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#2**+2 **

I happen to remember what this decimal is represented in a fraction form, but here is the method one can use to convert \(0.\overline{142857}\) into a fraction.

This first step is very simple; just set it equal to a variable. I'll use the normal x as my variable for this example. Therefore, \(0.\overline{142857}=x\)

The next goal is to get the repeating portion into the whole numbers part. It is probably easier showing by example than by explaining in words:

\(0.\overline{142857}=x\) | Multiply by 1000000 on both sides. |

\(142857.\overline{142857}=10000000x\) | As you can see, the repeating section is now in whole numbers. Now, subtract both equations from each other. |

\(142857=999999x\) | Now, divide by 999999 on both sides. |

\(x=\frac{142857}{999999}\div\frac{142857}{142857}\) | It is probably hard to realize here, but the GCF of the numerator and denominator is 142857. |

\(x=\frac{1}{7}\) | |

Look at that! \(0.\overline{142857}=\frac{1}{7}\). That's quite a nice fraction.

TheXSquaredFactor Sep 9, 2017