\[\boxed{\mathbf{1073}\mathbf{.}\mathbf{\ }}\]
\[1)\ 4\frac{4}{9} \cdot \frac{5}{8} + \frac{5}{8} \cdot 3\frac{5}{9} =\]
\[= \frac{5}{8} \cdot \left( 4\frac{4}{9} + 3\frac{5}{9} \right) = \frac{5}{8} \cdot 8 = 5\]
\[2)\ 3\frac{7}{8} \cdot \frac{4}{5} - 1\frac{5}{8} \cdot \frac{4}{5} =\]
\[= \frac{4}{5} \cdot \left( 3\frac{7}{8} - 1\frac{5}{8} \right) = \frac{4}{5} \cdot 2\frac{2}{8} =\]
\[= \frac{4}{5} \cdot 2\frac{1}{4} = \frac{4}{5} \cdot \frac{9}{4} =\]
\[= 1\frac{1}{19} \cdot \left( 2\frac{11^{\backslash 2}}{15} - \frac{3^{\backslash 3}}{10} - 1\frac{1^{\backslash 5}}{6} \right) =\]
\[= \frac{20}{19} \cdot \left( 2\frac{22}{30} - \frac{9}{30} - 1\frac{5}{30} \right) =\]
\[= \frac{20}{19} \cdot 1\frac{8}{30} = \frac{20}{19} \cdot 1\frac{4}{15} =\]
\[= \frac{20}{19} \cdot \frac{19}{15} = \frac{20}{15} = \frac{4}{3} = 1\frac{1}{3}\]