Basic Mathematics for College Students

The figure below shows and expressed as equivalent fractions with a
denominator of 15. Once the denominators are the same, the fractions are similar
objects and can be added easily.

We can use the following steps to add or subtract fractions with different
denominators.

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.

––^9

15

––^5

15

(^3) –
5
(^1) –
3
(^14) ––
+= 15
1
3
3
5
3.4 Adding and Subtracting Fractions 245
Self Check 4
Add:
Now TryProblem 35

1

2



2

5

Self Check 5

Subtract:

Now TryProblem 37

6

7



3

5

EXAMPLE 4

Add:

Strategy We will express each fraction as an equivalent fraction that has the LCD
as its denominator. Then we will use the rule for adding fractions that have the
same denominator.

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

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

1 1

To build and so that their denominators are 21,

multiply each by a form of 1.

Since 17 and 21 have no common factors other

 (^) than 1, this fraction is in simplest form.

17

21

Add the numerators and write the sum

 (^) over the common denominator 21.

3  14

21

Multiply the numerators. Multiply the denominators.

The denominators are now the same.



3

21



14

21

2

3

1

^27

3



7

7

1

7



2

3



1

7



3

3

1

7



2

3

EXAMPLE 5

Subtract:

Strategy We will express each fraction as an equivalent fraction that has the LCD
as its denominator. Then we will use the rule for subtracting fractions that have the
same denominator.

5

2



7

3