## What you need to know

A recurring decimal is a decimal number which has a pattern than repeats over and over after the decimal place. Every recurring decimal can also be written as a fraction. In this topic, we’ll look at how to go from a recurring decimal to and fraction and vice versa.

Two examples of recurring decimals, one quite common and one less common, are

$\dfrac{1}{3} = 0.\dot{3} = 0.33333...$

$\dfrac{6}{11} = 0.\dot{5}\dot{4} = 0.54545454...$

Notice the dot on top of some of the digits; this tells us what is repeated. The first dot denotes the start of the repeated section, and the second dot denotes the end of it.

Convert Fractions to Recurring Decimals

To convert a fraction to a recurring decimal we must treat the fraction like it is a division and use some method of division to divide the numerator by the denominator. Here we will use short division, also known as the “bus stop method” (I highly recommend this method of division for all purposes). In this example, we’ll see how it soon becomes obvious that the result of your division is a recurring decimal.

Example: Write $\dfrac{7}{33}$ as a decimal.

So, we’re going to try dividing 7 by 33. When setting up the bus stop method, you should put in a whole lot of zeros after the decimal place – chances are you’ll only need a few, but it’s better to put more than you need. So, the set up short division should look like

Doing the short division (which you can brush up on here (https://mathsmadeeasy.co.uk/gcse-maths-revision/multiplying-dividing-gcse-maths-revision-worksheets/)), we get

Since the last remainder was 7, we looked for how many times 33 goes into 70, but we already did that once. The answer is twice, and then the remainder is 4, which means we’re now going to be asking the question of how many times 33 goes into 40, but we’ve already done that, too. Carrying this on, the pattern becomes obvious.

Therefore, we must have that $\frac{7}{33} = 0.\dot{2}\dot{1}$. Practice a few of these and you’ll quickly get used to spotting when they start repeating.

Convert Recurring Decimals to Fractions

This is really what this topic is about. Let’s dive into an example to see how it goes.

Example: Write $0.\dot{1}\dot{4}$ as a fraction.

Firstly, set $x = 0.\dot{1}\dot{4}$, the thing we want to convert to a fraction. Then,

$100x = 14.\dot{1}\dot{4}$.

Now that we have two numbers, $x$ and $100x$, with the same digits after the decimal point, if we subtract one from another, the numbers after the decimal point will cancel.

$100x - x = 99x = 14.\dot{1}\dot{4} - 0.\dot{1}\dot{4} = 14$

Removing the working out steps from this line, we have

$99x = 14$

Then, if we divide both sides of this by 99, we get

$x = \dfrac{14}{99}$.

Okay, so how does this method work? Firstly, always assign the thing your converting to be $x$. Once you’ve done this, the aim is to end up with two numbers (both will be some multiple of ten times by $x$) that have exactly the same recurring digits after the decimal point. This really is key, because then when you subtract one from the other (in the main step of the process), the digits after the decimal point will cancel and you’ll be left with a nice whole number.

At that point, it’s just a case of solving a straightforward linear equation by doing one division, and you’ve got your answer.

Let’s see another example – remember, the aim is to get two multiples of $x$ both with the same thing after the decimal place.

Example: Write $0.8\dot{3}$ as a fraction.

Firstly, notice how the 8 has no dot above it so isn’t repeating. This will make a difference.

So, let $x = 0.8\dot{3}$. This time, if we multiply this by 10, 100, 1,000 etc, we’re not going to end up with something that has the same digits after the decimal point as $x$.

However, if instead we take $10x = 8.\dot{3}$ and $100x = 83.\dot{3}$, then we have two multiplies of $x$ that do have the same digits after the decimal point.

So, subtracting one from the other, we get

$100x - 10x = 90x = 83.\dot{3} - 8.\dot{3} = 83 - 8 = 75$

Removing the working out steps from this line, we have

$90x = 75$

Dividing both sides by 90, we get that

$x = \dfrac{75}{90} = \dfrac{5}{6}$.

This time we were unable to subtract $x$ from anything to get the desired outcome, so we had to be clever and multiply $x$ by 10 and 100 before doing the subtraction. You will need to do something like this anytime there’s a decimal digit in your number that isn’t involved in the recurring part.

## Example Questions

Treating this fraction like a division, we will use short division (or otherwise) to find the result of dividing 10 by 11. The result of the short division should look like

Set $x = 0.\dot{3}9\dot{0}$. Then, we get

$1,000x = 390.\dot{3}9\dot{0}$

Both of $x$ and $1,000x$ have the same thing after the decimal place, so they are suitable to subtract from one another. Doing so, we get

$1,000x - x = 999x = 390.\dot{3}9\dot{0} - 0.\dot{3}9\dot{0} = 390$

Removing the working-out steps, we have

$999x = 390$

Then finally, divide both sides by 999 to get

$x = \dfrac{390}{999}$.

Set $x = 1.5\dot{4}$. Then, we get

$10x = 15.\dot{4}$

This does not have the same thing after the decimal place as $x$ so we cannot subtract yet. Instead, consider

$100x = 154.\dot{4}$

Then, of $10x$ and $100x$ have the same thing after the decimal place, so they are suitable to subtract from one another. Doing so, we get

$100x - 10x = 90x = 154.\dot{4} - 15.\dot{4} = 154 - 15 = 139$

Removing the working-out steps, we have

$90x = 139$

Then finally, divide both sides by 90 to get

$x = \dfrac{139}{90}$.

## Fractions and Recurring Decimals Revision and Worksheets

Fractions to recurring decimals
Level 6-7
Recurring Decimals to Fractions 1
Level 6-7
Recurring Decimals to Fractions 2
Level 6-7
Recurring Decimals to Fractions 3
Level 6-7

## Fractions and Recurring Decimals Teaching Resources

For teachers and GCSE Maths tutors looking for fraction worksheets and decimal questions then the worksheets and revision materials above are a great resource to use with your students. You can also test their progress by getting them to take an online fractions test.