jmc

To express the number $0.0\overline{57}$ as a fraction, we call it $x$ and subtract it from $100x$: \begin{array}{r r c r@{}l} &100x &=& 5&.7575757\ldots \\ - &x &=& 0&.0575757\ldots \\ \hline &99x &=& 5&.7 \end{array} This shows that $0.0\overline{57}=\frac{5.7}{99}=\frac{57}{990}=\boxed{\frac{19}{330}}$.