The problem is to evaluate the expression: $\frac{3}{5} \div \frac{7}{10} \times (-1\frac{5}{9})$.

1. Problem Description

The problem is to evaluate the expression: $\frac{3}{5} \div \frac{7}{10} \times (-1\frac{5}{9})$.

2. Solution Steps

First, we rewrite the mixed number $-1\frac{5}{9}$ as an improper fraction:

$-1\frac{5}{9} = -\frac{1 \times 9 + 5}{9} = -\frac{9+5}{9} = -\frac{14}{9}$.

Next, we rewrite the division as multiplication by the reciprocal:

$\frac{3}{5} \div \frac{7}{10} = \frac{3}{5} \times \frac{10}{7}$.

Now, we perform the multiplication:

$\frac{3}{5} \times \frac{10}{7} = \frac{3 \times 10}{5 \times 7} = \frac{30}{35} = \frac{6}{7}$.

Finally, we multiply the result by $-\frac{14}{9}$:

$\frac{6}{7} \times (-\frac{14}{9}) = -\frac{6 \times 14}{7 \times 9} = -\frac{84}{63}$.

We can simplify the fraction $-\frac{84}{63}$ by dividing both numerator and denominator by their greatest common divisor, which is 21:

$-\frac{84}{63} = -\frac{84 \div 21}{63 \div 21} = -\frac{4}{3}$.

3. Final Answer

The final answer is $-\frac{4}{3}$.