Question #1a4ca

1 Answer
Jun 9, 2017

\frac{62}{333}

Explanation:

First lets call our recurring decimal x: 0.dot18dot6 = x = 0.186186186186186........

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

1000x = 1000xx0.dot18dot6 = 186.dot18dot6

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

1000x - x = 999x = 186.dot18dot6 - 0.dot18dot6 = 186

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

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

\frac{186}{999} = \frac{3xx62}{3xx333} = \frac{62}{333}