311. 1) \frac{5}{6} \cdot 8; \frac{23}{20} \cdot 10; \frac{23}{40} \cdot 8; \frac{11}{12} \cdot 6; \frac{8}{15} \cdot 3; \frac{9}{35} \cdot 7; \frac{8}{9} : 4; \frac{3}{4} : 6; \frac{2}{7} : 2; \frac{8}{15} : 4; \frac{2}{5} : 3; \frac{6}{7} : 6.

1) Выполним умножение дробей.

1. $$\frac{5}{6} \cdot 8 = \frac{5 \cdot 8}{6} = \frac{40}{6} = \frac{20}{3} = 6\frac{2}{3}$$
2. $$\frac{23}{20} \cdot 10 = \frac{23 \cdot 10}{20} = \frac{230}{20} = \frac{23}{2} = 11\frac{1}{2}$$
3. $$\frac{23}{40} \cdot 8 = \frac{23 \cdot 8}{40} = \frac{184}{40} = \frac{23}{5} = 4\frac{3}{5}$$
4. $$\frac{11}{12} \cdot 6 = \frac{11 \cdot 6}{12} = \frac{66}{12} = \frac{11}{2} = 5\frac{1}{2}$$
5. $$\frac{8}{15} \cdot 3 = \frac{8 \cdot 3}{15} = \frac{24}{15} = \frac{8}{5} = 1\frac{3}{5}$$
6. $$\frac{9}{35} \cdot 7 = \frac{9 \cdot 7}{35} = \frac{63}{35} = \frac{9}{5} = 1\frac{4}{5}$$

Выполним деление дробей.

1. $$\frac{8}{9} : 4 = \frac{8}{9} : \frac{4}{1} = \frac{8}{9} \cdot \frac{1}{4} = \frac{8 \cdot 1}{9 \cdot 4} = \frac{8}{36} = \frac{2}{9}$$
2. $$\frac{3}{4} : 6 = \frac{3}{4} : \frac{6}{1} = \frac{3}{4} \cdot \frac{1}{6} = \frac{3 \cdot 1}{4 \cdot 6} = \frac{3}{24} = \frac{1}{8}$$
3. $$\frac{2}{7} : 2 = \frac{2}{7} : \frac{2}{1} = \frac{2}{7} \cdot \frac{1}{2} = \frac{2 \cdot 1}{7 \cdot 2} = \frac{2}{14} = \frac{1}{7}$$
4. $$\frac{8}{15} : 4 = \frac{8}{15} : \frac{4}{1} = \frac{8}{15} \cdot \frac{1}{4} = \frac{8 \cdot 1}{15 \cdot 4} = \frac{8}{60} = \frac{2}{15}$$
5. $$\frac{2}{5} : 3 = \frac{2}{5} : \frac{3}{1} = \frac{2}{5} \cdot \frac{1}{3} = \frac{2 \cdot 1}{5 \cdot 3} = \frac{2}{15}$$
6. $$\frac{6}{7} : 6 = \frac{6}{7} : \frac{6}{1} = \frac{6}{7} \cdot \frac{1}{6} = \frac{6 \cdot 1}{7 \cdot 6} = \frac{6}{42} = \frac{1}{7}$$

Ответ: 1) $$\frac{5}{6} \cdot 8 = 6\frac{2}{3}$$; $$\frac{23}{20} \cdot 10 = 11\frac{1}{2}$$; $$\frac{23}{40} \cdot 8 = 4\frac{3}{5}$$; $$\frac{11}{12} \cdot 6 = 5\frac{1}{2}$$; $$\frac{8}{15} \cdot 3 = 1\frac{3}{5}$$; $$\frac{9}{35} \cdot 7 = 1\frac{4}{5}$$;

$$\frac{8}{9} : 4 = \frac{2}{9}$$; $$\frac{3}{4} : 6 = \frac{1}{8}$$; $$\frac{2}{7} : 2 = \frac{1}{7}$$; $$\frac{8}{15} : 4 = \frac{2}{15}$$; $$\frac{2}{5} : 3 = \frac{2}{15}$$; $$\frac{6}{7} : 6 = \frac{1}{7}$$.