Higher GCSE

# How to represent a recurring decimal as a fraction

## What is a recurring decimal?

Some decimal numbers are easy to deal with – 0.5, 0.3, 0.25 etc. These are easy because they do not have recurring digits. Converting between these numbers and fractions is easy. Recurring decimals are numbers which have a repeated digit or sequence of digits – meaning that the digit or sequence repeat forever. When we are writing recurring decimals we obviously don’t write all the digits in the number, because there are infinitely many of them and it would take you forever… literally! Thankfully there is a shortcut – we put a dot above the number(s) that repeat. For example:

If more than one digit repeats, we put a dot above the first and last numbers in the repeated sequence. For example:

Now that you understand how recurring decimals work, I am going to show you how to convert a recurring decimal into a fraction. This is examinable on the higher tier only, so if you are studying foundation GCSE maths this content will not come up in your exams. The best way to show you how to do this is through some examples.

## Example 1

Algebra is going to help us to do this. First of all, we set our recurring decimal as x.

Now, I multiply x by 10. When you multiply a number by 10, this has the same effect as moving the decimal place one place to the right. For example, 14.5 x 10 = 145.

Now, the key to dealing with recurring decimals is to eliminate the recurring decimal. Once we do that, things become a lot easier. To show you how this can be achieved, I am going to rewrite x and 10x.

You can see that both 10x and x have the number 1 repeating. So, if I take away one from the other, I will eliminate the recurring decimal completely. See below.

So what happens here is when you take these two away from each other, you get 1. So we have:

Now, I can find x by dividing by 9 (if you need a reminder on solving equations, check out our guide to solving linear equations):

That’s it! Remember that we set x as 0.. So we have just worked out that:

And we have answered the question. We have represented the recurring decimal as a fraction.

## Example 2

We are going to use a very similar, though not identical, approach for this example.

Now, remember what our goal is here – we want to eliminate the recurring decimal. We want to generate two numbers, both of which have the same repeated digits.

If we do what we did in Example 1, this will not work. Look:

If you look carefully, you will notice that the recurring decimals don’t quite match, so they won’t cancel out if we attempt to subtract them from each other.

So what we need to do on this occasion is multiply x by 100. As this will give us the same repeated digits.

Now if we take away x from 100x, the recurring decimal will be eliminated.

Now we follow the same steps as Example 1.

## Example 3

This time we need to multiply by 1000.

You may be noticing a pattern here. The numerator of the fraction is always the recurring part of the decimal, and the denominator of the fraction is always a string of 9s. The number of 9s is the same as the number of recurring digits. The fraction can then sometimes be simplified. However, this pattern is not going to work for this next example…

## Example 4

This time we need to be a bit smarter. If we try to do it the same way as the previous examples, it isn’t going to work. Because of the 2 that is not recurring, it means that x cannot be used in our calculation. So what we need to do here is find 10x and 100x and take them away from each other  as this will allow the recurring part of the decimal to cancel out. Like so:

So that’s how to convert recurring decimals into fractions. If you want to try some on your own, try our worksheet. If you are in Bristol and need someone to show you this method in person, book in a free taster session.

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