Recurring Decimals and Fractions

## Recurring Decimals and Fractions

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

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

Set x 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 equation from the second equation:

100x = 32.32323232...

x = 0.3232323232.... , and subtract:

99x = 32, and make a fraction such that

x = frac(32)(99).

If the number repeats every digit, multiply by 10; every third digit, multiply by 103 etc

## Example 1

Convert 0.dot1 2 dot3 to a fraction.

 x = 0.123123123... Recurs every 3 digits   Multiply by 103 1000x =123.123123123... Subtract equation 999x =123 x =frac(123)(999) Simplify: x =frac(41)(333)

Answer: frac(41)(333)

## Example 2

Convert 2.dot4 dot5 to a fraction.

 x = 2.45454545... Recurs every 2 digits Multiply by 102 100x = 245.454545... Subtract 99x = 243. x = frac(243)(99). Simplify x = 2frac(5)(11).

Answer: 2frac(5)(11)