10). $$4 \frac{1}{6} \cdot (1 \frac{1}{2} - \frac{3}{5}) + (\frac{3}{4} + \frac{5}{6}) \cdot 6$$.

Решим пример по действиям:

1. $$1 \frac{1}{2} - \frac{3}{5} = \frac{3}{2} - \frac{3}{5} = \frac{15}{10} - \frac{6}{10} = \frac{9}{10}$$
2. $$\frac{3}{4} + \frac{5}{6} = \frac{9}{12} + \frac{10}{12} = \frac{19}{12}$$
3. $$4 \frac{1}{6} \cdot \frac{9}{10} = \frac{25}{6} \cdot \frac{9}{10} = \frac{25 \cdot 9}{6 \cdot 10} = \frac{5 \cdot 3}{2 \cdot 2} = \frac{15}{4} = 3 \frac{3}{4}$$
4. $$\frac{19}{12} \cdot 6 = \frac{19 \cdot 6}{12} = \frac{19}{2} = 9 \frac{1}{2}$$
5. $$3 \frac{3}{4} + 9 \frac{1}{2} = 3 \frac{3}{4} + 9 \frac{2}{4} = 12 \frac{5}{4} = 13 \frac{1}{4}$$

Ответ: $$13 \frac{1}{4}$$