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! :)