## Finding Common Denominators

The numerator is the top part of a fraction, and the denominator is the bottom part of a fraction. Before you can add or subtract fractions, you have to be able to get a common denominator. Before you can get a common denominator, you have to understand some basics about fractions, so that’s where we’re going to start.

Consider the following two pieces of information: 1) If you multiply anything by 1, you get what you started with, and 2) Anything divided by itself equals 1. With that in mind, take a look at this:

$\frac{2}{3} * \frac{4}{4} = \frac{8}{12}$

We know from #2 above that 4/4 is 1. From #1, we know that if we multiply 2/3 by 1, we get an equivalent value to 2/3. With that in mind, we know for sure that 2/3 = 8/12. Notice that with this process, we changed the denominator of the fraction from 3 to 12 without changing the actual value of the fraction. This is the key idea in getting a common denominator for a fraction.

When we go to add or subtract fractions, we need the following sort of thing to happen:

$\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$
$\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}$

Note that the denominator of each fraction we’re adding or subtracting is the same. So if we want to add or subtract two fractions with a different denominator, then we have to use a technique like the one we looked at above to change the denominators around so that they are the same.

To get a common denominator, we need to find the least common multiple (LCM) of the two denominators in question. The LCM is the lowest number that is a perfect multiple of two numbers. For example, the LCM of 2 and 5 is 10, and the LCM of 4 and 6 is 12. The LCM of the two denominators is what our new denominator will be. Consider the following process:

$\frac{3}{4} + \frac{5}{6}$

Suppose we want to add these two fractions. First we’ll need to find the LCM of 4 and 6, which we noted earlier is 12. We want to make the denominator of each fraction 12, so we’ll need to do some multiplying by 1 to get that. So here we go:

$\frac{3}{4} * \frac{3}{3} = \frac{9}{12}$
$\frac{5}{6} * \frac{2}{2} = \frac{10}{12}$

We know from the above explanation that 3/4 = 9/12 and that 5/6 = 10/12, so we can substitute these values in like so:

$\frac{3}{4} + \frac{5}{6}$
$\frac{9}{12} + \frac{10}{12}$
$\frac{9+10}{12}$
$\frac{19}{12}$

And we’re finished.