Решим пример по действиям:
1) $$5\frac{9}{14} = \frac{5 \cdot 14 + 9}{14} = \frac{70 + 9}{14} = \frac{79}{14}$$
2) $$7\frac{1}{8} = \frac{7 \cdot 8 + 1}{8} = \frac{56 + 1}{8} = \frac{57}{8}$$
3) $$\frac{4}{7} \cdot 7\frac{1}{8} = \frac{4}{7} \cdot \frac{57}{8} = \frac{4 \cdot 57}{7 \cdot 8} = \frac{228}{56} = \frac{57}{14}$$
4) $$1\frac{1}{4} = \frac{1 \cdot 4 + 1}{4} = \frac{4 + 1}{4} = \frac{5}{4}$$
5) $$1\frac{1}{4} : \frac{5}{8} = \frac{5}{4} : \frac{5}{8} = \frac{5}{4} \cdot \frac{8}{5} = \frac{5 \cdot 8}{4 \cdot 5} = \frac{40}{20} = 2$$
6) $$5\frac{9}{14} - \frac{4}{7} \cdot 7\frac{1}{8} - 1\frac{1}{4} : \frac{5}{8} = \frac{79}{14} - \frac{57}{14} - 2 = \frac{79 - 57}{14} - 2 = \frac{22}{14} - 2 = \frac{11}{7} - 2 = \frac{11}{7} - \frac{14}{7} = \frac{11 - 14}{7} = \frac{-3}{7} = -\frac{3}{7}$$
Ответ: $$-\frac{3}{7}$$.