Adding Fractions With Different Denominators

It is one of the basic math problems. While it is relatively simple to add two numbers with the same denominator - where you just have to go on and add them; two fractions with different denominators is a slightly tougher job. But it's not like it can't be done! Adding fractions with unlike denominators is one of the most important functions in math.

Adding Fractions with Unlike Denominators

When we add two numbers, for e.g. 5+9, we are actually adding two numbers with the same denominator, 1. That means, we are adding the two numbers 5/1 + 9/1. Hence, since the denominator is the same, adding the two numbers becomes fairly straightforward. The addition goes thus:

5/1 + 9/1 = 14/1.

or simply,

5 + 9 = 14.

Now on to two numbers with unlike denominators.

LCM Method

In the first method, you have to find out the LCM of the two denominators. Say, you are trying to add the two numbers 20/10 + 30/20. Now the first denominator (10) is a factor of the other denominator (20). Hence, you take the greater number 20, which is the lowest common multiple (LCM) of the two numbers (10 and 20) as the denominator for the final answer. Then, since 20 = 10 * 2, you also multiply the numerator by 2 and then add the two numbers. Basically, whatever you multiply the lesser of the two denominators, to get to the LCM, you multiply to the numerator.

At the end of this step, we get two fractions with the same denominator, hence you can add them easily.

20/10 + 30/20

= (20*2/10*2) + 30/20

= 40/20 + 30/20

= 70/20.

Another case is where there is no LCM between the two denominators, but the LCM needs to be found out.

85/15 + 95/10

Now none of these two denominators is the LCM of the two numbers. Hence, we need to find out the LCM first. The LCM of the two numbers is 30 (15 * 2 and 10 * 3). The next step is to multiply the numerators of the two fractions. Since we are multiplying the first number's denominator (15) by 2, we multiply the numerator by 2 as well. Since we are multiplying the denominator of the second fraction (10) by 3, we multiply its numerator by 3 as well. Then once we have the same denominator (30), we add the two results in the numerators to get our final answer.

85/15 + 95/10

= (85*2/15*2) + (95*3/10*3)

= 170/30 + 285/30

= 455/30.

Cross-multiplication Method

Sometimes, the two denominators have no LCM and the only way we can get a common denominator is by multiplying the denominators by each other. It is the simplest and most straightforward method, but the numbers get pretty big.

11/7 + 13/9

Now 7 and 9 do not have an LCM, so we need to multiply one by the other to get a common denominator. The next step is to multiply the numerators by the same number which you multiplied the denominators by. Hence, you multiple 11 into 9 (because, we multiplied its denominator 7 into 9) and 13 into 7. And then we add the two fractions.

11/7 + 13/9

= (11*9/7*9) +(13*7/9*7)

= 99/63 + 91/63

= 190/63.

This method can also be used with numbers that have an LCM. Frankly, I find this method very simple as it avoids the necessity of finding an LCM and then multiplying. Cross-multiplication method is faster and much more straightforward.

You can practice this math lesson with some more similar sums so that you know this method perfectly well. After all, practice makes man perfect!