\( (((\frac{1}{2}.1)\div \frac{3}{2})-((5.\frac{2}{3}).\frac{5}{7}))\div (\frac{7}{3}\div 1) \)

To solve the expression \[ (((\frac{1}{2} \cdot 1) \div \frac{3}{2}) - ((5 \frac{2}{3}) \cdot \frac{5}{7})) \div (\frac{7}{3} \div 1) \] we will break it down step by step. 1. **Convert mixed numbers to improper fractions**: - \( 5 \frac{2}{3} = \frac{5 \cdot 3 + 2}{3} = \frac{15 + 2}{3} = \frac{17}{3} \) 2. **Substituting back into the expression**: \[ (((\frac{1}{2} \cdot 1) \div \frac{3}{2}) - ((\frac{17}{3}) \cdot \frac{5}{7})) \div (\frac{7}{3} \div 1) \] 3. **Calculate \(\frac{1}{2} \cdot 1\)**: \[ \frac{1}{2} \cdot 1 = \frac{1}{2} \] 4. **Calculate \(\frac{1}{2} \div \frac{3}{2}\)**: \[ \frac{1}{2} \div \frac{3}{2} = \frac{1}{2} \cdot \frac{2}{3} = \frac{1 \cdot 2}{2 \cdot 3} = \frac{2}{6} = \frac{1}{3} \] 5. **Calculate \((\frac{17}{3}) \cdot \frac{5}{7}\)**: \[ \frac{17}{3} \cdot \frac{5}{7} = \frac{17 \cdot 5}{3 \cdot 7} = \frac{85}{21} \] 6. **Now substitute back into the expression**: \[ \left(\frac{1}{3} - \frac{85}{21}\right) \div \left(\frac{7}{3} \div 1\right) \] 7. **Calculate \(\frac{7}{3} \div 1\)**: \[ \frac{7}{3} \div 1 = \frac{7}{3} \] 8. **Now we need to calculate \(\frac{1}{3} - \frac{85}{21}\)**. To do this, we need a common denominator: - The least common multiple of 3 and 21 is 21. - Convert \(\frac{1}{3}\) to have a denominator of 21: \[ \frac{1}{3} = \frac{1 \cdot 7}{3 \cdot 7} = \frac{7}{21} \] 9. **Now perform the subtraction**: \[ \frac{7}{21} - \frac{85}{21} = \frac{7 - 85}{21} = \frac{-78}{21} \] 10. **Now we have**: \[ \frac{-78}{21} \div \frac{7}{3} \] 11. **Dividing by a fraction is the same as multiplying by its reciprocal**: \[ \frac{-78}{21} \cdot \frac{3}{7} = \frac{-78 \cdot 3}{21 \cdot 7} \] 12. **Calculate the numerator and denominator**: - Numerator: \(-78 \cdot 3 = -234\) - Denominator: \(21 \cdot 7 = 147\) 13. **Now we have**: \[ \frac{-234}{147} \] 14. **Simplifying \(\frac{-234}{147}\)**: - The greatest common divisor (GCD) of 234 and 147 is 39. - Dividing both the numerator and denominator by 39: \[ \frac{-234 \div 39}{147 \div 39} = \frac{-6}{3} = -2 \] Thus, the final result is \[ \boxed{-2} \]