To solve the expression \( \frac{8}{9} \times \left(-\frac{3}{4}\right) \div \left(\frac{1}{9} \times \left(-\frac{4}{5}\right) \div \frac{3}{5}\right) \), we will break it down using the properties of operations step by step.
Step 1: Solve the parts inside the parentheses
First part: \[ \frac{1}{9} \times \left(-\frac{4}{5}\right) = -\frac{4}{45} \]
Second part: \[ -\frac{4}{45} \div \frac{3}{5} \] To divide by a fraction, we multiply by its reciprocal: \[ -\frac{4}{45} \times \frac{5}{3} = -\frac{4 \cdot 5}{45 \cdot 3} = -\frac{20}{135} \] Now, simplify \(-\frac{20}{135}\): \[ \text{GCD of } 20 \text{ and } 135 = 5 \] \[ -\frac{20 \div 5}{135 \div 5} = -\frac{4}{27} \]
Step 2: Substitute back and evaluate the entire expression
Now we substitute back into the main expression: \[ \frac{8}{9} \times \left(-\frac{3}{4}\right) \div \left(-\frac{4}{27}\right) \]
Multiply \( \frac{8}{9} \) and \( -\frac{3}{4} \): \[ \frac{8 \cdot -3}{9 \cdot 4} = -\frac{24}{36} \] Now simplify: \[ -\frac{24 \div 12}{36 \div 12} = -\frac{2}{3} \]
Now divide by \( -\frac{4}{27} \): To divide by a fraction, again we multiply by its reciprocal: \[ -\frac{2}{3} \div -\frac{4}{27} = -\frac{2}{3} \times -\frac{27}{4} = \frac{2 \cdot 27}{3 \cdot 4} = \frac{54}{12} \] Now simplify: \[ \frac{54 \div 6}{12 \div 6} = \frac{9}{2} \]
Thus, the final answer is: \[ \boxed{\frac{9}{2}} \]