Decimals

In order to do this, we can set some variables.

Let's set $x=0.\overline{2123}$

If x is equal to that, then notice that $10000 x = 2123.\overline{2123}$

This is very important to the problem. 

Subtracting x from 10000x, we get

$\eqalign{10000 X &= &\hfill2123.2123...\cr X &= &\hfill0.2123...\cr \hline 9999X &= &2123\cr}$

Notice that the repeating decimal cancels out. 

Now, we solve for x. We get

$X = \frac{2123}{9999}\\X = \frac{193}{909}$

So our final answer is $\frac{193}{909}$

Thanks! :)