+4622 What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to \(.3\overline{25}\) ?
+129899 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
+2446 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. |
| \(x=\frac{322}{990}=\frac{161}{495}\) | |
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