3 using an exponent.
W
The base is 3. Since 3 is used as a factor four times, the
exponent is 4.
3 × 3 × 3 × 3 = 34
Write as a power.
FOOD The number of Calories
in two pancakes can be written
as 73. Write 73 as a product of
the same factor. Then find
the value.
Write 73 as 7 × 7 × 7.
7 × 7 × 7 = 343
Two pancakes have 343 Calories.
Lesson 1E Prime Factorization and Exponents 305
You can calculate
expressions with a
base of 10 mentally.
⎧
⎨
⎩
104 = 10,000
4 zeros
ENVIRONMENT In a recent year, about 104 youth across
the United States participated in activities and events to
care for Earth’s environment. What is this number?
104 = 10 × 10 × 10 × 10
Write 104 as a product.
= 10,000
Multiply.
About 10,000 youth participated in these events.
Prime Factorization Using Exponents
Write the prime factorization of 72 using exponents.
W
72
9 × 8
3×3×2×4
3×3×2×2×2
2×2×2×3×3
23
32
Order factors from least to greatest.
Write products of identical factors
using exponents.
So, 72 = 23 × 32.
Write each product using an exponent
exponent. See Examples 11–44
1. 2 × 2 × 2 × 2
2. 6 × 6 × 6
Write each power as a product of the same factor. Then find
the value.
3. 26
4. 37
Write the prime factorization of each number using exponents.
5. 20
6. 48
8. There are nearly 35 species of monkeys on Earth.
What is the value of 35?
9.
E
TALK MATH Explain how a factor tree helps
you to write the prime factorization of a number
using exponents.
306
Factors, Fractions, and Decimals
7. 90
EXTRA
%
)# E
# T4 IC
!C
2A
0R
P
Begins on page EP2.
Write
i each
h product
d
using
i an exponent. See Examples 1–4
10. 9 × 9
11. 8 × 8 × 8 × 8
12. 3 × 3 × 3 × 3 × 3 × 3 × 3
13. 5 × 5 × 5 × 5 × 5
Write each power as a product of the same factor. Then find the value.
14. 103
15. 32
16. 54
17. 105
18. 93
19. 65
20. 101
21. 17
22. A single tusk that weighed just over 28 poundss
from an African elephant is the largest tooth ever
recorded from any modern animal. About how
w
many pounds did the tusk weigh?
Write the prime factorization of each number
using exponents.
23. 25
24. 56
25. 50
26. 68
27. 88
28. 98
29. 560
30. 378
31. 2,205
32. To find the amount of space a cube-shaped
bird cage occupies, find the cube of the
measure of one edge of the bird cage.
Express the amount of space occupied
by the bird cage shown as a power. Then
find the amount in cubic units.
18 units
18 units
18 units
33. OPEN ENDED Write a power whose value is greater than 100.
34. NUMBER SENSE Which is greater: 35 or 53? Explain your reasoning.
35.
E
WRITE MATH Explain how to find 106 mentally.
Lesson 1E Prime Factorization and Exponents 307
Multi-Part
Lesson
2
PART
Fractions in Simplest Form
A
B
C
Common Factors
Main Idea
I will find common
factors using Venn
diagrams.
Get ConnectED
A Venn diagram uses circles to display elements of different sets.
Overlapping circles show common elements.
S
SUMMER
CAMP The Venn diagram shows which activities
each
camper participated in on Monday. Who participated
e
in both swimming and crafts?
GLE 0506.2.2 Write
natural numbers (to 50) as a
product of prime factors and
understand that this is unique
(apart from order). SPI 0506.2.2
Write the prime factorization of
numbers through 50 using
both exponential and standard
notation.
Swimming
This circle
represents
swimming.
Crafts
4BWBOOBI
5ZMFS
0XFO
*TBCFM
-VJT
4POJB
.JLP
This circle
represents
crafts.
This part represents
both swimming
and crafts.
Owen and Isabel are in both circles.
So, they participated in both swimming and crafts.
Factors that are shared by two or more numbers are called
common factors.
The common factors
of 12 and 20 are:
1, 2, and 4.
About It
1. Use a Venn diagram to find the common factors of
30 and 45.
2.
E
TALK MATH Explain what it means if a factor is in both
circles of a Venn diagram.
308
Factors, Fractions, and Decimals
Multi-Part
Lesson
2
Fractions in Simplest Form
PART
A
Main Idea
Find the greatest
common factor of two
or more numbers.
Vocabulary
V
common factor
greatest common
factor (GCF)
B
C
Greatest Common
Factor
Factors shared by two or more numbers are called
common factors . The greatest of the common factors of
two or more numbers is the greatest common factor (GCF)
of the numbers.
Get ConnectED
Identify Common Factors
GLE 0506.2.2 Write
natural numbers (to 50) as a
product of prime factors and
understand that this is unique
(apart from order). SPI 0506.2.2
Write the prime factorization of
numbers through 50 using
both exponential and standard
notation. Also addresses
GLE 0506.1.7.
Identify the common factors of 16 and 24.
First, list the factors by pairs
for each number. Then,
circle the common factors.
Factors of 16
The common factors are
1, 2, 4, and 8.
Factors of 24
1 × 16
1
2
3
4
2×8
4×4
×
×
×
×
24
12
8
6
Find the GCF by Listing Factors
Find the GCF of 60 and 54.
Make an organized list of the factors for each number.
factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
factors of 54: 1, 2, 3, 6, 9, 18, 27, 54
The common factors are 1, 2, 3, and 6. So, the greatest
common factor, or GCF, of 60 and 54 is 6.
Check Use a Venn diagram to show
the factors. The common factors
are 1, 2, 3, and 6. 'BDUPSTPG 'BDUPSTPG
Lesson 2B Fractions in Simplest Form 309
You can use prime factorization to determine the greatest
common factor.
Divisibility tests are a
good way to find
factors. 18 and 30 are
even numbers. They
are both divisible by 2.
Find the GCF by Using
Prime Factorization
Find the GCF of 18 and 30.
Write the prime factorization.
18
30
2×9
2 × 15
2×3×3
2×3×5
2 and 3 are common factors.
The common prime factors are 2 and 3.
So, the GCF of 18 and 30 is 2 × 3 or 6.
FOOD A bakery arranges
Muffins
three different types of
Type
Number
muffins in a display case.
blueberry
40
There should be an equal
cinnamon raisin
24
number of muffins in
chocolate chip
32
each row in the case.
What is the greatest possible
number of muffins in each row?
prime factorization of 40: 2 × 2 × 2 × 5
prime factorization of 24: 2 × 2 × 2 × 3
prime factorization of 32: 2 × 2 × 2 × 2 × 2
The common prime factors are 2, 2, and 2.
The GCF of 40, 24, and 32 is 2 × 2 × 2 or 8. So, the greatest
number of muffins that could be placed in each row is 8.
How many rows of muffins are there if there are 8 in
each row?
There are a total of 40 + 24 + 32, or 96 muffins.
So, the number of rows of muffins is 96 ÷ 8, or 12.
310
Factors, Fractions, and Decimals
Identify the common factors of each set of numbers.
numbers See Example 1
1. 11, 44
2. 12, 21, 30
Find the GCF of each set of numbers. See Examples 2–4
3. 8, 32
4. 24, 60
5. 3, 12, 18
6. 4, 10, 14
Solve. See Example 5
7. Oliver has 14 chocolate cookies and 21 iced cookies. Oliver
gives each of his friends an equal number of each type of cookie.
What is the greatest number of friends with whom he can share
his cookies?
8.
E
TALK MATH Refer to Exercise 7. Explain how you could find
how many cookies each friend would receive. Then solve.
EXTRA
%
)# E
# T4 IC
!C
2A
0R
P
Begins on page EP2.
Identify
d
if the
h common factors
f
off each
h set off numbers.
b
See Example 1
9. 45, 75
11. 6, 21, 30
10. 36, 90
12. 16, 24, 40
Find the GCF of each set of numbers. See Examples 2–4
13. 12, 18
14. 18, 42
15. 48, 60
16. 30, 72
17. 14, 35, 84
18. 9, 18, 42
19. 16, 52, 76
20. 12, 30, 72
21. Annika is placing photos in a scrapbook. Each page
will have only one size of photo. She also wants to
place the same amount of photos on each page.
