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# How do you convert 0.18 (18 repeating) to a fraction?

Mar 17, 2016

$0.18181818 \ldots = \frac{2}{11}$

#### Explanation:

Let's begin by setting $x$ equal to our repeating fraction

$x = \setminus \textcolor{red}{0.18181818} \ldots$

The next step is a bit of a trick where we multiply by a power of 10 such that what remains to the right of the decimal is the repeating fraction itself, i.e.

$100 x = 18.1818181818 \ldots = 18 + \setminus \textcolor{red}{0.18181818} \ldots$

which can be re-written

$100 x = 18 + x$

subtracting $x$ from both sides we get

$99 x = 18$

and finally

$x = \frac{18}{99} = \frac{2}{11}$