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}#