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: 2frac(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 2frac(45)(99) then to 2frac(5)(11)