What is the greatest number of photos that could
be on each page? Justify your response.
Scrapbooking
PHOTO SIZE
Large
Medium
Small
AMOUNT
8
12
16
Lesson 2B Fractions in Simplest Form 311
22. A grocery store sells boxes of juice in equal-size packs. Carlos
bought 18 boxes, Rico bought 36 boxes, and Winston bought
45 boxes. What is the greatest number of boxes in each pack?
How many packs did each person buy?
23. The table shows the number of each
type of toy in a store. The toys will be
placed on shelves so that each shelf
has the same number of each type of
toy. How many shelves are needed for
each type of toy so that it has the greatest
number of toys?
Toy
dolls
Number
45
footballs
105
small cars
75
24. The table shows the amount of money Ms. Ayala
made over three days selling 4-by-6-inch prints at
an arts festival. Each print costs the same amount.
What is the most each print could have cost?
Ms. Ayala’s Artwork
Day
Amount ($)
Friday
Saturday
Sunday
25. What is the GCF of all the numbers in the pattern
9, 18, 27, 36, . . .? Explain your reasoning.
Use the information to solve the problem.
26. What is the length of the longest piece of sandwich that can
be cut so that all 16 guests get the same-sized sandwich? Explain.
312
Factors, Fractions, and Decimals
60
144
96
27. CHALLENGE Determine whether each statement is true or false.
If true, explain why. If false, give a reason.
a. The GCF of any two even numbers is always even.
b. The GCF of any two odd numbers is always odd.
c. The GCF of an odd number and an even number is always even.
28. WHICH ONE DOESN’T BELONG? Which number can you take
away so that 8 will be the GCF?
16
29.
E
8
24
20
WRITE MATH Which method would you prefer to use to find
the GCF of 48, 64, and 144? Explain your reasoning.
Test Practice
30.
SHORT RESPONSE Find the
greatest common factor of the
numbers below.
28, 42, 70
32. Jeremiah will share his collection
with his brother so that they each
have the same number of each
type of card. What is the greatest
number of baseball cards they will
each have?
31. Which number is NOT a common
factor of 24 and 36?
Sports Cards
Type
Number
A. 2
B. 6
baseball
32
football
48
C. 12
F. 4 cards
H. 12 cards
D. 24
G. 8 cards
I. 16 cards
33. California has 52 area codes. What is the value of 52? (Lesson 1E)
Find the prime factorization of each number. (Lesson 1C)
34. 63
35. 46
36. 56
37. 90
Lesson 2B Fractions in Simplest Form 313
Multi-Part
Lesson
2
Fractions in Simplest Form
PART
A
Main Idea
I will use the GCF to
write a fraction in
simplest form.
Vocabulary
V
B
C
Simplest Form
A fraction is written in simplest form when the GCF of the
numerator and the denominator is 1. The simplest form of a
fraction is one of its many equivalent fractions.
simplest form
Get ConnectED
GLE 0506.1.1
Use mathematical language,
symbols, and definitions
while developing mathematical
reasoning. SPI 0506.2.7
Recognize equivalent
representations for the same
number.
MEASUREMENT A praying mantis
is 12 centimeters long, and a
walking stick is 22 centimeters
long. So, a praying mantis is
12
_
of
22
the length of a walking stick. Write
the fraction in simplest form.
Step 1 Find the GCF of the
numerator and the
denominator.
factors of 12: 1, 2, 3, 4, 6, 12
factors of 22: 1, 2, 11, 22
The GCF of 12 and 22 is 2.
Step 2 Divide both the numerator and the denominator by
the GCF. Dividing both the numerator and the
denominator by the same number is equivalent
to dividing by one.
12 ÷ 2
6
12
_
=_=_
22
22 ÷ 2
11
The GCF of 6 and 11 is 1.
So, a praying mantis’s length is _ of the length of a
11
walking stick.
6
Check
Use models.
12
22
6
12
So, _ = _. 6
11
22
314
Factors, Fractions, and Decimals
11
Simplest Form
Write
W
Equivalent fractions are
fractions that have the
same value.
s
3 These fraction
18 = _
_
nt.
ale
5 are equiv
30
18
_
in simplest form.
30
One Way:
Divide by Common Factors
18 ÷ 2
18
_
_
_9 Divide 18 and 30 by the common factor 2.
=
=
30
30 ÷ 2
15
9÷3
3
_9 = _
= _5
15
15 ÷ 3
Divide 9 and 15 by the common factor 3.
Since 3 and 5 have no common factors other than 1, stop
dividing.
Another Way:
Divide by the GCF
factors of 18: 1, 2, 3, 6, 9, 18
factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
The GCF of 18 and 30 is 6.
18 ÷ 6
18
_
_
_3
=
=
5
30
30 ÷ 6
Divide by the GCF 6.
18
3
Using either method, _ written in simplest form is _.
5
30
Write each fraction in simplest form.
form If the fraction is already in
simplest form, write simplified. See Examples 1 and 2
4
1. _
2
2. _
8
3. _
8
4. _
9
5. _
4
6. _
15
7. _
21
8. _
6
18
12
14
9. Kara buys 24 bagels. Ten are whole
wheat. What fraction of the bagels are
whole wheat, in simplest form?
24
20
10.
E
9
35
TALK MATH Use at least two
sentences to explain how to find the
simplest form of any fraction.
Lesson 2C Fractions in Simplest Form
315
EXTRA
%
)# E
# T4 IC
!C
2A
0R
P
Begins on page EP2.
W it each
Write
h ffraction
ti
iin simplest
i l t fform. If th
the ffraction
ti
iis already
l d iin
simplest form, write simplified. See Examples 1 and 2
6
11. _
6
12. _
3
13. _
2
14. _
4
15. _
12
16. _
6
17. _
21
18. _
12
19. _
40
4
20. ___
11
8
21. _
9
22. _
3
23. _
25
24. _
18
25. _
36
26. _
8
10
24
16
36
5
18
25
30
24
28
45
30
27. A basket of fruit has 10 oranges,
12 apples, and 18 peaches. Express in
simplest form the fraction of fruit that
are oranges.
48
28. Measurement Andeana is 4 feet
tall. Her brother Berto is 38 inches tall.
What fractional part of Andeana’s
height is Berto’s height?
29. In a typical symphony orchestra, 16 out of every
100 musicians are first and second violin players. Express the
fraction of the orchestra that are violinists in simplest form.
30. The table shows the results of a survey about
favorite movie theater snacks. Write a fraction
in simplest form that compares the number
of people who chose popcorn to the total
number of people surveyed.
Favorite Movie Snack
k
Snack
Frequency
cy
popcorn
24
hot dog
12
nachos
11
chocolate
8
licorice
5
31. OPEN ENDED Write a real-world problem that uses _ in the
18
problem. Write the fraction in simplest form.
14
32. WHICH ONE DOESN’T BELONG? Identify the fraction that does
not belong with the other three. Explain your reasoning.
3
_
12
33.
316
E
4
_
16
WRITE MATH Explain how you would write
Factors, Fractions, and Decimals
5
_
25
24
_
in simplest form.
36
6
_
24
Test Practice
34. Gil’s aunt cut his birthday cake into
32 equal pieces, as shown below.
Eighteen pieces were eaten at his
birthday party. What fraction of the
cake was left?
5
2 3 4
36. The fractions _, _, _, and _ can
8 12 16
1
20
all be simplified to _. What is the
4
relationship between the numerator
and denominator in each fraction?
F. The numerator is 4 times the
denominator.
G. The denominator is 4 times the
numerator.
7
A. _
16
9
B. _
16
35.
H. The numerator is 4 more than
the denominator.
7
C. _
12
9
D. _
14
I. The denominator is 4 more than
the numerator.
GRIDDED RESPONSE Amelia
12
rode _ mile on the bike trail.
37.
20
What is the greatest common factor
of 12 and 20?
SHORT RESPONSE Joshua
answered 95 out of 100 test
questions correctly. Express the
fraction of correct answers in
simplest form.
