Writing recurring decimals as fractions

In a nutshell

Not all fractions can be expressed as a terminating decimals (decimals with only a finite number of digits). In some cases, their decimal form may be infinitely long and so you cannot write down the entire number.

Recurring decimals

Definition

A recurring decimal is a number containing an infinitely repeating digit or series of digits. The repeating digit(s) are indicated with a $\dot{}$ above the numbers.

Example 1

The following are examples of recurring decimals:

$0.1111... = 0.\dot{1},\text{ } 0.345345...=0.\dot{3}4\dot{5},\text{ } 0.298454545... = 0.298\dot{4}\dot{5}$

Converting between recurring decimals and fractions

Fraction to recurring decimal

The process of converting a fraction to a recurring decimal is the same as before, except ensure that the $\dot{}$ goes above the correct numbers to indicate which are repeating.

Recurring decimal to fraction - higher only

Procedure

1.	Multiply the recurring decimal by a multiple of $10$ , for example $10 , 100, or \space1000$ etc. such that the recurring digits overlap in the same position.
2.	Subtract the recurring decimal part from another, leaving a $9x, \text{ } 99x$ etc.
3.	Rearrange to obtain the number as a fraction.
4.	Simplify the fraction (if required).

Example 2

Write $0.4545... = 0.\dot{4}\dot{5}$ as a fraction in its simplest form.

$\begin{aligned}x &= 0.454545... &\qquad \textcircled{1} \\100x &= 45.454545... &\qquad \textcircled{2} \\ 99x &= 45 &\qquad \textcircled{2}- \textcircled{1}\\x &= \dfrac {45}{99}\end {aligned}$

Therefore, $\underline{ x= \dfrac {5}{11}}$ .