Let's break down the expression step by step. We need to solve the following:
\[ \frac{(8/9) \times (-3/4)}{(1/9) \times (-4/5)} \div \frac{3}{5} \]
First, calculate the numerator:
\[ (8/9) \times (-3/4) = \frac{8 \times -3}{9 \times 4} = \frac{-24}{36} \]
We can simplify \(\frac{-24}{36}\):
\[ -24 \div 12 = -2 \quad \text{and} \quad 36 \div 12 = 3 \]
So,
\[ \frac{-24}{36} = \frac{-2}{3} \]
Now calculate the denominator:
\[ (1/9) \times (-4/5) = \frac{1 \times -4}{9 \times 5} = \frac{-4}{45} \]
Now we substitute these into the expression:
\[ \frac{-2/3}{-4/45} \]
Dividing a fraction by another fraction involves multiplying by the reciprocal:
\[ \frac{-2}{3} \div \frac{-4}{45} = \frac{-2}{3} \times \frac{45}{-4} \]
Cancelling the negatives gives:
\[ \frac{2}{3} \times \frac{45}{4} \]
Now we multiply the fractions:
\[ \frac{2 \times 45}{3 \times 4} = \frac{90}{12} \]
We can simplify \(\frac{90}{12}\):
\[ 90 \div 6 = 15 \quad \text{and} \quad 12 \div 6 = 2 \]
So,
\[ \frac{90}{12} = \frac{15}{2} \]
We still need to divide this by \(\frac{3}{5}\):
\[ \frac{15}{2} \div \frac{3}{5} = \frac{15}{2} \times \frac{5}{3} \]
Multiplying the fractions gives:
\[ \frac{15 \times 5}{2 \times 3} = \frac{75}{6} \]
We can simplify this:
\[ 75 \div 3 = 25 \quad \text{and} \quad 6 \div 3 = 2 \]
Thus,
\[ \frac{75}{6} = \frac{25}{2} \]
In conclusion, the answer is:
\[ \frac{25}{2} \]
or as a mixed number, \(12 \frac{1}{2}\).