Recurring Decimals and Fractions

Recurring Decimals and Fractions

GCSE(H),

A recurring decimal can be converted to a fraction using powers of 10.

For example: `0. dot3 dot2` could be written as 0.3232323232....

The number recurs every second digit.

Multiply both x and the number by 102 such that 100x = 32.32323232...

Subtract the original value from the multiplied value:

100x = 32.32323232...

x = 0.3232323232.... , then subtract:

99x = 32

`x = frac(32)(99)`.

If the number repeats every digit, multiply by 10; every third digit, multiply by 1000 (etc)

Examples

1. Convert `0.dot1 2 dot3` to a fraction.

Answer: `frac(41)(333)`

x = 0.123123123...

1000x = 123.123123123...

999x = 123, giving a fraction of `frac(123)(999)` which can be simplified.

2. Convert `2.dot4 dot5` to a fraction.

Answer: 2`frac(5)(11)`

The number recurs every 2 digits, so multiply by 102

x = 2.45454545...

100x = 245.454545...

99x = 243

x = `frac(243)(99)`

Simplify to 2`frac(45)(99)` then to 2`frac(5)(11)`