e) \( \left(\frac{2}{3}-2\right) \times\left(\frac{1}{2}+5\right)-\left(4+\frac{1}{3}\right) \div\left(2-\frac{1}{3}\right)= \)

To solve the expression \( \left(\frac{2}{3}-2\right) \times\left(\frac{1}{2}+5\right)-\left(4+\frac{1}{3}\right) \div\left(2-\frac{1}{3}\right) \), we can break it down step-by-step. 1. Simplifying inside the parentheses: - \( \frac{2}{3} - 2 = \frac{2}{3} - \frac{6}{3} = \frac{-4}{3} \) - \( \frac{1}{2} + 5 = \frac{1}{2} + \frac{10}{2} = \frac{11}{2} \) - \( 4 + \frac{1}{3} = \frac{12}{3} + \frac{1}{3} = \frac{13}{3} \) - \( 2 - \frac{1}{3} = \frac{6}{3} - \frac{1}{3} = \frac{5}{3} \) 2. Now substitute back into the expression: - \( \left(-\frac{4}{3}\right) \times \left(\frac{11}{2}\right) - \left(\frac{13}{3}\right) \div \left(\frac{5}{3}\right) \) 3. Perform the multiplications and division: - \( -\frac{4}{3} \times \frac{11}{2} = -\frac{44}{6} = -\frac{22}{3} \) - \( \frac{13}{3} \div \frac{5}{3} = \frac{13}{3} \times \frac{3}{5} = \frac{13}{5} \) 4. Complete the expression: - Convert \( \frac{13}{5} \) to have a common denominator with \( -\frac{22}{3} \): - \( -\frac{22}{3} = -\frac{110}{15} \) - \( \frac{13}{5} = \frac{39}{15} \) 5. Now, combine the two fractions: - \( -\frac{110}{15} - \frac{39}{15} = -\frac{149}{15} \) Thus, the final answer is: \[ -\frac{149}{15} \]