Basic Mathematics for College Students

3.1 An Introduction to Fractions 215

EXAMPLE 8

Simplify each fraction, if possible: a. b.

StrategyWe begin by prime factoring the numerator, 90, and denominator, 105.
Then we look for any factors common to the numerator and denominator and
remove them.

WHYWhen the numerator and/or denominator of a fraction are large numbers,
such as 90 and 105, writing their prime factorizations is helpful in identifying any
common factors.

Solution

a.
To prepare to simplify, write 90 and 105
in prime-factored form.
Remove the common factors of 3 and 5 from
the numerator and denominator. Slashes and 1's
are used to show that and are replaced
by the equivalent fraction. A factor equal to
1 in the form of was removed.

Multiply the remaining factors in the numerator: 2  1  3 1 = 6.

Multiply the remaining factors in the denominator: 1  1 7 = 7.

Since 6 and 7 have no common factors (other than 1), is in simplest form.

b. Write 25 and 27 in prime-factored form.

Since 25 and 27 have no common factors, other than 1,

the fraction is in simplest form.

25

27

25

27



5  5

3  3  3

6

7



6

7

3  5

3  5 

15

15

1

1

5

5

3

 3

2  3

1

 3  5

1

3

1

 5

1

 7

90

105



2  3  3  5

3  5  7

25

27

90

105

Self Check 8

Simplify each fraction, if

possible:

a.

b.

Now TryProblems 77 and 81

16

81

70

126

90

910

~^3 ~^3 ~^2 ~^5

105

~^521

~^3 ~^7

27

~^39

~^3 ~^3

25

~^5 ~^5

EXAMPLE 9

Simplify:

StrategyWe will prime factor the numerator and denominator. Then we will look
for any factors common to the numerator and denominator and remove them.

WHYWe need to make sure that the numerator and denominator have no
common factors other than 1. If that is the case, then the fraction is in simplest form.

Solution
To prepare to simplify, write 63 and 36 in
prime-factored form.

Simplify by removing the common factors

of 3 from the numerator and denominator.

Multiply the remaining factors in the numerator: 1  1  7 7.

Multiply the remaining factors in the denominator: 2  2  1  1 4.

Success Tip If you recognized that 63 and 36 have a common factor of 9, you

may remove that common factor from the numerator and denominator without

writing the prime factorizations. However, make sure that the numerator and

denominator of the resulting fraction do not have any common factors. If they

do, continue to simplify.

Factor 63 as 7 9 and 36 as 4 9, and then remove the

common factor of 9 from the numerator and denominator.

63

36



7  9

1

4  9

1



7

4



7

4



3

1

 3

1

 7

2  2  3

1

 3

1

63

36



3  3  7

2  2  3  3

63

36

Self Check 9

Simplify:

Now TryProblem 89

162

72

3

3 ƒ 9

2 ƒ 18

2 ƒ 36

7

3 ƒ 21

3 ƒ 63