To solve the expression \((8/9) \times (-3/4) \div (1/9) \times (-4/5) \div (3/5)\), we will follow the order of operations (multiplication and division left to right) and simplify as we go.
Let's rewrite the expression for clarity:
\[ \left( \frac{8}{9} \times \frac{-3}{4} \right) \div \left( \frac{1}{9} \times \frac{-4}{5} \right) \div \left( \frac{3}{5} \right) \]
We can turn the divisions into multiplications by using the reciprocal:
\[ \frac{8}{9} \times \frac{-3}{4} \times \frac{9}{1} \times \frac{-5}{4} \times \frac{5}{3} \]
Now let's simplify the expression in steps.
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Calculate the first multiplication: \[ \frac{8}{9} \times \frac{-3}{4} = \frac{8 \times -3}{9 \times 4} = \frac{-24}{36} = \frac{-2}{3} \]
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Next, multiply by \(\frac{9}{1}\): \[ \frac{-2}{3} \times \frac{9}{1} = \frac{-2 \times 9}{3 \times 1} = \frac{-18}{3} = -6 \]
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Now multiply by \(\frac{-5}{4}\): \[ -6 \times \frac{-5}{4} = \frac{-6 \times -5}{1 \times 4} = \frac{30}{4} = \frac{15}{2} \]
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Finally, multiply by \(\frac{5}{3}\): \[ \frac{15}{2} \times \frac{5}{3} = \frac{15 \times 5}{2 \times 3} = \frac{75}{6} = \frac{25}{2} \]
Thus, the final result is
\[ \frac{25}{2} \]
So, the solution to the expression is \(\mathbf{\frac{25}{2}}\).