The problem asks to divide the fractions:

$\frac{5}{9} \div \frac{2}{3}$

To solve a division of fractions, you multiply the first fraction by the reciprocal (or inverse) of the second fraction. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$.

Thus, the equation becomes:

$\frac{5}{9} \times \frac{3}{2}$

Now, multiply the numerators and the denominators:

$\frac{5 \times 3}{9 \times 2} = \frac{15}{18}$

The fraction $\frac{15}{18}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

$\frac{15 \div 3}{18 \div 3} = \frac{5}{6}$

So, the answer is:

$\frac{5}{6}$

Would you like any more details or explanations on this?

Here are 5 related questions for practice:

1. What is $\frac{4}{7} \div \frac{5}{3}$?
2. Simplify $\frac{8}{12}$ to its lowest terms.
3. Multiply $\frac{7}{8} \times \frac{3}{4}$.
4. Divide $\frac{6}{5} \div \frac{2}{7}$.
5. What is the reciprocal of $\frac{9}{10}$?

Tip: When dividing fractions, remember to always multiply by the reciprocal of the second fraction.