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Least Common Denominators (LCDs)

It is difficult to add or subtract fractions when the denominators are not same. So, we use a common denominator. It is usually easiest to use the least common denominator. The least common denominator is simply the least common multiple (LCM) of the two denominators.

We need to find the least common multiple of 6 and 8. To do this, we can list the multiples:

6: 6, 12, 18, 24, 30, 36, 42, 48, ...

8: 8, 16, 24, 32, 40, 48, ...

We see that 24 is the LCM. So we use this as our common denominator.

6 × 4 = 24   and 8 × 3 = 24. So multiply the first fraction by 1 in the form of 4/4, and the second fraction by 1 in the form of 3/3:

Another way to find the LCM of two numbers is to divide their product by their greatest common factor (GCF).

In this example, the GCF of 6 and 8 is 2. The product of 6 and 8 is 48.

So, the LCM is 48 ÷ 2 = 24.

THE SAME IDEA can be used to add or subtract rational expressions.

The least common denominator (LCD) in this case is 16x. So, multiply the first expression by 1 in the form x/x, and multiply the second expression by 1 in the form 2/2.