jmc

We want to find $\frac{1}{2}+\frac{2}{7}-\frac{5}{8}$. The common denominator is 56, so we try to rewrite each of the three fractions with 56 on the bottom.

$\frac{1}{2}=\frac{1}{2}\cdot \frac{28}{28}=\frac{28}{56}$. Also, $\frac{2}{7}=\frac{2}{7}\cdot \frac{8}{8}=\frac{16}{56}$. Finally, $\frac{5}{8}=\frac{5}{8}\cdot \frac{7}{7}=\frac{35}{56}$.

Thus, we have $\frac{1}{2}+\frac{2}{7}-\frac{5}{8}=\frac{28}{56}+\frac{16}{56}-\frac{35}{56}=\frac{28+16-35}{56}=\frac{44-35}{56}=\frac{9}{56}$. Because 9 and 56 have no common divisors, this is the simplest form, and our answer is $\boxed{\frac{9}{56}}$.