Basic Mathematics for College Students

302 Chapter 3 Summary and Review

SECTION 3.4 Adding and Subtracting Fractions

To add (or subtract) fractions that have the

same denominator, add (or subtract) the

numerators and write the sum (or difference)

over the common denominator. Simplify the

result, if possible.

DEFINITIONS AND CONCEPTS EXAMPLES

Add:

The resulting fraction can be simplified.

Multiply the remaining factors in the denominator:

 2  1 2.

1

2

To simplify, factor 16 as 2 8.

Then remove the common factor of 8

from the numerator and denominator.



8

1

2  8

1



8

16

Add the numerators and write the sum

over the common denominator 16.

3

16



5

16



3  5

16

3

16



5

16

Adding and subtracting fractions that have

different denominators

1. Find the LCD.


2. Rewrite each fraction as an equivalent
   fraction with the LCD as the
   denominator. To do so, build each fraction
   using a form of 1 that involves any factors
   needed to obtain the LCD.


3. Add or subtract the numerators and write
   the sum or difference over the LCD.


4. Simplify the result, if possible.

Subtract:

Since the smallest number the denominators 7 and 3 divide exactly is 21,

the LCD is 21.

To build and so that their denominators

are 21, multiply each by a form of 1.

 This fraction is in simplest form.

5

21

Subtract the numerators and write the

 difference over the common denominator 21.

12  7

21

Multiply the numerators. Multiply the denominators.

 The denominators are now the same.

12

21



7

21

1

3

4

(^47)
7



1

3



4

7



3

3



1

3



7

7

4

7



1

3

The least common denominator (LCD)of a set

of fractions is the least common multiple

(LCM)of the denominators of the fractions.

Two ways to find the LCM of the denominators

are as follows:

• Write the multiples of the largest
  denominator in increasing order, until
  one is found that is divisible by the
  other denominators.


• Prime factor each denominator. The
  LCM is a product of prime factors,
  where each factor is used the greatest
  number of times it appears in any one
  factorization.

Add and simplify:

To find the LCD, find the prime factorization of both denominators and

use each prime factor the greatestnumber of times it appears in any one

factorization:

~

~

To build and so that their

denominators are 60, multiply

each by a form of 1.

This fraction is not in simplest form.

Multiply the remaining factors in the

 numerator and in the denominator.

11

12

To simplify, prime factor 55 and 60. Then

remove the common factor of 5 from the

numerator and denominator.



5

1

 11

2  2  3  5

1



55

60

Add the numerators and write the sum

 over the common denominator 60.

27  28

60

Multiply the numerators.

Multiply the denominators.

The denominators are now the same.



27

60



28

60

7

15

9

9 20

20



7

15



9

20



3

3



7

15



4

4

20  2  2  5

15  3  5

fLCD 2  2  3  5  60

9

20



7

15