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 10^{2} such that 100*x* = 32.32323232...

Subtract the original equation from the second equation:

100*x* = 32.32323232...

*x* = 0.3232323232.... , and subtract:

99*x* = 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 10^{3} etc

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

`x` | `= 0.123123123...` | |

Recurs every 3 digits | `` | `` |

Multiply by 10^{3} |
`1000x` | `=123.123123123...` |

Subtract equation | `999x` | `=123` |

`x` | `=frac(123)(999)` | |

Simplify: | `x` | `=frac(41)(333)` |

Answer: `frac(41)(333)`

Convert `2.dot4 dot5` to a fraction.

`x` | `= 2.45454545...` | |

Recurs every 2 digits | ||

Multiply by 10^{2} |
`100x` | `= 245.454545...` |

Subtract | `99x` | `= 243`. |

`x` | `= frac(243)(99)`. | |

Simplify | `x` | `= 2frac(5)(11)`. |

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

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