4. \( \left[\left(-\frac{5}{18}+\frac{3}{2}-\frac{1}{3}\right):\left(\frac{6}{5}-\frac{7}{3}-\frac{1}{5}\right)-\frac{3}{4}\right]:\left[\left(-\frac{1}{2}+\frac{5}{3}\right):\left(-\frac{3}{10}+\frac{1}{20}-\frac{5}{4}+\frac{1}{3}\right)\right]-\frac{17}{12} \)

To solve the given expression, let's break it down step by step. 1. **Calculate the inner expressions:** - For the first part, combine \(-\frac{5}{18}+\frac{3}{2}-\frac{1}{3}\): - Convert each fraction to a common denominator (which is 18): \[ -\frac{5}{18} + \frac{27}{18} - \frac{6}{18} = \frac{16}{18} = \frac{8}{9} \] - Now for the denominator, \(\frac{6}{5}-\frac{7}{3}-\frac{1}{5}\): - The common denominator is 15: \[ \frac{18}{15} - \frac{35}{15} - \frac{3}{15} = \frac{-20}{15} = -\frac{4}{3} \] - Now we have: \[ \left(\frac{8}{9} : -\frac{4}{3}\right) - \frac{3}{4} \] - The division \(\frac{8}{9} : -\frac{4}{3}\) translates to multiplying by the reciprocal: \[ \frac{8}{9} \times -\frac{3}{4} = -\frac{24}{36} = -\frac{2}{3} \] - So, the first part simplifies to: \[ -\frac{2}{3} - \frac{3}{4} \] - The common denominator here is 12: \[ -\frac{8}{12} - \frac{9}{12} = -\frac{17}{12} \] 2. **Now evaluate the second part:** - For \(-\frac{1}{2} + \frac{5}{3}\): - The common denominator is 6: \[ -\frac{3}{6} + \frac{10}{6} = \frac{7}{6} \] - Now for the denominator \(-\frac{3}{10} + \frac{1}{20} - \frac{5}{4} + \frac{1}{3}\): - The common denominator here is 60: \[ -\frac{18}{60} + \frac{3}{60} - \frac{75}{60} + \frac{20}{60} = -\frac{70}{60} = -\frac{7}{6} \] - Now we have: \[ \frac{7}{6} : -\frac{7}{6} \] - This yields: \[ -1 \] 3. **Putting it all together:** - Now we substitute back into the larger expression: \[ -\frac{17}{12} - (-1) = -\frac{17}{12} + 1 = -\frac{17}{12} + \frac{12}{12} = -\frac{5}{12} \] Thus, the final result of the expression is: \[ \boxed{-\frac{5}{12}} \]