7. \( \left(\frac{3}{4}-\frac{5}{6}\right) \div\left(\frac{7}{8}+\frac{8}{9}\right)= \)

To solve the expression \( \left(\frac{3}{4}-\frac{5}{6}\right) \div\left(\frac{7}{8}+\frac{8}{9}\right) \), we need to address the operations step by step. First, calculate \( \frac{3}{4}-\frac{5}{6} \): 1. Find a common denominator, which is 12. 2. Rewrite the fractions: \[ \frac{3}{4} = \frac{9}{12} \quad \text{and} \quad \frac{5}{6} = \frac{10}{12} \] 3. Now subtract: \[ \frac{9}{12} - \frac{10}{12} = \frac{-1}{12} \] Next, calculate \( \frac{7}{8}+\frac{8}{9} \): 1. Find a common denominator, which is 72. 2. Rewrite the fractions: \[ \frac{7}{8} = \frac{63}{72} \quad \text{and} \quad \frac{8}{9} = \frac{64}{72} \] 3. Now add: \[ \frac{63}{72} + \frac{64}{72} = \frac{127}{72} \] Finally, divide \( \left(\frac{-1}{12}\right) \div \left(\frac{127}{72}\right) \): 1. Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{-1}{12} \times \frac{72}{127} = \frac{-1 \times 72}{12 \times 127} = \frac{-72}{1524} \] 2. Simplifying \( \frac{-72}{1524} \): \[ \frac{-72 \div 12}{1524 \div 12} = \frac{-6}{127} \] Thus, the solution is: \[ \left(\frac{3}{4}-\frac{5}{6}\right) \div\left(\frac{7}{8}+\frac{8}{9}\right) = \frac{-6}{127} \]