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# What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to ?

+1
256
5
+2780

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

tertre  Dec 23, 2017
#1
+87714
+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
#2
+2780
+1

thanks so much!

tertre  Dec 23, 2017
#3
+2143
+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
#4
+87714
0

Thanks, X2.....!!!

CPhill  Dec 23, 2017
#5
+2
0

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rebeccahic  Dec 25, 2017