# Question #1a4ca

Jun 9, 2017

$\setminus \frac{62}{333}$

#### Explanation:

First lets call our recurring decimal $x$: $0. \dot{18} \dot{6} = x = 0.186186186186186 \ldots \ldots .$.

Because there are three digits which repeat, we're going to multiply $x$ by 10 to the power of three:

$1000 x = 1000 \times 0. \dot{18} \dot{6} = 186. \dot{18} \dot{6}$

Now notice we can make our recurring decimal an integer by subtracting $x$ off $1000 x$:

$1000 x - x = 999 x = 186. \dot{18} \dot{6} - 0. \dot{18} \dot{6} = 186$

So now we don't have a recurring decimal, but an expression:

$999 x = 186$, so, dividing by $999$ we get $x = \setminus \frac{186}{999}$ which can be simplified by removing a common factor of 3:

$\setminus \frac{186}{999} = \setminus \frac{3 \times 62}{3 \times 333} = \setminus \frac{62}{333}$