Adding fractions with the same denominator is much like adding non-fractions. You add the numerator as you would normally, and keep the denominator the same. For example:
Adding fractions when they have a different denominator is slightly more involved. To add fractions, the denominator of the fractions must be the same. If the denominator is not the same, the fractions being added need to be turned into equivalent fractions with the same denominator, then added. To do this we need to find a common denominator. Because of the nature of fractions, it is possible to find numerous common denominators, but generally, we want to find the least common denominator to make the arithmetic simpler.
1. Add the fractions :
Notice that in (1) we cannot just add the numerators because the denominators are not the same. So we must first find a common denominator. To do this, we need to form equivalent fractions for , that share the same denominator. We can do this by multiplying each of the fractions by some multiple of the fraction such that these fractions will have the same denominator:
Now that all the fractions share the same denominator, we can add them as we would normally, as shown in (2). Note that in this particular problem, we did not need to change all the fractions, since one of the fractions has a denominator of 12, a multiple that all the fractions share.
2. Add the fractions :
In this case, the first common multiple that 4 and 7 share is 4 × 7 = 28. Multiplying all of the denominators is one way to find a common denominator, but it often won't be the least common denominator. In Example 1, we could have multiplied 3 × 6 × 12 = 216, and formed the equivalent fractions:
and are equivalent fractions, but using the least common denominator of 12 makes the problem easier to work with.