Let's simplify each expression step by step:

1. $\frac{2}{3}(1 - 5)$
   $1 - 5 = -4$ $\frac{2}{3} \times (-4) = \frac{-8}{3}$ Answer: $\frac{-8}{3}$

2. $\frac{3}{4}(6 - 8)$
   $6 - 8 = -2$ $\frac{3}{4} \times (-2) = \frac{-6}{4} = \frac{-3}{2}$ Answer: $\frac{-3}{2}$

3. $\frac{1}{2}\left(\frac{1}{3} - 2\right)$
   $\frac{1}{3} - 2 = \frac{1}{3} - \frac{6}{3} = \frac{-5}{3}$ $\frac{1}{2} \times \frac{-5}{3} = \frac{-5}{6}$ Answer: $\frac{-5}{6}$

4. $\frac{1}{5}(3 + \frac{1}{2})$
   $3 + \frac{1}{2} = \frac{6}{2} + \frac{1}{2} = \frac{7}{2}$ $\frac{1}{5} \times \frac{7}{2} = \frac{7}{10}$ Answer: $\frac{7}{10}$

5. $\frac{2}{5}(1 + \frac{3}{4})$
   $1 + \frac{3}{4} = \frac{4}{4} + \frac{3}{4} = \frac{7}{4}$ $\frac{2}{5} \times \frac{7}{4} = \frac{14}{20} = \frac{7}{10}$ Answer: $\frac{7}{10}$

6. $\frac{3}{4}\left[2 - (1 + 3) + 5\right]$
   $1 + 3 = 4$ $2 - 4 + 5 = 3$ $\frac{3}{4} \times 3 = \frac{9}{4}$ Answer: $\frac{9}{4}$

7. $\frac{1}{2}\left[4 + 2(1 + 5) - 6\right]$
   $1 + 5 = 6$ $2 \times 6 = 12$ $4 + 12 - 6 = 10$ $\frac{1}{2} \times 10 = 5$ Answer: 5

8. $\frac{2}{3}\left[2 - \frac{1}{3}(1 - 5) - 2\right]$
   $1 - 5 = -4$ $\frac{1}{3} \times (-4) = \frac{-4}{3}$ $2 - \frac{-4}{3} - 2 = 0 + \frac{4}{3} = \frac{4}{3}$ $\frac{2}{3} \times \frac{4}{3} = \frac{8}{9}$ Answer: $\frac{8}{9}$

9. $3\left\{\frac{1}{2} - 4\left[2 - \frac{1}{3}(1 - 5) - 5\right]\right\}$
   First, handle the inner expression: $1 - 5 = -4$ $\frac{1}{3} \times (-4) = \frac{-4}{3}$ $2 - \frac{-4}{3} - 5 = 2 + \frac{4}{3} - 5 = \frac{6}{3} + \frac{4}{3} - \frac{15}{3} = \frac{-5}{3}$ Now, $4 \times \frac{-5}{3} = \frac{-20}{3}$ $\frac{1}{2} - \frac{-20}{3} = \frac{1}{2} + \frac{20}{3} = \frac{3}{6} + \frac{40}{6} = \frac{43}{6}$ Finally, $3 \times \frac{43}{6} = \frac{129}{6} = \frac{43}{2}$ Answer: $\frac{43}{2}$

10. $\frac{3}{4}\left\{\frac{2}{5} - 3\left[4 + 5(1 - 2) - \frac{1}{3}\right] + 6\right\}$
    This expression has several steps, so I'll break it down in parts. Let's start with the innermost expression: $1 - 2 = -1$ $5 \times (-1) = -5$ $4 + (-5) = -1$ $-1 - \frac{1}{3} = -1 - \frac{1}{3} = \frac{-3}{3} - \frac{1}{3} = \frac{-4}{3}$ Now, moving outward: $3 \times \frac{-4}{3} = -4$ $\frac{2}{5} - (-4) + 6 = \frac{2}{5} + 4 + 6 = \frac{2}{5} + 10$ Converting $10$ to a fraction with denominator 5: $\frac{2}{5} + \frac{50}{5} = \frac{52}{5}$ Finally, $\frac{3}{4} \times \frac{52}{5} = \frac{156}{20} = \frac{39}{5}$ Answer: $\frac{39}{5}$

11. $\frac{5}{6}\left\{\frac{1}{2} - \frac{1}{4}\left[3 - \frac{1}{2}(1 + 3)\right] - \frac{1}{5}\right\}$
    First, simplify the inner terms: $1 + 3 = 4$ $\frac{1}{2} \times 4 = 2$ $3 - 2 = 1$ Now, $\frac{1}{4} \times 1 = \frac{1}{4}$ [ \frac{1}{2} - \frac{1}{4} - \frac{1}{5} = \frac{5}{10} - \frac{2.5}{10} = <ratio