\[\boxed{\mathbf{349}\mathbf{.}}\]
\[1)\ 2\frac{5}{8} \cdot \frac{2}{3} + \frac{7}{8} = \frac{21}{8} \cdot \frac{2}{3} + \frac{7}{8} =\]
\[= \frac{7 \cdot 2}{8} + \frac{7}{8} = \frac{14}{8} + \frac{7}{8} = \frac{21}{8} = 2\frac{5}{8}\]
\[2)\ \left( 6 - 7\frac{7}{9} \cdot \frac{3}{5} \right) \cdot \frac{7}{12} =\]
\[= 6 \cdot \frac{7}{12} - \frac{70}{9} \cdot \frac{3}{5} \cdot \frac{7}{12} =\]
\[= \frac{7}{2} - \frac{14 \cdot 7}{3 \cdot 12} = \frac{7}{2} - \frac{49}{18} =\]
\[= 3\frac{1^{\backslash 9}}{2} - 2\frac{13}{18} = 3\frac{9}{18} - 2\frac{13}{18} =\]
\[= 2\frac{27}{18} - 2\frac{13}{18} = \frac{14}{18} = \frac{7}{9}\]