To solve the expression \((\frac{8}{9} \times \frac{-3}{4}) \div (\frac{1}{9} \times \frac{-4}{5} \div \frac{3}{5})\), we will break it down step by step.
-
Calculate \(\frac{8}{9} \times \frac{-3}{4}\): \[ \frac{8}{9} \times \frac{-3}{4} = \frac{8 \times -3}{9 \times 4} = \frac{-24}{36} = \frac{-2}{3} \]
-
Calculate \(\frac{1}{9} \times \frac{-4}{5}\): \[ \frac{1}{9} \times \frac{-4}{5} = \frac{1 \times -4}{9 \times 5} = \frac{-4}{45} \]
-
Now divide \(\frac{-4}{45}\) by \(\frac{3}{5}\): When dividing by a fraction, we multiply by its reciprocal: \[ \frac{-4}{45} \div \frac{3}{5} = \frac{-4}{45} \times \frac{5}{3} = \frac{-4 \times 5}{45 \times 3} = \frac{-20}{135} \] We can simplify \(\frac{-20}{135}\): \[ \frac{-20}{135} = \frac{-4}{27} \quad \text{(dividing the numerator and denominator by 5)} \]
-
Now we have \(\frac{-2}{3} \div \frac{-4}{27}\): Again, we multiply by the reciprocal: \[ \frac{-2}{3} \div \frac{-4}{27} = \frac{-2}{3} \times \frac{27}{-4} = \frac{-2 \times 27}{3 \times -4} = \frac{54}{12} \] We can simplify \(\frac{54}{12}\): \[ \frac{54}{12} = \frac{9}{2} \quad \text{(dividing the numerator and denominator by 6)} \]
Thus, the final answer is: \[ \frac{9}{2} \]