Basic Mathematics for College Students

3.4 Adding and Subtracting Fractions 249

EXAMPLE 9

Subtract and simplify:

Strategy We begin by expressing each fraction as an equivalent fraction that has
the LCD for its denominator. Then we use the rule for subtracting fractions with
likedenominators.

WHY To add (or subtract) fractions, the fractions must have like denominators.

Solution

To find the LCD, we find the prime factorization of both denominators and use
each prime factor the greatestnumber of times it appears in any one factorization:

~

~

The LCD for and is 84.
We will compare the prime factorizations of 28, 21, and the prime factorization
of the LCD, 84, to determine what forms of 1 to use to build equivalent fractions
for and with a denominator of 84.

Cover the prime factorization of 28. Cover the prime factorization of 21.

Since 3 is left uncovered, Since 2 2 4 is left uncovered,

use to build use to build

To build and so that their denominators are 84,

multiply each by a form of 1.

This fraction is not in simplest form.

Multiply the remaining factors in the

numerator: 5  1 5. Multiply the

remaining factors in the denominator:

2  2  3  1 12.



5

12

To simplify, factor 35 and 84. Then

remove the common factor of 7 from

the numerator and denominator.



5  7

1

2  2  3  7

1



35

84

Subtract the numerators and write the difference

 over the common denominator.

39  4

84

Multiply the numerators. Multiply the denominators.

 The denominators are now the same.

39

84



4

84

1

21

13

(^1328)
28



1

21



13

28



3

3



1

21



4

4

1

21.

4

4

13

28.

3

3

LCD 2  2  3  7 LCD 2  2  3  7

1

21

13

28

1

21

13

28

2 appears twice in the factorization of 28.

3 appears once in the factorization of 21.

7 appears once in the factorizations of 28

and 21.

fLCD 2  2  3  7  84

28  2  2  7

21  3  7

13

28



1

21

Self Check 9

Subtract and simplify:

Now TryProblem 53

21

56



9

40

Identify the greater of two fractions.

If two fractions have the same denominator, the fraction with the greater numerator
is the greater fraction.
For example,

because because

If the denominators of two fractions are different, we need to write the fractions
with a common denominator (preferably the LCD) before we can make a
comparison.

  1  2

1

3



2

3

7  3

7

8



3

8

4

84

~^242

~^221

~^3 ~^7