How do you convert 0.27 (27 repeating) to a fraction?

Question

lovagwb · Accepted Answer

Solution:The fraction&amp;nbsp;0.27―&amp;nbsp;can be written as an infinite sum:&amp;nbsp;0.27―=0.27+0.0027+0.000027+…&amp;nbsp;The right side is the total of a geometric progression where&amp;nbsp;a1=0.27&amp;nbsp;and&amp;nbsp;r=0.01&amp;nbsp;In the sequence the ratio satisfies condition&amp;nbsp;|r|&amp;lt;1, as a result, it is convergent, and the sum is as follows:S=a11−r=0.271−0.01=0.270.99=2799=311&amp;nbsp;Thus, we can state:0.27=311

Archie Jones · Accepted Answer

Explanation:
The standard notation for a repeating decimal is to put a bar over the repeating digits, that is,

0.27272727 \dots = \overset{―}{0.27}

Let

x = \overset{―}{0.27}

\Rightarrow 100 x = 27 . \overset{―}{27}

\Rightarrow 100 x - x = 27 . \overset{―}{27} - 0 . \overset{―}{27}

\Rightarrow 99 x = 27

\Rightarrow x = \frac{27}{99} = \frac{3}{11}

This trick works in general for repeating decimals. The idea is to multiply by a power of

10

which will result in the same repeating decimal after multiplication, that is,

10^{n}

where there are

n

digits in the sequence which repeats.

How do you convert 0.27 (27 repeating) to a fraction?

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