38. Thirty-six fourth graders, 48 fifth graders, and 24 sixth graders
will attend a play. An equal number of students must sit in
each row, and only students from the same grade can sit in
a row. What is the greatest number of fifth graders that can
sit in each row? (Lesson 2B)
Write each product using an exponent. (Lesson 1E)
39. 4 × 4 × 4
40. 9 × 9
41. 6 × 6 × 6 × 6 × 6
42. Grant has $225 in his savings account. Write the prime
factorization of 225. (Lesson 1C)
43. A tangerine has about 37 Calories. Is 37 prime or
composite? (Lesson 1B)
To assess mastery of SPI 0506.2.2, see your Tennessee Assessment Book.
317
Mid-Chapter
Check
Tell whether each number is prime or
composite. (Lesson 1B)
Find the GCF of each set of numbers.
(Lesson 2B)
1. 15
2. 36
15. 9, 21
16. 12, 26
3. 19
4. 28
17. 20, 30, 40
18. 8, 24, 32
5. MULTIPLE CHOICE Which model does
NOT represent a composite number?
(Lesson 1B)
A.
19. MULTIPLE CHOICE Devin recorded the
shirt color of the 30 students who rode
his bus on Monday. The results are
shown below.
B.
C.
D.
Find the prime factorization of each
number. (Lesson 1C)
6. 16
7. 50
8. 63
9. 120
Identify the common factors of each set
of numbers. (Lesson 2B)
10. 5, 15
11. 12, 30
12. 24, 32, 40
13. 10, 22, 30
14. MULTIPLE CHOICE Which group shows
all the numbers that are common
factors of 24 and 40? (Lesson 2B)
318
F. 1, 2, 4
H. 1, 2, 4, 6
G. 1, 2, 4, 8
I.
Mid-Chapter Check
1, 2, 4, 6, 8, 12
Which fraction of shirts were red?
(Lesson 2C)
1
A. _
3
1
B. _
5
1
C. _
4
1
D. _
6
Write each fraction in simplest form. If
the fraction is already in simplest form,
write simplified. (Lesson 2C)
8
20. _
6
21. _
9
22. _
25
23. _
24
20
24.
E
43
14
30
WRITE MATH Explain how to write
as a product of its factors. Then find
its value. (Lesson 1E)
Multi-Part
Lesson
3
Write Multiples and Compare Fractions
PART
A
B
C
D
Least Common Multiple
Main Idea
I will explore finding
the least common
multiple of two
numbers.
Step 1
S
Draw a number line from 0 to 15.
0
Materials
color tiles
Step 2
0
Get ConnectED
Step 3
GLE 0506.2.2 Write
natural numbers (to 50) as a
product of prime factors and
understand that this is
unique (apart from order).
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
Find the product of 2 and each of the numbers
1, 2, 3, 4, 5, 6, and 7. Place a red tile above each
of the products on the number line.
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
Find the product of 3 and each of the numbers
1, 2, 3, 4, and 5. Place a blue tile above each of
the products on the same number line.
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
About It
1. Which numbers have both a red and a blue tile?
2. What is the least number that has a red and blue tile?
and Apply It
Use a number line and color tiles to find the least number
that is a product of each of the numbers.
3. 2, 4
6.
E
4. 3, 6
5. 2, 6
WRITE MATH Explain how you can use color tiles to find
the common products of 2, 4, and 5.
Lesson 3A Write Multiples and Compare Fractions 319
Multi-Part
Lesson
3
Write Multiples and Compare Fractions
PART
A
Main Idea
Find the least common
multiple of two or
more numbers.
Vocabulary
V
B
C
D
E
F
G
Least Common Multiple
A multiple of a number is the product of the number and any
other whole number (0, 1, 2, 3, 4, . . . ). Multiples that are shared
by two or more numbers are common multiples .
multiple
Identify Common Multiples
common multiples
least common multiple
(LCM)
Identify the first three common multiples of 4 and 8.
First, list the nonzero multiples of each number.
Get ConnectED
GLE 0506.2.2 Write
natural numbers (to 50) as a
product of prime factors and
understand that this is unique
(apart from order).
multiples of 4: 4, 8, 12, 16, 20, 24, . . . 1 × 4, 2 × 4, 3 × 4, …
multiples of 8: 8, 16, 24, 32, 40, 48, . . . 1 × 8, 2 × 8, 3 × 8, …
The first three common multiples of 4 and 8 are 8, 16, and 24.
The least common multiple (LCM) is the least multiple, other
than 0, common to sets of multiples.
FOOD Ben’s Burgers gives away a free order
of fries every 2 days, a free milkshake
every 3 days, and a free hamburger every
4 days. If they gave away all three items
today, in how many days will they give
away all three items again?
S hake
F ries
Find the LCM of 2, 3 and 4.
multiples of 2: 2, 4, 6, 8, 10, 12 . . . 2 × 1, 2 × 2, 2 × 3, 2 × 4, …
multiples of 3: 3, 6, 9, 12, 15, 18 . . . 3 × 1, 3 × 2, 3 × 3, 3 × 4, …
multiples of 4: 4, 8, 12, 16, 20 . . . 4 × 1, 4 × 2, 4 × 3, 4 × 4, …
Notice that 12 is the least common multiple of 2, 3, and 4.
So, Ben’s Burgers will give away all three items again in 12 days.
Draw a number line to check.
H
M
F
Day 0
320
H
F M F
1
Factors, Fractions, and Decimals
2
3
4
M
F
5
6
H
F M
7
8
F
H
M
F
F
M
9 10 11 12 13 14 15
Find the LCM
Find the LCM of 15 and 40.
Step 1
You can use a factor
tree to find the prime
factorization.
Write the prime
factorization of
each number.
15
40
3×5
8×5
4×2 ×5
2×2 ×2 ×5
Step 2 Identify all common
prime factors.
15 = 3 × 5
40 = 2 × 2 × 2 × 5
5 is a common prime factor.
Step 3 Find the product of the prime factors using
each common prime factor only once and any
remaining factors.
The LCM is 2 × 2 × 2 × 3 × 5 or 120.
Identify the first three common multiples of each set of
numbers. See Examples 1–3
1. 7, 14
2. 2, 8
3. 2, 4, 6
4. 3, 6, 12
Find the LCM of each set of numbers.
5. 6, 10
6. 2, 13
7. 4, 7, 10
8. 6, 7, 9
9. Juan gets an allergy shot every 3 weeks. Percy gets an allergy shot
every 5 weeks. If Juan and Percy meet while getting an allergy shot,
how many weeks will it be before they see each other again?
10.
E
TALK MATH Could the LCM of two numbers be one of
the numbers? Explain. Support your answer with an example.
Lesson 3B Write Multiples and Compare Fractions
321
EXTRA
%
)# E
# T4 IC
!C
2A
0R
P
Begins on page EP2.
Identify
d
if the
h fi
first three
h
common multiples
l i l off each
h set off
numbers. See Examples 1–3
11. 2, 10
12. 1, 7
13. 6, 9
14. 3, 8
15. 4, 8, 10
16. 3, 9, 18
Find the LCM of each set of numbers.
17. 3, 4
18. 7, 9
19. 16, 20
20. 15, 12
21. 15, 25, 75
22. 9, 12, 15
23. A full moon occurs about every 30 days. If the last full
moon occurred on a Friday, how many days will pass
before a full moon occurs again on a Friday?
24. The cycles for two different events are shown in the table.
Each of these events happened in the year 2000.
What is the next year in which both will happen?
Event
Summer Olympics
United States Census
Cycle (yr)
4
10
25. FIND THE ERROR Maria is finding the LCM
of 6 and 8. Help find and correct her mistake.
istake.
6=2×3
8=2×2×2
The LCM of 6 and
8 is 2.
26. CHALLENGE Is the statement below always, sometimes, or never
true? Give at least two examples to support your reasoning.
The LCM of two numbers is the product of the two numbers.
27.
E
WRITE MATH Write a real-world problem in which it would be
helpful to find the least common multiple.
322
Factors, Fractions, and Decimals
Test Practice
28. Micah is buying items for a birthday
party. If he wants to have the same
amount of each item, what is the
least number of packages of cups he
needs to buy?
