# Common denominator

In a fraction, a denominator is the number on the bottom half of the fraction, below the fraction bar.

A common denominator is a common multiple of the denominators of two or more fractions. For example, for the fractions and , 24 and 48 are two of the common denominators for denominators 8 and 12. The least common denominator is 24.

When performing fraction addition and subtraction, the denominators of the fractions must be the same; it is necessary to find a common denominator.

There are a number of ways to find the common denominator of a fraction.

### Multiplying the denominators

This may be the quickest way to find a common denominator, but in many cases will make the actual addition/subtraction problem more tedious.

Example

If we want to add the fractions and , we could multiply 8 and 16 to find a common denominator of 128. Then, we would convert the fractions to equivalent fractions with a denominator of 128. To do this, we multiply each fraction by equivalent fractions of the number 1 that would make the denominator 128:  is not in simplest form since 72 and 128 share a factor of 8 and can be reduced to .

In this case, the least common denominator is 16, since 8 × 2 = 16, so we only needed to change one fraction. This is the reason that fraction addition and subtraction are usually performed using the least common denominator. Otherwise, unless the least common denominator happens to be the product of the denominators, the result will always share a factor and need to be reduced.

### Listing multiples

For fractions with denominators that are easy to work with, listing multiples is one of the simplest ways to find a common denominator.

Example

Add and :

We can start by listing the multiples of the denominators, 3 and 4.

Multiples of 3: 3, 6, 9, 12,...

Multiples of 4: 4, 8, 12,...

Since 12 is the first multiple that 3 and 4 share, 12 is the least common multiple, or least common denominator of and .

We could keep going and find more common multiples, but using the least common multiple whenever possible helps keep the problem simple. There are other methods for finding the least common denominator such as using a formula involving the greatest common divisor, using prime factorization, or other algorithms, but the above are two of the simpler methods that should suffice for fractions you're likely to work with by hand.