We need to evaluate the expression $2\frac{2}{3} \div (-\frac{8}{9}) \times (-1\frac{3}{5})$.

1. Problem Description

We need to evaluate the expression $2\frac{2}{3} \div (-\frac{8}{9}) \times (-1\frac{3}{5})$.

2. Solution Steps

First, we convert the mixed numbers to improper fractions:

$2\frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{6+2}{3} = \frac{8}{3}$

$-1\frac{3}{5} = -\frac{1 \times 5 + 3}{5} = -\frac{5+3}{5} = -\frac{8}{5}$

So, the expression becomes:

$\frac{8}{3} \div (-\frac{8}{9}) \times (-\frac{8}{5})$

Next, we change the division to multiplication by taking the reciprocal of the second fraction:

$\frac{8}{3} \times (-\frac{9}{8}) \times (-\frac{8}{5})$

Now, we can multiply the fractions:

$\frac{8}{3} \times (-\frac{9}{8}) \times (-\frac{8}{5}) = \frac{8 \times (-9) \times (-8)}{3 \times 8 \times 5}$

Simplify the expression:

$\frac{8 \times (-9) \times (-8)}{3 \times 8 \times 5} = \frac{8 \times 9 \times 8}{3 \times 8 \times 5}$ (since a negative times a negative is a positive)

We can cancel out the common factor of 8:

$\frac{8 \times 9 \times 8}{3 \times 8 \times 5} = \frac{9 \times 8}{3 \times 5}$

We can simplify by dividing 9 by 3 to get 3:

$\frac{9 \times 8}{3 \times 5} = \frac{3 \times 8}{5}$

Multiply the remaining numbers:

$\frac{3 \times 8}{5} = \frac{24}{5}$

Finally, convert the improper fraction back to a mixed number:

$\frac{24}{5} = 4\frac{4}{5}$

3. Final Answer

$4\frac{4}{5}$