A recurring decimal is just a decimal that repeats forever. Some fractions can only be expressed as recurring decimals.

E.g. 1/3 = 0.333333333…, 24/99 = 0.24242424…

They are often written with little dots over them to indicate how they recur (as shown above).

Turning Recurring Decimals Into Fractions

To do this you’re going to need algebra (sorry about that!).

Example: Write 0.234234234234… as a decimal.

Let x = 0.234234234…

Multiply by 10,100, 1000 or whatever power of ten to get the recurring bit past the decimal point. So here you would multiply by 1000.

So 1000x = 234.234234234….

So now just take x away from 1000x

999x = 234.234234234… – 0.234234234… = 234

999x = 234

So x = 234/999

Simplify it down.

x = 26/11

But it’s a bit harder when the recurring bit doesn’t come directly after the decimal.

Example: Turn 0.166666666… into a fraction.

So let x = 0.166666666…

Multiply by 10,100, 1000 or whatever power of ten to get one full repeated lump past the decimal point. So here you would multiply by 100.

100x = 16.66666666…

Subtracting x isn’t going to help here….you’ll end up with a horrible number….but what if you subtracted 10x?

10x = 1.66666666…

So 90x = 16.66666666… – 1.66666666… = 15

So x = 15/90 = 1/6

Neat huh?

Turning Fractions Into Recurring Decimals

Easy with a calculator…without it you are going to have to do some long division until you spot the recurring pattern.