Party Supplies
Number in
Item
Each Package
cups
6
plates
8
29. What is the least common multiple
of 5, 9, and 15?
F. 3
H. 45
G. 29
I. 60
30. Look at the patterns in each
sequence below. Each sequence is
an example of which kind of
numbers?
3, 6, 12, 24, 48
A. 2 packages
5, 10, 20, 40, 80
B. 3 packages
8, 16, 32, 64, 128
C. 4 packages
D. 5 packages
A. even numbers
C. multiples
B. odd numbers
D. prime numbers
31. A container of bagels has 10 plain, 5 blueberry, 6 poppy seed,
and 3 mixed grain bagels. What fraction of bagels are poppy seed?
Write in simplest form. (Lesson 2C)
Find the GCF of each set of numbers. (Lesson 2B)
32. 9, 12
33. 32, 24
34. 27, 36
35. 16, 40
36. 22, 55
37. 14, 28, and 42
Write each product using an exponent. (Lesson 1E)
38. 6 × 6 × 6 × 6
39. 10 × 10 × 10
40. 7 × 7 × 7 × 7
41. Denzel has a collection of 149 magnets. Is 149
a prime or composite number? (Lesson 1B)
Lesson 3B Write Multiples and Compare Fractions
323
Multi-Part
Lesson
3
Write Multiples and Compare Fractions
PART
A
B
C
D
Problem-Solving Strategy:
Look for a Pattern
Main Idea I will solve problems by looking for a pattern.
Shawna is saving money to buy an
airplane ticket to visit her aunt. Each
month she puts money into her
savings account. Based on the
pattern in the table, determine how
much money Shawna will have in July.
Understand
Month
Total in Savings
January
February
March
April
$35
$70
$105
$140
What facts do you know?
• We know how much money Shawna has saved for four months.
• The amount in her account increases according to a pattern.
What do you need to find?
• The amount of money in Shawna’s account in July.
Plan
One way to solve the problem is by looking for a pattern. Then extend the
pattern to find the amount of money in her account in July.
Solve
Use your plan to solve the problem.
Jan
$35
Feb
$70
+35
Mar
$105
+35
Apr
$140
May
June
July
+35
The amount in Shawna’s savings account increases each month by $35.
Continue the pattern to find the total in July.
Jan
$35
Feb
$70
Mar
$105
Apr
$140
May
$175
+35
June
$210
+35
July
$245
+35
In July, Shawna will have $245 in her savings account.
Check
Since July is the 7th month find the first seven multiples of 35. They are 35,
70, 105, 140, 175, 210, and 245. GLE 0506.2.2 Write natural numbers (to 50) as a product of prime factors and understand that this is unique
(apart from order). GLE 0506.1.2 Apply and adapt a variety of appropriate strategies to problem solving, including
estimation, and reasonableness of the solution. Also addresses GLE 0506.1.5.
324
Factors, Fractions, and Decimals
Refer to the problem on the previous page.
1. How much money will Shawna have in
her account in August?
3. Explain when to use the look for a
pattern strategy to solve a problem.
2. If the airline ticket costs $315, when
can Shawna stop saving?
4. Can you always use the look for a
pattern strategy when solving a
problem?
EXTRA
%
)# E
# T4 IC
!C
2A
0R
P
Begins on page EP2.
Solve. Use the look for a pattern
strategy.
5. Draw the next two figures in the
pattern.
6. Stefano is buying a few pencils. The
table shows the price of different
numbers of pencils.
For Exercises 8–10, use the following
information.
Gavin rode his bike for a longer distance
each day while training. Here is his record
of the number of miles he rode.
Mon
Tues
Wed
Thurs
Fri
3.5 mi
4.2 mi
5.0 mi
6.9 mi
8. Based on Gavin’s pattern, how long did
he ride on Thursday?
9. Algebra If the pattern continues, how
far will Gavin ride on Saturday?
10. Explain how to find the number of
miles Gavin will ride on Sunday, if the
pattern continues.
What is the relationship between the
number of pencils and price?
7. Measurement Cheryl is filling a
pool. She measures the depth in feet
every 5 minutes. Her measurements
are 2.5, 3.6, 4.7, and 5.8. If this pattern
continues, how deep will the water be
the next time she measures?
11. The Fibonacci sequence is a famous
pattern of numbers. The first seven
numbers in the Fibonacci sequence are
1, 1, 2, 3, 5, 8, and 13. Find the next
three numbers. Explain the pattern.
12.
E
WRITE MATH Write a real-world
problem that uses the look for a
pattern strategy. Use the pattern below.
2.45, 2.8, 3.15, 3.5, . . .
Lesson 3C Write Multiples and Compare Fractions 325
Multi-Part
Lesson
3
PART
Write Multiples and Compare Fractions
A
Main Idea
I will compare fractions
using common
denominators.
Vocabulary
V
least common
denominator (LCD)
Get ConnectED
GLE 0506.1.1
Use mathematical language,
symbols, and definitions
while developing mathematical
reasoning. SPI 0506.2.9
Compare whole numbers,
decimals, and fractions using
the symbols <, >, and =.
B
D
C
E
Compare Fractions
If two fractions have the same denominator, you can compare
them by comparing the numerators. If the fractions have different
denominators, first write equivalent fractions with the least
common denominator.
The least common denominator (LCD) is the least common
multiple of the denominators of the fractions.
Compare Fractions
Compare
C
_3 and _1 using the least common denominator.
5
2
Step 1 Find the LCM of the denominators.
The LCM of 5 and 2 is 10.
Step 2 Find equivalent fractions with a denominator of 10.
6
3
_
=_
THINK 5 × 2 = 10, 3 × 2 = 6
5
1
_
=_
THINK 2 × 5 = 10, 1 × 5 = 5
5
2
10
10
Step 3 Compare the numerators.
6
5
3
1
Since 6 > 5, then _ > _. So, _ > _.
10
5
10
2
3
1
Check The models show that _
> _. 5
2
1
5
1
5
1
5
1
2
You can always multiply the denominators of two fractions to
find a common denominator. But, this method does not always
give the LCD.
326
Factors, Fractions, and Decimals
Compare Fractions Using the LCD
Compare
You can also find a set
of equivalent fractions
5
_
7
and 6 to find
for _
9
the LCD.
_7 and _5 using the least common denominator.
9
6
Step 1 Find the LCM of the denominators.
The LCM of 9 and 6 is 18.
Step 2 Find equivalent fractions with a denominator of 18.
7
14
_
=_
21 , . . .
14 , _
7 _
_
,
9 18 27
THINK 9 × 2 = 18, 7 × 2 = 14
9
18
15
5
_
=_
6
18
20 . . .
15 _
10 _
5 _
_
,
,
,
,
6 12 18 24
Step 3
THINK 6 × 3 = 18, 5 × 3 = 15
Compare the numerators.
15
5
14
7
Since 14 < 15, then _ < _. So, _ < _.
18
18
6
9
SPORTS Trevor made 2 out of
3 field goals and Tyler made 5 out
of 6 field goals. Who made a greater
fraction of field goals?
Step 1 Find the LCM of the
denominators. The LCM
of 3 and 6 is 6.
Step 2 Find equivalent fractions with a denominator of 6.
2
4
_
=_
3
6
5
5
_
=_
6
6
THINK 3 × 2 = 6, 2 × 2 = 4
THINK 6 × 1 = 6, 5 × 1 = 5
Step 3 Compare the numerators.
5
5
4
2
Since 5 > 4, then _ > _. So, _ > _.
6
6
3
6
Tyler made a greater fraction of field goals.
5
2
Check The models show that _
> _.
3
6
1
3
1
3
2
3
1
6
1
6
1
6
1
6
1
6
5
6
Lesson 3D Write Multiples and Compare Fractions 327
Compare each pair of fractions using models or the LCD
LCD.
See Examples 1–3
1
1
1. _ and _
5
1
1
2. _ and _
3
2
3
7
3. _ and _
4
6
2
7
4. _ and _
3
8
10
Algebra Replace each with <, >, or = to make a true
statement. See Examples 1–3
5
1
5. _ _
3
2
7
6. _ _
3
9
1
1
7. _ _
12
5
9. A recipe calls for _ cup of brown sugar
8
2
and _ cup of flour. Which ingredient is
3
greater?
