Adding Fractions with Unlike Denominators

Adding fractions, like any other mathematical problem, may seem a bit difficult at first. However, once you know the basic formula and you start practicing, it would not take more than a couple of minutes.

Always start with a revision of the basics of fractions. Let's take an example:

In the fraction 3/4, 3 is the numerator and 4 is the denominator.

There is one rule in fractions, which applies whether we are adding, subtracting, or comparing fractions,which is―the denominator is always the same. So, if we have to add up fractions 3/4 and 2/4, the denominator is already same, i.e., 4. In such a case, adding both these fractions would be like:

3/4 + 2/4 = 6/4

This was pretty easy, as all we had to do, was to add up the numerators. Now, let's take two such fractions, in which the denominator is not the same. For example, 3/5 and 6/7. In such a case, we need to make the denominator same first, only then we can add up the numerators. A simple way to do that is to follow the cross multiplication method.

Let's start with an example :If we have to add up 3/5 and 6/7, to make denominator same, we multiply both the denominators, which will be 5 × 7 = 35.

The denominator is 35. Now, before adding up the numerators, they firstly have to be multiplied with the same number the denominator has been multiplied with. In the fraction 3/5, denominator 5 was multiplied by 7 to arrive at 35, the common denominator. So, the numerator 3 too will be multiplied by 7, hence, the numerator becomes 3 × 7 = 21.

Likewise, in the fraction 6/7, the numerator is multiplied by 5, as the denominator 7 was multiplied by 5 too, hence, the numerator becomes 30. Let's see this problem mathematically:

3/5 + 6/7 = 3 × 7/5 × 7 + 6 × 5/7 × 5 = 21/35 + 30/35 = 51/35

There is another method, which is called the LCM method. This method, however, can only be used if either one the denominators is a factor of the other, or if their LCM can be found. Let's take an example here of the fractions 7/10 + 9/20.

In this case, if we have to add both these fractions, we need to make the denominator same. It can be done through the cross multiplication method. However an easy method here will be the LCM method, since we can see that the denominator 10 is a factor of the denominator 20.

The LCM of the denominators 10 and 20, which are factors of each other, is the bigger number out of the two, i.e., 20. So, 20 is now the common denominator. As in the previous method, before adding up numerators, they are multiplied by the same number the denominator is multiplied to arrive at the common denominator.

So, for fraction 7/10, to arrive at 20, denominator 10 is multiplied by 2, hence, numerator 7 is multiplied by 2 as well. Likewise, for fraction 9/20, to arrive at 20, denominator 20 is multiplied by 1, so numerator is multiplied by 1 as well, hence, numerator remains the same as 9 × 1 = 9.

Let us see the problem mathematically:

7/10 + 9/20 = 7 × 2/10 × 2 + 9 × 1/20 × 1 = 14/20 + 9/20 = 23/20

You need to practice everyday if you want to perfect this mathematical concept. So, solve as many fraction problems as you can and be a master!