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Equivalent Fractions
BEFORE
Now
WHY?
You factored whole numbers.
You’ll write equivalent fractions.
So you can compare the numbers
of bones in a skeleton, as in Ex. 38.
In the Real World
Word Watch
fraction, p. 169
numerator, p. 169
denominator, p. 169
equivalent fractions, p. 169
simplest form, p. 170
Aquarium There are 10 fish
in an aquarium, and 2 of
them are goldfish. What
fraction of the fish in the
aquarium are goldfish?
A fraction is a number
a
b
of the form (b 0)
where a is called the
numerator and b is called
the denominator . A fraction is used to describe equal parts of a whole.
Fractions that represent the same part-to-whole relationship are called
equivalent fractions .
EXAMPLE
1
Identifying Equivalent Fractions
The fish in the aquarium described above are arranged in the diagram.
Using the diagram, you can write two equivalent fractions:
Number of goldfish
2
10
Number of fish
Number of groups of 2 goldfish
1
5
Number of groups of 2 fish
2
10
1
5
The fractions and are equivalent fractions because they represent
the same part-to-whole relationship.
EXAMPLE
with
Solving
For any fraction, if you tried
to list all the equivalent
fractions, the list would
continue without end.
This is because you can
multiply the numerator
and denominator by any
nonzero number.
2
Writing Equivalent Fractions
6
8
Write two fractions that are equivalent to .
Multiply or divide the numerator and denominator by the same
nonzero number to find an equivalent fraction.
6
63
18
8
83
24
Multiply numerator and denominator by 3.
6
62
3
8
82
4
Divide numerator and denominator by
2, a common factor of 6 and 8.
Lesson 4.3
Equivalent Fractions
169
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Simplest Form A fraction is in simplest form if its numerator and
denominator have 1 as their greatest common factor. Writing a fraction
in simplest form is called simplifying the fraction. To simplify a fraction,
you divide its numerator and denominator by their GCF.
EXAMPLE
with
Reading
8
The fraction is read
28
“eight twenty eighths.”
3
Simplifying Fractions
Write the fraction in simplest form.
8
8
a. 28
b. 15
Solution
2p41
8
a. 28 7p41
Divide out GCF
of 8 and 28.
8
b. 15
2
7
The GCF is 1.
The fraction is in simplest form.
Your turn now
Write two fractions that are equivalent to the
given fraction.
3
2. 10
1
1. 6
8
3. 18
30
4. 36
Write the fraction in simplest form.
12
5. 16
EXAMPLE
15
6. 35
4
7
7. 28
14
8. 34
Using Fractions in Simplest Form
Basketball At a girls’ basketball game, the home team made 14 out of
22 free throw attempts. The away team made 10 out of 25 free throw
attempts. Write the number of free throws made by each team as a
fraction. Are the fractions equivalent?
Home Team
1
7p2
7
14
Free throws made
11 p 2
11
22
Free throws attempted
1
Away Team
1
10
2p5
2
Free throws made
25
5p5 1 5
Free throws attempted
7
11
2
5
ANSWER No, and are not equivalent fractions.
170
Chapter 4
Number Patterns and Fractions
Page 3 of 6
INTERNET
Exercises
eWorkbook Plus
CLASSZONE.COM
More Practice, p. 708
Getting Ready to Practice
1. Vocabulary How can you tell if a fraction is in simplest form?
Tell whether the fractions are equivalent.
5 15
2. , 6 18
20 5
3. , 24 8
14 21
4. , 15 24
7. Find the Error Describe and
4 8
5. , 6 12
24
42
12 8
6. , 18 32
8p3 1 8
24
14 p 3 1 14
42
correct the error in simplifying .
Write the fraction in simplest form.
32
8. 72
6
9. 21
15
10. 21
12
11. 35
28
12. 48
13. Reading Books You read 35 pages of a chapter in a book and have
15 pages left. What fraction of the chapter have you read? Write the
fraction in simplest form.
Practice and Problem Solving
with
Example
1
2
3
4
Homework
Write two equivalent fractions that describe the model.
Exercises
14–16
17–21, 33–37
22–27
28–32, 38
14.
Online Resources
CLASSZONE.COM
• More Examples
• eTutorial Plus
15.
16.
Write two fractions that are equivalent to the given fraction.
25
17. 120
18
18. 21
14
19. 34
30
20. 52
28
21. 32
33
25. 81
49
26. 105
Write the fraction in simplest form.
30
22. 45
24
23. 32
22
24. 27
27. U.S. Presidents During the 1800s, there were 22 different Presidents of
the United States, and 6 of them were born in Virginia. Write a fraction
comparing the number of Presidents born in Virginia with the total
number of Presidents in the 1800s.
Lesson 4.3
Equivalent Fractions
171
Page 4 of 6
Science
Write the fractions in simplest form. Tell whether they are equivalent.
