Recurring Decimals to Fractions | Maths Made Easy

# Recurring Decimals to Fractions Worksheets, Questions and Revision

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## What you need to know

### Recurring Decimals to Fractions

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. Having a good understanding of basic algebra and rearranging formulae will help with this topic.

Two examples of recurring decimals, 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.

### 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”.

### Convert Recurring Decimals to Fractions

This is really what this topic is about. Converting a recurring decimal to a fraction can be a stand alone exam question so it is certainly a skill you want to master. In order to do this, understanding basic algebra and how to rearrange equations is a useful skill to have. The following examples will show an algebra method to help convert recurring decimals to fractions

### Example 1: Fraction to Decimal

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

Then look to complete the bus stop method.

As the decimal repeats itself, it is a recurring decimal which can be written as $0.\dot{2}\dot{1}$

### Example 2: Recurring Decimal to Fraction

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

Always assign the thing your converting to be $x$. Once you’ve done this, the aim is to multiply this number so that you end up with two numbers that have exactly the same recurring digits after the decimal point. When you subtract one from the other, the digits after the decimal point will cancel and you’ll be left with a nice whole number.

So, 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.

$100x -x = 99x$

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

$99x = 14$

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

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

### Example 3: Recurring Decimal to Fraction

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

So, let $x = 0.8\dot{3}$. This time, 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

$100x - 10x = 90x$

$83.\dot{3} - 8.\dot{3} = 83 - 8 = 75$

$90x = 75$

Dividing both sides by 90, we get that

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

### Example Questions

Treating this fraction as a division, we will use short division to find the result of dividing 1 by 9. The result of the short division should look like,

$\;\;\;\;0.\;1\;1\;1\;1\;1\;\\9\overline{\left)1.0{}^10{}^10{}^10{}^10\right.}$

Hence,

$\dfrac{1}{9} = 0.\dot{1}$

If we let $x=0.\dot{3}$ and $10x=3.\dot{3}$

Then $x$ and $10x$ are identical after the decimal place. Hence if we subtract one from the other then everything after the decimal point shall cancel.

$10x - x = 9x = 3.\dot{3} - 0.\dot{3} = 3$

Thus if we rearrange to make $x$ the subject then,

$x=\dfrac{3}{9}=\dfrac{1}{3}$

If we let $x = 0.\dot{3}9\dot{0}$ and $1,000x = 390.\dot{3}9\dot{0}$

Then $x$ and $1,000x$ are identical after the decimal place. Hence if we subtract one from the other then everything after the decimal point shall cancel.

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

Thus if we rearrange to make $x$ the subject then,

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

This simplifies to,

$x = 0.\dot{3}9\dot{0} = \dfrac{130}{333}$

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

$\;\;\;\;\;0\;\;.9\;\;0\;\;9\;\;0\;\;9\;\;0\;\;9\;\;0\\11\overline{\left)10.0{}^10{}^{10}0{}^10{}^{10}0{}^10{}^{10}0{}^10\right.}$

Hence,

$\dfrac{10}{11} = 0.\dot{9}\dot{0}$

If we let $x = 1.5\dot{4}$ then $10x = 15.\dot{4}$ and $100x = 154.\dot{4}$

Then, both $10x$ and $100x$ are identical after the decimal place. Hence, we are able to subtract one from the other in order that everything after the decimal point shall cancel.

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

Thus if we rearrange to make $x$ the subject then,

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

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