10.
6
2
8. _ _
4
6
5
E
TALK MATH Explain how the LCM
15
and the LCD are alike. How are they
different?
EXTRA
%
)# E
# T4 IC
!C
2A
0R
P
Begins on page EP2.
Compare each
C
h pair
i off fractions
f ti
using
i models
d l or the
th LCD.
LCD
See Examples 1–3
3
2
11. _ and _
3
1
12. _ and _
1
1
13. _ and _
3
2
14. _ and _
5
4
15. _ and _
7
2
16. _ and _
3
1
17. _ and _
5
4
18. _ and _
3
4
5
6
5
15
3
8
3
6
10
5
12
6
4
9
Algebra Replace each with <, >, or = to make a true
statement. See Examples 1–3
3
2
19. _ _
3
3
20. _ _
1
1
21. _ _
6
1
22. _ _
3
2
23. _ _
5
11
24. _ _
3
5
25. _ _
15
3
26. _ _
5
10
7
6
4
12
7
5
8
8
16
15
Which sport was liked less?
30. The fifth graders were given sandwiches
for lunch during their field trip. Nathan
5
7
ate _ of his sandwich, Leroy ate _
8
of his sandwich, and Sofia ate _ of her
8
sandwich. Who had the least amount of
sandwich left to eat?
328
6
12
8
liked soccer and 0.4 liked basketball.
29. The amounts of water four runners
drank are shown at the right. Who
drank the most?
5
2
7
28. A survey showed that _ of the class
27. A trail mix has 0.5 cup of raisins
2
and _ cup of peanuts. Which
3
ingredient is greater?
6
4
Factors, Fractions, and Decimals
Evita
Jack
Keisha
Sirjo
3
5
5
8
3
4
5
10
31. OPEN ENDED Replace with a number to make _ > _ a true
4
24
statement.
1
32. NUMBER SENSE Suppose two fractions have the same numerator
and different denominators. How can you decide which fraction is
greater without finding the LCD?
33.
E
WRITE MATH Write a real-world problem that can be
solved by comparing two fractions with different denominators.
Then solve. Support your answer with a model.
Test Practice
34. The table shows the cost of renting a
bicycle. If the pattern continues, how
much will it cost to rent a bicycle for
6 hours?
Number of Hours
Cost ($)
2
3
4
5
12
18
24
30
A. $6
C. $36
B. $32
D. $42
35. Eighteen out of 24 of Emil’s CDs are
country music. Five out of 8 of
Imani’s CDs are country music.
Which is a true statement?
F. Both of their CD collections are
half country music.
G. Both of their CD collections are
less than half country music.
H. Emil’s collection is closer to half
country than Imani’s collection.
I. Imani’s collection is closer to half
country than Emil’s collection.
36. Find the missing number in the pattern 1, 2, 4, 7, , 16, . . . .
(Lesson 3C)
Find the first two common multiples of each pair of numbers. (Lesson 3B)
37. 4, 6
38. 3, 9
41. The table shows the number of games
lost by the girls’ basketball team in three
4
months. The fraction _ represents the
16
losses. Write this fraction in simplest form.
39. 2, 5
40. 8, 20
Number
of Games
Number
of Losses
16
4
(Lesson 2C)
To assess partial mastery of SPI 0506.2.9, see your Tennessee Assessment Book.
329
Multi-Part
Lesson
4
PART
Fractions and Decimals
A
B
C
D
E
Fractions and Decimals
Main Idea
I will explore using
models to write
fractions as decimals.
You can use models to write fractions in their equivalent
decimal form.
Get ConnectED
Use a model to write
GLE 0506.1.4 Move
flexibly between concrete
and abstract representations
of mathematical ideas in
order to solve problems,
model mathematical ideas,
and communicate solution
strategies.
Step 1
_1 as a decimal.
2
1
Write _ as a fraction with a denominator of 100.
2
× 50
50
1
_
=_
2
Since 2 × 50 = 100, multiply 1 × 50.
100
× 50
Step 2
Shade a model of
50
_
.
100
Since 50 out of the 100 squares are shaded, the model
1
shows 50 hundredths or 0.50. So, _ = 0.50.
2
About It
1
1. How would the Activity change if _ was written as a
2
fraction with a denominator of 10? Would the result be the
same? Explain.
Use a model to write each fraction as a decimal.
1
2. _
4
330
Factors, Fractions, and Decimals
2
3. _
5
7
4. _
10
3
5. _
20
Multi-Part
Lesson
4
PART
Fractions and Decimals
A
Main Idea
I will use equivalent
fractions to write
fractions as decimals.
I will write decimals
as fractions.
Get ConnectED
GLE 0506.1.1
Use mathematical language,
symbols, and definitions while
developing mathematical
reasoning. SPI 0506.2.7
Recognize equivalent
representations for the same
number.
B
C
D
E
Fractions and Decimals
Fractions with denominators that are factors of 10, 100, or
1,000 can be written as decimals by writing equivalent fractions.
Write Fractions as Decimals
W
Write
W
_3 as a decimal.
4
Since 4 is a factor of 100, write an equivalent fraction with
a denominator of 100.
× 25
3
75
_
=_
4
100
Since 4 × 25 = 100, multiply 3 × 25.
× 25
= 0.75 Read 0.75 as seventy-five hundredths.
HONEYBEES The average length
of a honeybee is 0.8 inch. Write
this length as a fraction in
simplest form.
You can use a place-value chart.
The place value of the last decimal
is tenths.
Ones
Tenths
0
8
8
0.8 = _
10
8÷2
=_
10 ÷ 2
4
=_
5
Hundredths
Say eight tenths.
Divide the numerator and denominator by the
GCF, 2.
Simplify.
4
The length of a honeybee is _ inch.
5
Lesson 4B Fractions and Decimals 331
Write each fraction as a decimal
decimal. See Example 1
1
1. _
3
2. _
5
1
3. _
4
10
6
4. _
10
Write each decimal as a fraction in simplest form. See Example 2
5. 0.25
6. 0.6
7. 0.5
9. Yesterday it rained 0.45 inch. Write
0.45 as a fraction in simplest form.
10.
8. 0.7
E TALK MATH Explain how to write
a fraction as a decimal using
equivalent fractions.
EXTRA
%
)# E
# T4 IC
!C
2A
0R
P
Begins on page EP2.
Write
W
it each
h fraction
f ti
as a decimal.
d i l See Example
l 1
2
11. _
5
4
_
15.
25
8
12. _
1
13. _
20
8
_
17.
25
10
1
_
16.
10
17
14. _
20
13
18. _
25
Write each decimal as a fraction in simplest form. See Example 2
19. 0.40
20. 0.35
21. 0.04
22. 0.9
23. 0.48
24. 0.55
25. 0.36
26. 0.65
19
27. At basketball practice, Savannah spent _ of an hour practicing
19
free throws. Write _ as a decimal.
20
20
28. Paolo made a model of his house that is 0.08 the size of
his actual house. What fraction of the actual house length
is the model? Write the fraction in simplest form.
The smallest known female spider is 0.46 millimeter
long. The smallest male spider is 0.37 millimeter long.
Write each decimal as a fraction in simplest form.
29. 0.46
332
30. 0.37
Factors, Fractions, and Decimals
31. OPEN ENDED Write a real-world problem that uses a
20
25
decimal between _ and _.
100
100
32. CHALLENGE Write each mixed number as a decimal and each
decimal as a mixed number in simplest form.
3
a. 5_
5
33.
E
6
b. 2_
c. 6.48
25
WRITE MATH Explain why 0.04 is not equivalent to
d. 3.07
4
_
.
10
Test Practice
34. Which decimal represents the shaded
portion of the figure below?
35.
A. 0.13
C. 1.3
B. 0.52
D. 5.2
SHORT RESPONSE Write 0.16 as
a fraction in simplest form.
36. Emilia bought _ pound of sliced
4
salami at the deli counter. Which of
the following decimals did the
scale show?
3
F. 0.25
H. 0.75
G. 0.34
I. 3.4
37.
