Recurring Decimals (Cambridge (CIE) IGCSE Maths)

Revision Note

Written by: Mark Curtis

Reviewed by: Dan Finlay

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When writing a rational number as a decimal, it will either be:
- A decimal that stops, called a "terminating" decimal
  - $\frac{1}{4} = 0.25$
- Or a decimal that repeats with a pattern, called a "recurring" decimal
  - $\frac{32}{99} = 0.32323232 . . .$
The recurring part can be written with a dot above the digit that repeats
If multiple digits repeat, dots are used on the first and last digits that repeat
- $0.3333 . . . = 0 . \dot{3}$
- $0.121212 . . . = 0 . \dot{1} \dot{2}$
- $0.325632563256 . . . = 0 . \dot{3} 25 \dot{6}$

Write out the first few decimal places to show the recurring pattern and then:

STEP 1
Write the recurring decimal as $x = . . .$
- $x = 0.35353535 . . .$
STEP 2
Multiply both sides by 10 repeatedly until two lines have the same recurring decimal part
- $\begin{array}{rcl} x & = & 0.35353535 . . . \end{array}$
- $\begin{array}{rcl} 10 x & = & 3.5353535 . . . \end{array}$
- $\begin{array}{rcl} 100 x & = & 35.353535 . . . \end{array}$
  - Note that x and 100x have 35 repeating after the decimal point, the repeating pattern after 10x is 53 repeating
STEP 3
Subtract the two lines which have matching recurring decimal parts
- $\begin{array}{rcl} 100 x - x & = & 35.353535 . . . - 0.35353535 . . . \end{array}$
- $\begin{array}{rcl} 99 x & = & 35 \end{array}$
STEP 4
Divide both sides to get $x = . . .$
Cancel if necessary to get fraction in its lowest terms
- $x = \frac{35}{99}$

Write $0 . \dot{3} 0 \dot{7}$ as a fraction in its lowest terms.

Write as $x = . . .$ to show the pattern

$x = 0.307307307307 . . .$

Multiply both sides by 10 repeatedly until two lines have the same recurring decimal part

$\begin{array}{rcl} 10 x & = & 3.07307307307 . . . \\ 100 x & = & 30.730730730 . . . \\ 1000 x & = & 307.307307307 . . . \end{array}$

Notice that $x$ and $1000 x$ have matching recurring decimal parts

Subtract one from the other

$\begin{array}{rcl} 1000 x - x & = & 307.307307307 . . . - 0.307307307 . . . \\ 999 x & = & 307 \end{array}$

Divide both sides by 999

$x = \frac{307}{999}$

This cannot be simplified, so this is the final answer

$\frac{307}{999}$

Last updated: 17 October 2024

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