Question

Question #1a4ca

Answer 1

`\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}`