GRIDDED RESPONSE What
14
is _ as a decimal?
25
3
38. At the movies, Ariana ate _ of the tub of popcorn and Lauren ate
5
3
_
of the tub of popcorn. Who ate more of the popcorn?
10
(Lesson 3D)
3
1
1
39. Find the next three numbers: _, 0.2, _, 0.4, _, . . . . (Lesson 3C)
10
10
2
Find the LCM of each set of numbers. (Lesson 3B)
40. 6, 18
41. 8, 22
42. 15, 20
Lesson 4B Fractions and Decimals 333
That’s Not Proper
You will need: spinners
Converting Improper Fractions to Mixed Numbers
Get Ready!
Players: 3 players
Get Set!
Label equal sections of one
spinner with the numbers
13, 17, 23, 29, 37, 41, 57.
The numbers on this spinner
stand for the numerators of
improper fractions.
Label equal sections of a
second spinner with the
numbers 3, 4, 5, 6, 7, 8, 9.
These numbers stand for
the denominators of
improper fractions.
Go!
One student spins both
spinners.
The first of the other two
students to express the
resulting improper fraction
as an equivalent mixed
number wins the round.
Play several rounds, trading
roles each time.
334
Factors, Fractions, and Decimals
57 13
41
17
37
23
29
9
3
4
8
7
6
5
Multi-Part
Lesson
4
PART
Fractions and Decimals
A
Main Idea
I will use division to
write fractions as
terminating decimals.
Vocabulary
V
tterminating decimal
C
B
D
Terminating Decimals
In Lesson 4B, you wrote fractions as decimals using equivalent
fractions. Any fraction can be written as a decimal by dividing
the numerator by the denominator. Decimals whose division
ends, or terminates, are terminating decimals .
Get ConnectED
SPI 0506.2.7
Recognize equivalent
representations for the same
number. SPI 0506.2.8 Write
terminating decimals in the
form of fractions or mixed
numbers. Also addresses
GLE 0506.1.1.
E
Fractions as Decimals
F
Write
W
_3 as a decimal.
5
The numerator, 3, is the dividend. The denominator, 5,
is the divisor.
0.6
5 3.0
30
−−−−
0
3
_
5
Place a decimal point after 3 and
annex a zero. Place the decimal
point in the quotient directly above
the decimal point in the dividend.
Keep dividing until the remainder is zero.
3
So, _ = 0.6.
5
_3
DIRT BIKES Vicki rode mile on a
8
dirt bike track. Write the length
she rode as a decimal.
Divide 3 ÷ 8.
0.375
8 3.000
-2
4
−−−−
60
56
−−−−
40
40
−−−−
0
Keep annexing zeroes
until the division is
complete.
3
So, _ = 0.375. Vicki rode 0.375 mile.
8
Lesson 4C Fractions and Decimals
335
Write each fraction as a decimal
decimal. See Examples 11, 2
5
1. _
3
2. _
8
3
3. _
17
4. _
40
16
7
5. About _ of the students that
6.
40
E
40
TALK MATH Explain how to write a
fraction as a decimal using division.
completed a survey said they walk to
school every day. Write _ as a
40
decimal.
7
EXTRA
%
)# E
# T4 IC
!C
2A
0R
P
Begins on page EP2.
Write
i each
h fraction
f
i
as a decimal.
d i l See Examples 1, 2
5
7. _
9
8. _
1
9. _
33
11. _
11
12. _
13
13. _
40
16
40
1
10. _
16
16
8
11
14. _
40
16
15. The table shows the number of laps completed
out of 200 by each racecar driver. What fraction
of the race did Bowers complete? Write the fraction
as a decimal.
Driver
Laps Completed
(out of 200)
Cunningham
Gutierrez
Bowers
195
170
103
16. Last week, _ of Mrs. Palmer’s class downloaded a
16
podcast. What portion of the class had not downloaded
it? Write as a decimal.
15
Write each decimal as a fraction in simplest form.
17. 0.164
18. 0.485
19. 0.748
20. OPEN ENDED Write a decimal between 0.620 and 0.685.
Then write it as a fraction in simplest form.
21. CHALLENGE Write each mixed number as a decimal.
9
a. 7_
40
22.
336
E
b. 14_
c. 18_
5
16
7
8
WRITE MATH Explain how to write
Factors, Fractions, and Decimals
d. 73_
21
_
as a decimal using division.
40
23
40
Test Practice
23. Which of the following decimals is
7
equivalent to _?
24.
16
A. 0.4375
B. 0.716
SHORT RESPONSE Samir got
31 out of 40 spelling words correct
31
on his spelling test. What is _ as
40
a decimal?
C. 0.875
D. 7.16
Repeating Decimals
Any fraction can be written as a decimal by dividing. There are
times when the division continues without end. These decimals
are called repeating decimals. A bar is drawn above the numbers
that repeat.
Write
W
_1 as a decimal.
3
0.33 . . .
3 1.00 . . .
-9
____
10
9
____
1
Divide 1 ÷ 3.
Annex zeros after the decimal point.
The remainder will always be 1.
1
So, _ = 0.333 . . . or 0.3.
3
Write
W
_1 as a decimal.
6
Divide 1 ÷ 6.
0.16 . . .
6 1.00 . . .
- 6
_____
40
36
−−−−
4
Annex zeros after the decimal point.
The remainder will always be 4.
1
So, _ = 0.166 . . . or 0.16.
6
Write each fraction as a repeating decimal.
2
25. _
3
5
26. _
6
1
27. _
12
2
28. _
9
Lesson 4C Fractions and Decimals
337
Multi-Part
Lesson
4
PART
Fractions and Decimals
A
B
C
D
Problem-Solving Investigation
Main Idea I will choose the best strategy to solve a problem.
MYKAELA: My science teacher is giving
goldfish to some of my classmates. He
gave some of them to Gisela. Then he
gave twice as many to Chris. He gave
twice as many to Clara as he gave to
Chris. He gave away all 28 goldfish.
YOUR MISSION: What fraction of the
goldfish did the teacher give Gisela?
Understand
There are 28 goldfish. He gave some to Gisela. He gave twice as
many to Chris as to Gisela. He gave twice as many to Clara as to
Chris. You need to determine what fraction of goldfish Gisela
took home.
Plan
You can use the guess, check, and revise strategy to find how many
goldfish Gisela took home.
Solve
Make a guess as to how many goldfish the teacher gave Gisela.
Check if it is correct. Then revise the guess, if necessary.
Number of Goldfish
Gisela
Chris
Clara
Total
5
2 × 5 = 10
2 × 10 = 20
35
35 > 28; too many
3
2×3=6
2 × 6 = 12
21
21 < 28; too few
4
2×4=8
2 × 8 = 16
28
28 = 28; correct
4
1
So, the teacher gave 4 out of the 28 goldfish to Gisela. _ = _
28
Check
Since, 4 + 8 + 16 = 28, the answer is correct. GLE 0506.1.2 Apply and adapt a variety of appropriate strategies to problem solving,
including estimation, and reasonableness of the solution.
338
Factors, Fractions, and Decimals
7
EXTRA
%
)# E
# T4 IC
!C
2A
0R
P
Begins on page EP2.
•
•
•
•
Guess, check and revise.
Look for a pattern.
Solve a simpler problem.
Choose an operation.
Use any strategy shown to solve each
problem.
1. Lorraine and Yori have 12 plants
between them. Yori has 4 more than
Lorraine. How many does each
girl have?
2. Algebra When Cesar’s Pizza Parlor
makes a pizza, they use the following
amount of cheese.
Number of
Amount of
Pizzas
Cheese (ounces)
5. Annette put $5 in her bank account
each week for 14 weeks. Serefina put
$7 in her bank account each week for
11 weeks. Who saved more money?
How much more?
6. Liu has $53 in his bank account. Each
week for 5 weeks, he adds $2.50 to the
account. How much does he now have
in his bank account?
7. Maggie and her younger brother, Ty,
went to lunch. Ty had a sandwich and
soft drink. Maggie had a hamburger
and milkshake. How much more did
Maggie’s lunch cost than Ty’s?
