1.006. $$\frac{\left(13.75 + 9\frac{1}{6}\right) \cdot 1.2}{\left(10.3 - 8\frac{1}{2}\right) \cdot \frac{5}{9}} + \frac{\left(6.8 - 3\frac{3}{5}\right) \cdot \frac{5}{6}}{\left(3\frac{2}{3} - 3\frac{1}{6}\right) \cdot 56} - 27\frac{1}{6}$$

Решим пример по действиям:

1) Сначала упростим первое слагаемое:

$$13.75 + 9\frac{1}{6} = 13\frac{3}{4} + 9\frac{1}{6} = 13\frac{9}{12} + 9\frac{2}{12} = 22\frac{11}{12} = \frac{22 \cdot 12 + 11}{12} = \frac{264 + 11}{12} = \frac{275}{12}$$

$$2) 22\frac{11}{12} \cdot 1.2 = \frac{275}{12} \cdot \frac{12}{10} = \frac{275}{10} = 27.5$$

$$3) 10.3 - 8\frac{1}{2} = 10.3 - 8.5 = 1.8 = \frac{18}{10} = \frac{9}{5}$$

$$4) 1.8 \cdot \frac{5}{9} = \frac{9}{5} \cdot \frac{5}{9} = 1$$

$$5) \frac{\left(13.75 + 9\frac{1}{6}\right) \cdot 1.2}{\left(10.3 - 8\frac{1}{2}\right) \cdot \frac{5}{9}} = \frac{27.5}{1} = 27.5$$

Теперь упростим второе слагаемое:

$$6) 6.8 - 3\frac{3}{5} = 6.8 - 3.6 = 3.2 = \frac{32}{10} = \frac{16}{5}$$

$$7) 3.2 \cdot \frac{5}{6} = \frac{16}{5} \cdot \frac{5}{6} = \frac{16}{6} = \frac{8}{3}$$

$$8) 3\frac{2}{3} - 3\frac{1}{6} = 3\frac{4}{6} - 3\frac{1}{6} = \frac{3}{6} = \frac{1}{2}$$

$$9) \frac{1}{2} \cdot 56 = 28$$

$$10) \frac{\left(6.8 - 3\frac{3}{5}\right) \cdot \frac{5}{6}}{\left(3\frac{2}{3} - 3\frac{1}{6}\right) \cdot 56} = \frac{\frac{8}{3}}{28} = \frac{8}{3 \cdot 28} = \frac{8}{84} = \frac{2}{21}$$

Теперь вычитаем:

$$11) 27.5 + \frac{2}{21} - 27\frac{1}{6} = 27\frac{1}{2} + \frac{2}{21} - 27\frac{1}{6} = 27\frac{3}{6} - 27\frac{1}{6} + \frac{2}{21} = \frac{2}{6} + \frac{2}{21} = \frac{1}{3} + \frac{2}{21} = \frac{7}{21} + \frac{2}{21} = \frac{9}{21} = \frac{3}{7}$$

Ответ: $$\frac{3}{7}$$