Number Theory

We can use a clever trick to make \( 2.\overline{57}\) a fraction. 

 

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

If we multiply both sides by 100, we get \( 100x=257.\overline{57}\)

 

Subtracting x from 100x, we get \(100x-x=257.\overline{57}- 2.\overline{57}\)

 

\(99x= 257.\overline{57}-2.\overline{57}\\ 99x=257\\ x=257/99\\ x=2 \frac{19}{33}\)

 

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

 

Thanks! :)