1.\frac{2\frac{38}{45}-\frac{1}{15}:13\frac{8}{9}+3\frac{3}{65}\cdot\frac{26}{99}}{18\frac{1}{2}-13\frac{7}{9}}\cdot\frac{1}{85}-0,5

1. Выполним действия:

1) $$2\frac{38}{45}=\frac{2\cdot45+38}{45}=\frac{90+38}{45}=\frac{128}{45}$$

2) $$13\frac{8}{9}=\frac{13\cdot9+8}{9}=\frac{117+8}{9}=\frac{125}{9}$$

3) $$3\frac{3}{65}=\frac{3\cdot65+3}{65}=\frac{195+3}{65}=\frac{198}{65}$$

4) $$18\frac{1}{2}=\frac{18\cdot2+1}{2}=\frac{36+1}{2}=\frac{37}{2}$$

5) $$13\frac{7}{9}=\frac{13\cdot9+7}{9}=\frac{117+7}{9}=\frac{124}{9}$$

6) $$\frac{1}{15}:13\frac{8}{9}=\frac{1}{15}:\frac{125}{9}=\frac{1}{15}\cdot\frac{9}{125}=\frac{1}{5\cdot3}\cdot\frac{3\cdot3}{125}=\frac{3}{5\cdot125}=\frac{3}{625}$$

7) $$2\frac{38}{45}-\frac{3}{625}=\frac{128}{45}-\frac{3}{625}=\frac{128\cdot125}{45\cdot125}-\frac{3\cdot9}{625\cdot9}=\frac{16000}{5625}-\frac{27}{5625}=\frac{16000-27}{5625}=\frac{15973}{5625}$$

8) $$3\frac{3}{65}\cdot\frac{26}{99}=\frac{198}{65}\cdot\frac{26}{99}=\frac{2\cdot99}{5\cdot13}\cdot\frac{2\cdot13}{99}=\frac{2\cdot2}{5}=\frac{4}{5}$$

9) $$\frac{15973}{5625}+\frac{4}{5}=\frac{15973}{5625}+\frac{4\cdot1125}{5\cdot1125}=\frac{15973}{5625}+\frac{4500}{5625}=\frac{15973+4500}{5625}=\frac{20473}{5625}$$

10) $$18\frac{1}{2}-13\frac{7}{9}=\frac{37}{2}-\frac{124}{9}=\frac{37\cdot9}{2\cdot9}-\frac{124\cdot2}{9\cdot2}=\frac{333}{18}-\frac{248}{18}=\frac{333-248}{18}=\frac{85}{18}$$

11) $$\frac{20473}{5625}:\frac{85}{18}=\frac{20473}{5625}\cdot\frac{18}{85}=\frac{20473}{625\cdot9}\cdot\frac{2\cdot9}{85}=\frac{20473}{625}\cdot\frac{2}{85}=\frac{40946}{53125}$$

12) $$\frac{40946}{53125}\cdot\frac{1}{85}=\frac{40946}{4515625}$$

13) $$\frac{40946}{4515625}-0,5=\frac{40946}{4515625}-\frac{1}{2}=\frac{40946\cdot2}{4515625\cdot2}-\frac{4515625}{9031250}=\frac{81892}{9031250}-\frac{4515625}{9031250}=\frac{81892-4515625}{9031250}=\frac{-4433733}{9031250}$$

Ответ: $$\frac{-4433733}{9031250}$$