How to obtain the common denominator. Solved exercises

I’m going to show you how the common denominator is calculated in fractions.

What the common denominator is and what it is used for

The common denominator is used to add and subtract fractions with different denominator, since in that case, the denominators of all of them must be the same.

Therefore, in order for all denominators to be equal, we need to obtain a common denominator for all fractions.

We must distinguish between the common denominator and the lowest common denominator:

The common denominator must be a multiple of any of the denominators and furthermore, as its own name indicates, it must be common to all denominators.

And is it necessary that it be a multiple, can’t you put any denominator?

Yes, it must be a multiple common to denominators, because we need the fractions to be equivalent and with any denominator, the numerator would become a decimal number, thus complicating the fraction and what we want is to make it always easy.

We’ll see below when we learn to get the common denominator

The lowest common denominator is the lowest common multiple of all denominators and is obtained by calculating the lowest common multiple of denominators.

It is at the end and after two methods of obtaining the same denominator for the fractions and using one or the other we will arrive at the same final result, although in principle the common denominators are different.

How to draw common denominator

Once we are clear about what the common denominator is and we are clear about what the equivalent fractions are, we are going to join these two concepts to explain how the common denominator is calculated.

To reduce a fraction to a common denominator we need 2 steps:

1. Determine the common denominator, which may be:
   1. A multiple common to denominators
   2. The minimum common multiple of denominators
2. Transform each fraction to its corresponding equivalent fraction with its new denominator

Let’s see an example:

We have to make this sum of fractions with different denominator. Therefore, in order to add we need all denominators to be the same:

1 – What is the common denominator?

Well, on the one hand, it can be any multiple common to denominators. The best way to achieve this is to multiply all denominators:

Or it could also be the lowest common multiple of denominators, which corresponds to the lowest common denominator:

2- Transform fractions to their equivalent fractions, with the common denominator chosen.

First of all, we are going to choose 24.

as the common denominator.

We are going to transform the first fraction:

It must have 24 as denominator. Therefore, as we know that to transform this fraction into its equivalent, we must multiply up and down by the same number.

We already have the bottom one, but by what number do I multiply the numerator?

Then if before we had a 2 and now we have a 24, it is as if we had multiplied by 12 the denominator and this 12 is obtained by dividing the common denominator between the denominator of the original fraction:

24/2 = 12

Therefore, the numerator must also be multiplied by 12.

With the other fractions we do the same:

So in order to make the original addition, we would have to add the equivalent fractions:

With the lowest common denominator, they would look like this: