2.344 Выполните действия: a) \frac{9}{52} \cdot 4\frac{1}{3} \cdot \frac{1}{3}+(3\frac{2}{3}+2\frac{4}{5}) \cdot \frac{60}{97}+\frac{5}{36} \cdot 1\frac{4}{5}; б) (\frac{5}{9}+\frac{1}{5}) \cdot (28\frac{6}{7}-19\frac{5}{14}) \cdot \frac{9}{17} - \frac{1}{5}

a) $$\frac{9}{52} \cdot 4\frac{1}{3} \cdot \frac{1}{3}+(3\frac{2}{3}+2\frac{4}{5}) \cdot \frac{60}{97}+\frac{5}{36} \cdot 1\frac{4}{5}$$

1. Преобразуем смешанные дроби в неправильные:$$4\frac{1}{3} = \frac{4 \cdot 3 + 1}{3} = \frac{13}{3}; \quad 3\frac{2}{3} = \frac{3 \cdot 3 + 2}{3} = \frac{11}{3}; \quad 2\frac{4}{5} = \frac{2 \cdot 5 + 4}{5} = \frac{14}{5}; \quad 1\frac{4}{5} = \frac{1 \cdot 5 + 4}{5} = \frac{9}{5}$$
2. $$\frac{9}{52} \cdot \frac{13}{3} \cdot \frac{1}{3} = \frac{9 \cdot 13 \cdot 1}{52 \cdot 3 \cdot 3} = \frac{1}{4}$$
3. $$(\frac{11}{3} + \frac{14}{5}) = \frac{11 \cdot 5 + 14 \cdot 3}{15} = \frac{55 + 42}{15} = \frac{97}{15}$$
4. $$\frac{97}{15} \cdot \frac{60}{97} = \frac{97 \cdot 60}{15 \cdot 97} = 4$$
5. $$\frac{5}{36} \cdot \frac{9}{5} = \frac{5 \cdot 9}{36 \cdot 5} = \frac{1}{4}$$
6. $$\frac{1}{4} + 4 + \frac{1}{4} = 4 + \frac{2}{4} = 4 + \frac{1}{2} = 4\frac{1}{2}$$

Ответ: $$4\frac{1}{2}$$

б) $$(\frac{5}{9}+\frac{1}{5}) \cdot (28\frac{6}{7}-19\frac{5}{14}) \cdot \frac{9}{17} - \frac{1}{5}$$

1. $$\frac{5}{9} + \frac{1}{5} = \frac{5 \cdot 5 + 1 \cdot 9}{45} = \frac{25 + 9}{45} = \frac{34}{45}$$
2. Преобразуем смешанные дроби в неправильные:$$28\frac{6}{7} = \frac{28 \cdot 7 + 6}{7} = \frac{202}{7}; \quad 19\frac{5}{14} = \frac{19 \cdot 14 + 5}{14} = \frac{271}{14}$$
3. $$\frac{202}{7} - \frac{271}{14} = \frac{202 \cdot 2 - 271}{14} = \frac{404 - 271}{14} = \frac{133}{14}$$
4. $$\frac{34}{45} \cdot \frac{133}{14} \cdot \frac{9}{17} = \frac{34 \cdot 133 \cdot 9}{45 \cdot 14 \cdot 17} = \frac{2 \cdot 19 \cdot 17 \cdot 7 \cdot 9}{5 \cdot 9 \cdot 2 \cdot 7 \cdot 17} = \frac{19}{5}$$
5. $$\frac{19}{5} - \frac{1}{5} = \frac{18}{5} = 3\frac{3}{5}$$

Ответ: $$3\frac{3}{5}$$