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# converting a repeating decimal to a fraction

1. Set the repeating decimal equal to $x$.

2. 3. If applicable, rewrite the second equation so that its repeating part lines up with the repeating part in the original equation.

4. Subtract the original equation from the most recently obtained equation. (The repeating part should cancel at this step.)

5. 6. Divide both sides of the equation by the coefficient of $x$.

7. Reduce the fraction to lowest terms.

Below, this algorithm is demonstrated for $0.58\overline{3}$ with the steps indicated on the far right.

$x=0.58\overline{3}$ | (1) |

$10x=5.8\overline{3}$ | (2) |

$10x=5.83\overline{3}$ | (3) |

$9x=5.25$ | (4) |

$900x=525$ | (5) |

$x=\frac{525}{900}$ | (6) |

$x=\frac{7}{12}$ | (7) |

An important application of this algorithm is that it supplies a proof for the fact that $0.\overline{9}=1$:

$\displaystyle x$ | $\displaystyle=0.\overline{9}$ | ||

$\displaystyle 10x$ | $\displaystyle=9.\overline{9}$ | ||

$\displaystyle 9x$ | $\displaystyle=9$ | ||

$\displaystyle x$ | $\displaystyle=1$ |

Major Section:

Reference

Type of Math Object:

Algorithm

Parent:

## Mathematics Subject Classification

11A99*no label found*11-00

*no label found*

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