What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to \(.3\overline{25}\) ?

tertre
Dec 23, 2017

#1**+1 **

For this one

1) Ignore the decimal

2) Take the "whole" and subtract the non-repeating part = 325 - 3 = 322

3) Put this over a number that has the same number of 9's as the repeating part = 99, followed by the same number of 0's as the non-repeating part = 0

4) So we have

322 / 990 = 161 / 495

Verify for yourself that this is equal to the required decimal

CPhill
Dec 23, 2017

#3**+1 **

I have not seen a interminably repeating decimal be converted into a simplified fraction in the fashion Cphill described above, but I will present to you an alternate method. On further review, though, the method I have below appears to prove Cphill's method.

\(x=0.3\overline{25}\) | Firstly, I set the repeating decimal equal to a variable. I will use the standard choice, x. |

\(x=0.325252525...\) | A few more decimal places should be written out so that the method is clear. |

Now, multiply x by a factor of ten such that the repeating portion lines up with the first line.

\(10x=3.25252525...\\ \hspace{5mm}x=0.325252525...\) | Notice how the repeating portion does not line up here, so this is not the correct multiple of ten. Let's multiply both sides by ten again. |

\(100x=32.525252525...\\ -(\hspace{1mm}x=\hspace{2mm}0.325252525...)\) | Look at this! Notice how the repeating portion of both equations line up perfectly. Now, subtract the two equations from each other. |

\(99x=32.2\) | Now, solve for x. |

\(x=\frac{32.2}{99}\) | Apply a multiplication of 10/10 to simplify the fraction. |

\(x=\frac{322}{990}=\frac{161}{495}\) | |

TheXSquaredFactor
Dec 23, 2017

#5**0 **

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rebeccahic
Dec 25, 2017