jmc

To express the number $3.\overline{7}$ as a fraction, we let $x=3.\overline{7}$, so $10x=37.\overline{7}$ and: \begin{array}{r r c r@{}l} &10x &=& 37&.77777\ldots \\ - &x &=& 3&.77777\ldots \\ \hline &9x &=& 34& \end{array} This shows that $3.\overline{7}=\boxed{\frac{34}{9}}$.