When adding and subtracting fractions, it is necessary that the fractions have the same denominator - also known as a "common denominator". Once the fractions have the same denominator, their sum or difference can be found by adding/subtracting the numerators and placing the result over the common denominator. You do NOT add/subtract the denominators. In step form, the process is...

1. If denominators are different, find a common denominator (common multiple of the denominators)

2. Change the fractions so they have the common denominator, but in a way that does not change the value of the fraction (e.g. multiply by 1 - or by something equal to 1 -- identity property).

3. Add/subtract the numerators only. Place the result over the common denominator.

4. Reduce if necessary.

A common denominator is simply a multiple that each denominator has in common. There are a number of ways to find a common denominator, unfortunately, there is no one way that is always easiest. Sometimes the numbers are easy enough to work with that you instinctively know what the common multiple should be. For example, if your denominators were 4 and 6, you know a multiple common to both of these numbers is 12. There are, of course, others, but 12 is a common multiple.

If a common multiple is not easy to come up with, you have a couple of choices. First, you can try multiplying all the denominators together. That will give you a multiple of all of the denominators. This will always work, but usually this results in a larger value than you really need - which means you will have to reduce your answer. Using the numbers above again, if I multiply 4 and 6, I get a common multiple of 24. Although 24 is a common multiple, 12 is too - and lower as well. If I use the 24, I will can get a correct answer, but the end result will have to be reduced.

The LCM (least common multiple) of the denominators is always a good choice, for it is a common multiple, and is the lowest (that is why it is called the least common multiple). There are a number of ways to find a LCM, but we will not concentrate on that task here.

Once you have found a common multiple of your denominators, you are ready to change your fractions so that each one features this common multiple (so that all your denominators are the same). To do this, we use the identity property of multiplication. This property assures us, that if we multiply our fraction(s) by 1 (in this form, or any other), the value of our fraction(s) will not change - although the numbers in the fraction might look a bit different. Let's take a look!

We start with the problem . Since 4 and 6 are not the same, we cannot add these right away. We need to change these fractions so that they have the same denominator (without changing the value of the fraction). 4 and 6 have many common multiples (an infinite number actually), but only one is necessary. As we saw above, we could multiply them together and get a common multiple (24), but we can also use 12 (the LCM). Now that we have a common multiple in mind (12), we must change the fractions so that 12 is the denominator.

We can multiply the first fraction by , for it equals 1 (which won't change the value), but will cause the denominator to be 12. Likewise, we will multiply the second fraction by , for it also equals 1. This will cause the denominator of the second fraction to become 12 also. So, we take and it results in . Now, since we have the same denominator, we can add. Notice, the changes we made to the fractions are okay, for if we reduced them (which does not change the value), we would end up with the same fractions we started with.

To add these fractions we need to add their numerators, and place the result over the common denominator. 9 plus 10 equals 19 and the common denominator is 12, so the sum of these two fractions is . Now, if this fraction reduced, we would go ahead and reduce, however, this fraction is reduced. It is not necessary to change the fraction to a mixed number, however, if this result was the answer to a real life situation, the mixed number might make more sense.