5.491 Найдите значение выражения: a) $$\left(\frac{61}{64}-\frac{7}{12}-\frac{5}{14}\right) \cdot \left(\frac{13}{16}+\frac{1}{2}\right)$$; б) $$\left(1-\frac{11}{17}\right) \cdot \left(\frac{3}{4}-\frac{5}{12}+\frac{11}{18}\right);$$

Решение:

a) $$\left(\frac{61}{64}-\frac{7}{12}-\frac{5}{14}\right) \cdot \left(\frac{13}{16}+\frac{1}{2}\right)$$

1. $$\frac{13}{16} + \frac{1}{2} = \frac{13}{16} + \frac{1 \cdot 8}{2 \cdot 8} = \frac{13}{16} + \frac{8}{16} = \frac{13+8}{16} = \frac{21}{16}$$
2. $$\frac{61}{64} - \frac{7}{12} - \frac{5}{14} = \frac{61 \cdot 21}{64 \cdot 21} - \frac{7 \cdot 112}{12 \cdot 112} - \frac{5 \cdot 48}{14 \cdot 48} = \frac{1281}{1344} - \frac{784}{1344} - \frac{240}{1344} = \frac{1281 - 784 - 240}{1344} = \frac{257}{1344}$$
3. $$\frac{257}{1344} \cdot \frac{21}{16} = \frac{257 \cdot 21}{1344 \cdot 16} = \frac{5397}{21504} = \frac{257 \cdot 7 \cdot 3}{64 \cdot 21 \cdot 16} = \frac{257}{1024}$$

Ответ: $$\frac{257}{1024}$$

б) $$\left(1-\frac{11}{17}\right) \cdot \left(\frac{3}{4}-\frac{5}{12}+\frac{11}{18}\right)$$

1. $$1 - \frac{11}{17} = \frac{17}{17} - \frac{11}{17} = \frac{17-11}{17} = \frac{6}{17}$$
2. $$\frac{3}{4} - \frac{5}{12} + \frac{11}{18} = \frac{3 \cdot 9}{4 \cdot 9} - \frac{5 \cdot 3}{12 \cdot 3} + \frac{11 \cdot 2}{18 \cdot 2} = \frac{27}{36} - \frac{15}{36} + \frac{22}{36} = \frac{27-15+22}{36} = \frac{34}{36} = \frac{17}{18}$$
3. $$\frac{6}{17} \cdot \frac{17}{18} = \frac{6 \cdot 17}{17 \cdot 18} = \frac{6}{18} = \frac{1}{3}$$

Ответ: $$\frac{1}{3}$$