What is the quotient of $5 \frac{2}{3} \div \frac{4}{9} ?$

Step 1 ：Convert the mixed number $5 \frac{2}{3}$ to an improper fraction. The numerator of the improper fraction is calculated by multiplying the whole number by the denominator and then adding the numerator. So, $5 \times 3 + 2 = 17$. Therefore, $5 \frac{2}{3}$ is equivalent to $\frac{17}{3}$.

Step 2 ：Divide the two fractions by multiplying the first fraction by the reciprocal of the second fraction. The reciprocal of $\frac{4}{9}$ is $\frac{9}{4}$. So, $\frac{17}{3} \div \frac{4}{9} = \frac{17}{3} \times \frac{9}{4}$.

Step 3 ：Perform the multiplication to find the quotient. $\frac{17}{3} \times \frac{9}{4} = \frac{153}{12}$.

Step 4 ：Simplify the fraction $\frac{153}{12}$ by dividing both the numerator and the denominator by their greatest common divisor. The greatest common divisor of 153 and 12 is 3. So, $\frac{153}{12} = \frac{51}{4}$.

Step 5 ：Convert the improper fraction $\frac{51}{4}$ back to a mixed number. The whole number is the quotient of the division, and the remainder is the numerator of the fractional part. So, $\frac{51}{4} = 12 \frac{3}{4}$.

Step 6 ：Final Answer: The quotient of $5 \frac{2}{3} \div \frac{4}{9}$ is \(\boxed{12 \frac{3}{4}}\).