Example 1:
Find the common denominator of the fractions.
16 and 38
We need to find the least common multiple of 6 and 8 . One way to do this is to list the multiples:
6,12,18,24−−,30,36,42,48,...8,16,24−−,32,40,48,...
The first number that occurs in both lists is 24 , so 24 is the LCM. So we use this as our common denominator.
Listing multiples is impractical for large numbers. Another way to find the LCM of two numbers is to divide their product by their greatest common factor ( GCF ).
Example 2:
Find the common denominator of the fractions.
512 and 215
The greatest common factor of 12 and 15 is 3 .
So, to find the least common multiple, divide the product by 3 .
12⋅153=3 ⋅ 4 ⋅ 153=60
If you can find a least common denominator, then you can rewrite the problem using equivalent fractions that have like denominators, so they are easy to add or subtract.
Example 3:
Add.
512+215
In the previous example, we found that the least common denominator was 60 .
Write each fraction as an equivalent fraction with the denominator 60 . To do this, we multiply both the numerator and denominator of the first fraction by 5 , and the numerator and denominator of the second fraction by 4 . (This is the same as multiplying by 1=55=44 , so it doesn't change the value.)
512=512⋅55=2560215=215⋅44=860
512+215=2560+860 =3360
Note that this method may not always give the result in lowest terms. In this case, we have to simplify.
=1130
The same idea can be used when there are variables in the fractions—that is, to add or subtract rational expressions .
Example 4:
Subtract.
12a−13b
The two expressions 2a and 3b have no common factors, so their least common multiple is simply their product: 2a⋅3b=6ab .
Rewrite the two fractions with 6ab in the denominator.
12a⋅3b3b=3b6ab13b⋅2a2a=2a6ab
Subtract.
12a−13b=3b6ab−2a6ab =3b − 2a6ab
Example 5:
Subtract.
x16−38x
16 and 8x have a common factor of 8 . So, to find the least common multiple, divide the product by 8 .
16⋅8x8=16x
The LCM is 16x . So, multiply the first expression by 1 in the form xx , and multiply the second expression by 1 in the form 22 .
x16⋅xx=x216x38x⋅22=616x
Subtract.
x16−38x=x216x−616x =x2 − 616x\
