Number Theory

Let's use a handy trick to solve this problem. 

First off, let's set a variable. Let's set $X=2.\overline{57}$

For that value of X, then we have that

$100 X = 257.\overline{57}$

Now, we subtract x from 100x. Note that the repeated part of the decimal cancels out. Thus, we have

$\eqalign{100 X &= &\hfill257.57...\cr X &= &\hfill2.57...\cr \hline 99X &= &255\cr}$

We end up witht he equation

$99 X = 255$

Simplifying and isolating x, we get 

$X = \frac{255}{99}$

Thus,

The answer is $2.\overline{57} = 2 \frac{19}{33}$

Thanks! :)