14 24
28. , 21 36
45 8
29. , 54 18
56 132
31. , 196 462
15 40
30. , 36 96
34 136
32. , 44 144
Write two fractions that are equivalent to the given fraction.
95
34. 126
84
33. 96
54
35. 168
92
36. 115
38. Skeletons Use the table to write the
Body Region
Lower limbs
Upper limbs
Ankles
Wrists
number of bones in the wrists as a
fraction of the number of bones in the
upper limbs, and to write the number
of bones in the ankles as a fraction of
the number of bones in the lower
limbs. Are the fractions equivalent?
■
39
37. 169
Bones
62
64
14
16
39. Challenge You are burning music onto a CD. There are 74 minutes
Skeletons
There are about 206 bones
in an adult’s skeleton.
There are 62 bones in the
lower limbs and 64 bones
in the upper limbs. How
many bones in the body
are not in the upper and
lower limbs?
available on a CD. You have used 36 minutes. Write a fraction for
the part of the CD that is unused.
Mixed Review
Write the product as a power. (Lesson 1.3)
40. 3 p 3 p 3 p 3
41. 7 p 7 p 7 p 7
42. y p y p y p y p y p y
Find the greatest common factor of the numbers. (Lesson 4.2)
43. 64, 88, 144
44. 42, 70, 112
45. 81, 108, 117
Basic Skills Copy and complete.
46. 55 min _?_ sec
47. 2 h _?_ min
48. 98 d _?_ wk
Test-Taking Practice
INTERNET
State Test Practice
49. Multiple Choice Three ninths of the students in your class have brown
eyes. Which fraction pairs are equivalent to three ninths?
CLASSZONE.COM
9 1
10 3
A. , 3 6
19 38
B. , 1 5
3 20
C. , 6 2
18 6
D. , 50. Multiple Choice The U.S. highway speed limit in 20 states is
55 miles per hour. In simplest form, what fraction of states have a
speed limit of 55 miles per hour?
2
8
F. 172
Chapter 4
Number Patterns and Fractions
2
5
G. 4
10
H. 10
25
I. Page 5 of 6
4.1 TO
4. 3
Notebook Review
Check Your Definitions
Review the
vocabulary
definitions in
your notebook.
Copy the review
examples in
your notebook.
Then complete
the exercises.
common factor, p. 164
greatest common factor
(GCF), p. 164
relatively prime, p. 165
fraction, p. 169
numerator, p. 169
prime number, p. 157
composite number,
p. 157
prime factorization,
p. 158
factor tree, p. 158
denominator, p. 169
equivalent fractions,
p. 169
simplest form, p. 170
Use Your Vocabulary
Tell whether the statement is true or false.
1. The numerator and denominator of a fraction in simplest form are
relatively prime.
2. A prime number is a whole number greater than 1 whose only whole
number factors are 1 and itself.
4.1–4.2 Can you find the GCF using prime factorization?
EXAMPLE Use a factor tree to write the prime factorization of 72 and
84. Then find the greatest common factor of 72 and 84.
84
72
8
4 21
9
2 4 3 3
2 2 3 7
2 2 2 3 3
72 2 2 2 3 3 23 32
84 2 2 3 7 22 3 7
ANSWER The common prime factors of 72 and 84 are 2, 2, and 3. So,
the greatest common factor is 22 3 12.
✓ Find the greatest common factor of the numbers using prime
factorization.
3. 18, 32
4. 16, 49
5. 90, 135
Lessons 4.1–4.3
6. 96, 112
Notebook Review
173
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4.3 Can you tell whether fractions are equivalent?
36
42
48
64
EXAMPLE Write and in simplest form. Tell whether
they are equivalent.
1
1
36
6
6p6
42
7p61 7
6
7
48
3
3 p 16
64
4 p 16 1 4
3 36
4 42
48
64
ANSWER Because , and are not equivalent fractions.
✓
Write the fractions in simplest form. Tell whether they
are equivalent.
6 9
7. , 10 15
6 4
8. , 15 16
14 10
9. , 22 35
15 25
10. , 33 55
about Lessons 4.1–4.3
11. Writing Describe two different ways you can find the greatest
common factor of 45 and 56.
Review Quiz 1
Tell whether the number is prime or composite. Explain your reasoning.
1. 75
2. 53
3. 61
4. 98
5. Quilt Squares You are making a quilt. You made 36 squares to be sewn
together. How many different rectangular arrangements can you make?
Find the greatest common factor of the numbers using prime
factorization. Then tell whether the numbers are relatively prime.
6. 12, 30
7. 10, 21
8. 28, 50
9. 117, 195
Write the fractions in simplest form. Tell whether they are equivalent.
15 45
10. , 28 84
7 12
11. , 56 84
27 36
12. , 72 90
45 81
13. , 75 180
14. Critical Thinking A number is a common factor of 96 and 144.
The sum of the number’s digits is 7. Find the number.
174
Chapter 4
Number Patterns and Fractions