Adding Fractions With Different Denominators

Arjun Kulkarni Mar 23, 2019
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Math. A lifelong passion to some and an arch nemesis to others. No matter which way you're headed in life, there are some basic math problems that you need to learn how to solve. Math involving fractions is one of them.
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 tough. But it's not an impossible task! It involves a level of ingenuity at a small level. 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. Any number by default identification has a denominator. To make it simple, if a number does not display a denominator, it should not be assumed it does not have a denominator.
If a number does not display a denominator, it means that the denominator of that number is invariably 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
5 + 9 = 14.

Now, on to two numbers with unlike denominators.

LCM Method (Least Common Multiple)

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 has the lowest common multiple 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.
As an example to verify the given theory, we have two numbers with differing denominators. The objective is the addition of both the numbers.The first step would be to make the denominators as one, and the same, or in other words, work a way to make a denominator common for the numerators.
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. Solving math problems, one can say this method being i) very simple, as it prevents the necessity of finding an LCM and then multiplying. The 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!
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