Basic Mathematics for College Students

Add and subtract fractions that have different denominators.

Now we consider the problem  Since the denominators are different, we cannot

add these fractions in their present form.

three-fifths  one-third

Not similar objects

To add (or subtract) fractions with different denominators, we express them as

equivalent fractions that have a common denominator. The smallest common

denominator, called the leastor lowest common denominator,is usually the easiest

common denominator to use.

Least Common Denominator

The least common denominator (LCD)for a set of fractions is the smallest

number each denominator will divide exactly (divide with no remainder).

The denominators of and are 5 and 3. The numbers 5 and 3 divide many

numbers exactly (30, 45, and 60, to name a few), but the smallest number that they

divide exactly is 15. Thus, 15 is the LCD for and

To find we buildequivalent fractions that have denominators of 15. (This

procedure was introduced in Section 3.1.) Then we use the rule for adding fractions

that have the same denominator.

(^11) We need to multiply this denominator by 5 to obtain 15.
It follows that should be the form of 1 used to build
We need to multiply this denominator by 3 to obtain 15.
It follows that should be the form of 1 that is used to build
Since 14 and 15 have no common factors other
 (^) than 1, this fraction is in simplest form.

14

15

Add the numerators and write the sum

 (^) over the common denominator 15.

9  5

15

Multiply the numerators. Multiply the denominators.

 (^) Note that the denominators are now the same.

9

15



5

15

3

5.

3

3



1

3.

5

5





1

3



5

5

3

5



1

3



3

5



3

3

3

5 

1

3 ,

1

3.

3

5

1

3

3

5

 

1

3.

3

5

2

244 Chapter 3 Fractions and Mixed Numbers

Self Check 3

Perform the operations and

simplify:

Now TryProblem 29

2

9



2

9



2

9

EXAMPLE 3

Perform the operations and simplify: 

StrategyWe will use the rule for subtracting fractions that have the same

denominator.

WHYAll three fractions have the same denominator, 25.

Solution

This fraction can be simplified.

Multiply the remaining factors in the numerator: 3  1 3.

 (^) Multiply the remaining factors in the denominator: 1 5 5.

3

5

To simplify, factor 15 as 3 5 and 25 as 5 5. Then remove the

 (^) common factor of 5 from the numerator and denominator.

3  5

1

5

1

 5



15

25

Subtract the numerators and write the difference

(^) over the common denominator 25.

18

25



2

25



1

25



18  2  1

25

1

25

2

25

18

25