Menu
1
4
2
8
Sandwich
$2.25
Milkshake $3.00
3
12
Hamburger
$4.50
Soft drink
5
20
7
8
Complete the table and find how much
cheese Cesar’s uses to make 8 pizzas.
3. Mr. Whitmore bought tickets for a
movie. Adult tickets cost $5 each and
children’s tickets cost $3 each. He
spent a total of $22 for the tickets.
What fraction of adult tickets and
what fraction of children’s tickets did
he buy?
4. During the World Series, one pitcher
used a pattern when he pitched. The
pattern was two fastballs followed by
two sliders followed by a change-up.
If he continued this pattern, what was
his eleventh pitch?
Item
Cost
Item
Cost
$0.75
8. There are 48 students on the math
team. There are 12 more girls than
boys. How many boys and how many
girls are on the math team?
9. Oscar makes leather belts. If he
continues his pattern below, what will
be the design of the seventeenth link
in the belt?
10.
E
WRITE MATH Explain how you use
the guess, check, and revise strategy.
To assess mastery of SPI 0506.2.7 and SPI 0506.2.9, see your Tennessee Assessment Book.
339
Dreamcatchers were first made by
the Chippewa people, who hung
them over the beds of children to
trap bad dreams. The Chippewa
are one of the largest Native
American groups in North America.
In 1990, around 106,000 Chippewa
were living throughout their
original territories.
Each dreamcatcher is made with
many beads and feathers. A simple
dreamcatcher has 28 pony beads
and is made with 7 yards of string.
Today, Native Americans continue
to make dreamcatchers on more
than 300 reservations.
340
Factors, Fractions, and Decimals
The Chippewa
people have signed
51 treaties with the United
States government, the
most of any Native
American tribe.
Use the information on the previous page to
solve each problem.
In a simple dreamcatcher, how
many beads do you use for each
yard of string?
For each dreamcatcher you made,
you used 12 beads. If you had
144 beads, how many
dreamcatchers did you make?
Each time you add a feather to
a dreamcatcher, you add
3 turquoise beads. Use the multiples
of 3 to find out how many beads
you will need if you have 2, 3, 4,
or 5 feathers in your dreamcatcher.
You are making dreamcatchers that
require 6 beads for every 1 feather.
Use the multiples of 6 to find how
many beads you will need if you
have 2, 3, 4, or 5 feathers in your
dreamcatcher.
3
You have made 4_ dreamcatchers.
4
3
Write 4_ as an improper fraction.
4
Suppose you use 12 feathers and a
certain amount of beads to make a
dreamcatcher. If you had 48 feathers
and beads, how many beads did
you use?
7
You have made _ dreamcatchers.
2
7
Write _ as a mixed number.
2
Problem Solving in Social Studies 341
Chapter Study
Guide and Review
Be sure the following Key
Concepts are noted in your
Foldable.
Vocabulary
common factor
composite number
prime number
Lesson 1
P
Factoriza rime
and Exp tion
onents
simplest form
terminating decimal
Vocabulary Check
Complete. Use a word from the Key
Vocabulary list.
Key Concepts
Prime and Composite Numbers (Lesson 1)
• A prime number has exactly two factors,
1 and itself. A composite number has more
than two factors.
prime numbers: 2, 3, 11, 29
composite numbers: 4, 8, 15, 20
Fractions in Simplest Form (Lesson 2)
• A fraction is in simplest form when the
GCF of the numerator and denominator is 1.
3
1
_
=_
6
1. A whole number greater than
1 that has exactly two factors,
?
1 and itself, is called a(n)
2. 0.6 is called a
.
3. A whole number that has more
than two factors is called a(n)
? .
2
Compare Fractions (Lesson 3)
• To compare fractions with different
denominators, rename the fractions using
common denominators.
Fractions and Decimals (Lesson 4)
× 20
80
4
__
= _ = 0.80 or 0.8
5
100
× 20
?
4. A fraction is written in
when the numerator and
denominator have no common
factors greater than 1.
5. A whole number that is a factor of
two or more numbers is called
? .
a(n)
7
6. The fraction _ is in
8
342
3
?
.
Factors, Fractions, and Decimals
?
.
Multi-Part Lesson Review
Lesson 1
Prime Factorization and Exponents
Prime and Composite Numbers
(Lesson 1B)
Tell whether each number is prime or
composite.
7. 23
8. 48
9. 34
EXAMPLE 1
Tell whether 51 is prime or composite.
51 = 1 × 51
51 = 3 × 17
factors of 51: 1, 3, 17, 51
10. A goliath birdeater is a spider that can
grow up to 28 centimeters. Is 28 prime
or composite?
Prime Factorization
Since 51 has more than two factors, it is a
composite number.
(Lesson 1C)
Find the prime factorization of each
number.
EXAMPLE 2
11. 42
Make a factor tree.
12. 75
13. 96
Find the prime factorization of 18.
18
14. Prime numbers are used to encode
bank account information. Suppose
Suki’s bank account was encoded
with the number 273. What is the
prime factorization of Suki’s bank
code?
Powers and Exponents
2×9
2×3 ×3
The prime factorization of 18 is 2 × 3 × 3.
(Lesson 1E)
Write each product using an exponent.
Then find the value of the power.
15. 5 × 5 × 5 × 5
EXAMPLE 3
Write 4 × 4 × 4 × 4 × 4 × 4 using an
exponent. Then find the value of the
power.
16. 12 × 12 × 12
The base is 4. Since 4 is a factor 6 times,
the exponent is 6.
17. The average brain weight in grams for
a walrus is 210. Find this value.
4 × 4 × 4 × 4 × 4 × 4 = 46
= 4,096
Chapter Study Guide and Review 343
Chapter Study Guide and Review
Lesson 2
Fractions in Simplest Form
Greatest Common Factor
(Lesson 2B)
Find the GCF of each set of numbers.
EXAMPLE 4
18. 6, 8
19. 9, 21, and 24
Find the greatest common factor of
6 and 21.
20. 12, 30
21. 18, 45
List the prime factorizations of 6 and 21.
22. Twelve pens and 16 pencils will be
placed in bags with an equal number
of each item. What is the most number
of bags that can be made?
Simplest Form
6: 2 × 3
21: 3 × 7
The common factor is 3. So, the greatest
common factor of 6 and 21 is 3.
(Lesson 2C)
Write each fraction in simplest form. If
the fraction is already in simplest form,
write simplified.
EXAMPLE 5
4
Write
in simplest form.
20
2
23. _
10
4
24. _
18
factors of 20: 1, 2, 4, 5, 10, 20
6
25. _
8
26. _
12
27. _
20
28. _
25
Lesson 3
factors of 4: 1, 2, 4
The GCF of 4 and 20 is 4.
24
21
_
4÷4
4
1 Divide the numerator and
_
= _ = _ denominator by 4.
5
20
20 ÷ 4
4
1
_
So,
in simplest form is _.
32
20
5
Write Multiples and Compare Fractions
Least Common Multiple
(Lesson 3B)
Find the LCM of each set of numbers.
Make a table or a graph.
EXAMPLE 6
29. 5 and 9
List multiples of each number.
30. 4, 7, and 14
31. Every 7 days, a video store gives free
popcorn with movie rentals. Every
5 days, they offer a free movie. If they
gave away popcorn and a movie today,
when will they give away both again?
344
Factors, Fractions, and Decimals
Find the LCM of 12 and 16.
multiples of 12: 12, 24, 36, 48, 60 . . .
multiples of 16: 16, 32, 48, 64, 80 . . .
The LCM of 12 and 16 is 48.
Lesson 3
Write Multiples and Compare Fractions
Problem-Solving Strategy: Look for a Pattern
32. Find the next three numbers. 3, 7, 12,
18, 25, . . .
33. Bena makes bracelets. She uses red,
white, and gray beads. If she
continues her pattern, what color is
the 15th bead?
(continued)
(Lesson 3C)
EXAMPLE 7
This stairway is
made of cubes.
How many cubes
would be needed
to make it 7 steps
high?
Understand
34. Draw the next two figures in the
pattern.
You know how many cubes
are used for 1, 2, and 3 steps.
Plan
Look for a pattern.
