5. Найдите \frac{6}{11} от значения выражения $$(17\frac{5}{6} - 8\frac{2}{9} \cdot 1\frac{14}{37}) \cdot 1\frac{3}{5} : \frac{5}{39}$$.

Решение:

1. $$1\frac{14}{37} = \frac{1 \cdot 37 + 14}{37} = \frac{51}{37}$$
2. $$8\frac{2}{9} = \frac{8 \cdot 9 + 2}{9} = \frac{74}{9}$$
3. $$8\frac{2}{9} \cdot 1\frac{14}{37} = \frac{74}{9} \cdot \frac{51}{37} = \frac{2 \cdot 37}{9} \cdot \frac{3 \cdot 17}{37} = \frac{2 \cdot 3 \cdot 17}{9} = \frac{102}{9} = \frac{34}{3}$$
4. $$17\frac{5}{6} = \frac{17 \cdot 6 + 5}{6} = \frac{107}{6}$$
5. $$17\frac{5}{6} - 8\frac{2}{9} \cdot 1\frac{14}{37} = \frac{107}{6} - \frac{34}{3} = \frac{107}{6} - \frac{34 \cdot 2}{3 \cdot 2} = \frac{107}{6} - \frac{68}{6} = \frac{39}{6} = \frac{13}{2}$$
6. $$1\frac{3}{5} = \frac{1 \cdot 5 + 3}{5} = \frac{8}{5}$$
7. $$\frac{13}{2} \cdot \frac{8}{5} = \frac{13 \cdot 8}{2 \cdot 5} = \frac{13 \cdot 4}{5} = \frac{52}{5}$$
8. $$\frac{52}{5} : \frac{5}{39} = \frac{52}{5} \cdot \frac{39}{5} = \frac{52 \cdot 39}{5 \cdot 5} = \frac{2028}{25}$$
9. $$\frac{6}{11} \cdot \frac{2028}{25} = \frac{6 \cdot 2028}{11 \cdot 25} = \frac{12168}{275}$$

Ответ: $$\frac{12168}{275}$$.