Decimals

We can use variables and equations to solve this problem. 

First, let's let \(x=2.\overline{57}\)

 

If we have that for the value of x, we have \(100x = 257.\overline{57}\)

 

This is extremely important for our solution. 

 

Now, we subtract x from 100x. We have

\(100x-x=257.\overline{57}-2.\overline{57}\)

 

Notice the repeating decimal cancels out. We get

\(99x=255\\ x=255/99\\ x=\frac{85}{33}\\ x=2\frac{19}{33}\\\)

So our final answer is \(x=2\frac{19}{33}\)

 

Thanks! :)