Solve
35. Jackson wrote the following fractions
on the board.
2
3
4
7
1 _
2 3 4 5
_
, , _, _, _
2 5 8 11 14
If he continues writing fractions
according to the pattern, what will be
the next three fractions?
Compare Fractions
6
_
8
7
_
12
Check
Draw a picture and count the
cubes. There are 28 cubes, so the answer
is correct. (Lesson 3D)
Replace each with <, >, or = to make
a true statement.
1
36. _
4
2
_
38.
3
1 step: 1
steps: 2 + 1, or 3
steps: 3 + (2 + 1), or 6
steps: 4 + (3 + 2 + 1), or 10
steps: 7 + (6 + 5 + 4 + 3 + 2 + 1),
or 28
2
37. _
3
5
_
39.
8
3
_
5
7
_
12
40. Christine worked on social studies
3
homework _ of an hour. She worked
4
5
on math for _ of an hour. On which
6
subject did she spend more time?
EXAMPLE 8
Replace with <, >, or = to make
_2 _3 a true statement.
5
4
The LCM of 5 and 4 is 20. So, 20 is the LCD
3
2
of _ and _.
5
4
8
2
_
=_
5
20
3
15
_=_
4
20
THINK 5 × 4 = 20, 2 × 4 = 8
THINK 4 × 5 = 20, 3 × 5 = 15
8
15
3
2
Since 8 < 15, _ < _. So, _ < _.
20
20
5
4
Chapter Study Guide and Review 345
Chapter Study Guide and Review
Lesson 4
Fractions and Decimals
Fractions and Decimals
(Lesson 4B)
Write each fraction as a decimal. Use
equivalent fractions.
3
41. _
19
42. _
8
43. _
7
44. _
20
25
EXAMPLE 9
4
Write as a decimal.
5
_
Write an equivalent fraction with a
denominator of 10.
50
8
4
_
=_
50
5
10
= 0.8
Terminating Decimals
5
46. _
29
47. _
17
48. _
8
40
Read 0.8 as eight tenths.
(Lesson 4C)
Write each fraction as a decimal. Use
division.
1
45. _
Since 5 × 2 = 10, multiply
4 × 2.
EXAMPLE 10
9
Write
as a decimal.
20
_
Divide 9 ÷ 20.
0.45 Place the decimal point in the
20 9.00 quotient.
80
−−−−
1 00
1
00 Keep dividing until the
−−−−
0 remainder is zero.
9
_
So,
= 0.45.
16
40
20
Problem-Solving Investigation: Choose a Strategy
(Lesson 4D)
EXAMPLE 11
Solve. Use any strategy.
49. Tiffany has two hamster cages. When
3
she cleans them, she uses _ bag of
8
hamster bedding for one cage and
1
_
bag for the other cage. Does she
Pilar has $69 to spend on presents. CDs
cost $13, and DVDs cost $15. How many
CDs and DVDs can she buy?
Use the guess, check, and revise strategy.
4
need 1 bag or 2 bags when she cleans
the cages?
CDs
DVDs
Total
2:
2 × $13 = $26 3:
3 × $15 = $45 $71 too much
3:
3 × $13 = $39 2:
2 × $15 = $30 $69 correct
So, Pilar can buy 3 CDs and 2 DVDs.
346
Factors, Fractions, and Decimals
Practice
Chapter Test
Find the common factors of each set of
numbers.
1. 15, 45
2. 24, 32, and 40
12. The table shows when customers at
Maltey’s Burgers receive items free with
the purchase of a burger deal.
Find the GCF of each set of numbers.
3. 8, 28
4. 21, 24, and 27
C. 5
B. 4
D. 12
When
Milkshake
every 4 days
Tater tots
every 10 days
If they gave away both items today, in
how many days will a customer be able
to get both a milkshake and tater tots
free again?
5. MULTIPLE CHOICE Which is a prime
factor of the composite number 24?
A. 3
Free Item
Write each fraction as a decimal.
6. The table shows the countries with the
most wins for the Tour De France
cycling race. Which country’s wins can
also be represented as 2 × 2 × 3 × 3
or 22 × 32?
13
13. _
9
14. _
9
15. _
5
16. _
20
10
40
8
17. What is the least common multiple of
12 and 20?
18. Measurement Grasshoppers can
jump 40 times the length of their body.
If one foot equals 12 inches, how
many feet could a 3-inch grasshopper
jump?
7. Write the prime factorization of
150 using exponents.
19. MULTIPLE CHOICE Amber went to the
library after school 3 of the 5 school
days this week. Which fraction is less
3
than _?
5
Write each fraction in simplest form. If
the fraction is already in simplest form,
write simplified.
9
8. _
15
9. _
28
10. _
6
11. _
18
32
1
F. _
2
3
_
G.
4
16
27
20.
4
H. _
5
5
I. _
6
E
WRITE MATH Explain how to
21
write _ as a decimal.
25
Practice Chapter Test
347
Test Practice
9
In the grid below, _ of the squares are shaded.
12
Make sure to check the
reasonableness of your
answers.
9
Which fraction is equivalent to _ in simplest form?
12
A. _
1
2
3
B. _
4
C. _
2
3
5
D. _
6
Read the Test Item
9
You need to write the fraction _ in simplest form.
12
Solve the Test Item
You can find the simplest form of the fraction by
dividing the numerator and the denominator by the
greatest common factor (GCF).
9÷3
3
_
=_
4
12 ÷ 3
The answer is B.
Read each question. Then fill in the correct answer on the answer
sheet provided by your teacher or on a separate sheet of paper.
1. Paige cut a cake into
20 pieces. If 14 pieces
have been eaten,
what portion of the
cake remains?
348
A. 0.1
C. 0.3
B. 0.2
D. 0.4
Factors, Fractions, and Decimals
2. Emilia used 4 of her 8 stamps to
mail letters. Which fraction is less
4
than _?
8
5
F. _
8
1
G. _
2
H. _
3
4
3
I. _
7
3. Which is a prime factor of the
composite number 32?
A. 2
C. 4
B. 3
D. 5
7.
4. Clarence bought a 3-pound can of
mixed nuts for a party. One-fourth of
the can is made up of walnuts, and
two-fifths of the can is made up of
peanuts. Which of the following shows
1
the correct relationship between _
4
2
and _?
5
1
2
F. _ = _
5.
8. In one weekend, a florist made
$2,170 selling vases. If the florist sold
62 vases, how much did she charge for
each vase?
1
2
H. _ < _
4
5
1
2
G. _ > _
4
5
4
5
3
1
I. _ < _
5
10
9.
SHORT RESPONSE Jonas buys the
items shown at the grocery store. About
how much will all of the items cost?
Store Purchases
Item
Cost
Cereal
$4.05
Bread
$2.89
Milk
$3.04
GRIDDED RESPONSE Two
soccer balls are used during a
professional game. The first ball weighs
15.5 ounces. The second ball weighs
14.18 ounces. How many more ounces
is the first ball than the second ball?
F. $60
H. $50
G. $55
I. $35
SHORT RESPONSE Explain the
difference between a prime number
and a composite number. Be sure to
include examples of each.
10. Julian purchased the 3 packages of
T-shirts shown. What is the cost of one
of the packages of T-shirts?
A. $15
B. $16
C. $17
6. Which group shows the prime
factorization of the number 252?
D. $20
A. 2 × 3 × 3 × 7
B. 2 × 2 × 2 × 3 × 5
11.
C. 2 × 2 × 3 × 3 × 7
D. 2 × 2 × 2 × 3 × 3 × 7
GRIDDED RESPONSE What is
29.87 rounded to the nearest whole
number?
NEED EXTRA HELP?
If You Missed Question . . .
Go to Chapter-Lesson . . .
For help with . . .
1
2
3
4
5
6
7
8
9
10
11
1-2B
7-3D
7-1C
7-3D
5-1B
7-1C
5-3C
4-2B
7-1B
3-1C
5-1A
SPI 2.7 SPI 2.9 SPI 2.2 SPI 2.9 SPI 1.2 SPI 2.2 SPI 2.5 GLE 2.3 GLE 2.2 SPI 2.4 GLE 2.5
Test Practice 349