Решение:

1) $$1\frac{5}{16}\cdot(9-2\frac{1}{7}) = \frac{21}{16} \cdot (9 - \frac{15}{7}) = \frac{21}{16} \cdot (\frac{63}{7} - \frac{15}{7}) = \frac{21}{16} \cdot \frac{48}{7} = \frac{3}{1} \cdot \frac{3}{1} = 9$$

2) $$1\frac{1}{35}\cdot(7-5\frac{4}{9}) = \frac{36}{35} \cdot (7 - \frac{49}{9}) = \frac{36}{35} \cdot (\frac{63}{9} - \frac{49}{9}) = \frac{36}{35} \cdot \frac{14}{9} = \frac{4}{5} \cdot \frac{2}{1} = \frac{8}{5} = 1\frac{3}{5}$$

3) $$1\frac{5}{19}\cdot(6-3\frac{5}{8}) = \frac{24}{19} \cdot (6 - \frac{29}{8}) = \frac{24}{19} \cdot (\frac{48}{8} - \frac{29}{8}) = \frac{24}{19} \cdot \frac{19}{8} = \frac{3}{1} \cdot \frac{1}{1} = 3$$

4) $$1\frac{5}{17}\cdot(7-2\frac{4}{11}) = \frac{22}{17} \cdot (7 - \frac{26}{11}) = \frac{22}{17} \cdot (\frac{77}{11} - \frac{26}{11}) = \frac{22}{17} \cdot \frac{51}{11} = \frac{2}{1} \cdot \frac{3}{1} = 6$$

5) $$24\frac{2}{3}-(20,95-2\frac{1}{3})-3,4 = \frac{74}{3} - (20,95 - \frac{7}{3}) - 3,4 = \frac{74}{3} - (20,95 - 2,33) - 3,4 = 24,67 - 18,62 - 3,4 = 6,05 - 3,4 = 2,65$$

6) $$6\frac{2}{5}-(2,32-1\frac{1}{3})+1,02 = \frac{32}{5} - (2,32 - \frac{4}{3}) + 1,02 = 6,4 - (2,32 - 1,33) + 1,02 = 6,4 - 0,99 + 1,02 = 5,41 + 1,02 = 6,43$$

Ответ: 1) 9; 2) $$1\frac{3}{5}$$; 3) 3; 4) 6; 5) 2,65; 6) 6,43.