Let's begin by setting x equal to our repeating fraction
x=0.18181818...
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.
100x=18.1818181818...=18+0.18181818...
which can be re-written
100x=18+x
taking x to the left-hand side we get
99x=18
and finally
x=18/99=2/11
Hope this helps!
