1. The problem is to calculate $3 \frac{2}{3} \div \frac{5}{6}$ and express the answer as a simplified fraction. 2. First, convert the mixed number $3 \frac{2}{3}$ to an improper fraction. Recall that $a \frac{b}{c} = \frac{ac + b}{c}$. 3. Calculate the improper fraction: $$3 \frac{2}{3} = \frac{3 \times 3 + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3}$$ 4. Division of fractions is done by multiplying the first fraction by the reciprocal of the second: $$\frac{11}{3} \div \frac{5}{6} = \frac{11}{3} \times \frac{6}{5}$$ 5. Multiply the numerators and denominators: $$\frac{11 \times 6}{3 \times 5} = \frac{66}{15}$$ 6. Simplify the fraction by dividing numerator and denominator by their greatest common divisor (GCD). The GCD of 66 and 15 is 3: $$\frac{66 \div 3}{15 \div 3} = \frac{22}{5}$$ 7. The simplified answer is: $$\frac{22}{5}$$ This fraction is an improper fraction and cannot be simplified further.