How to convert the recurring decimal #0.bar(32)# to a fraction?

Question

Christine Sosa · Accepted Answer

So we have:      x    =    0    .                  32            ¯      Two digits are repeated:      100    x    =    100    ×    0    .                  32            ¯            100    x    =    32    .                  32            ¯            ⇒    x    =    0    .                  32            ¯       and       100    x    =    32    .                  32            ¯        :        100    x    -    x    =    32    .                  32            ¯        -    0    .                  32            ¯            99    x    =    32        x    =          32      99

Cierra Mclaughlin · Accepted Answer

A clever shortcut technique exists to convert recurrent decimals into fractions:
If each digit appears again
Write a fraction as :

\frac{the recurring digit(s)}{9 for each recurring digit}

Then, if it's possible, simplify for the simplest form.

0.55555 ... . . = 0 . \bar{5} = \frac{5}{9}

0.272727 ... = 0 . \bar{27} = \frac{27}{99} = \frac{3}{11}

0 . \bar{32} = \frac{32}{99}

3 . \bar{732} = 3 \frac{732}{999} = 3 \frac{244}{333}

If only a few digits are repeated
Write a fraction as:

\frac{all the digits - non-recurring digits}{9 for each recurring and 0 for each non-recurring digit}

0.654444 ... = 0.65 \bar{4} = \frac{654 - 65}{900} = \frac{589}{900}

0.85 \bar{271} = \frac{85271 - 85}{99900} = \frac{85186}{99900} = \frac{42593}{49950}

4.167 \bar{4} = 4 \frac{1673 - 167}{9000} = 4 \frac{1506}{9000} = 4 \frac{251}{1500}

How to convert the recurring decimal 0.32 to a fraction?

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