1. **State the problem:** Multiply the mixed numbers $3\frac{3}{4}$ and $5\frac{2}{3}$. 2. **Convert mixed numbers to improper fractions:** $3\frac{3}{4} = \frac{3 \times 4 + 3}{4} = \frac{12 + 3}{4} = \frac{15}{4}$ $5\frac{2}{3} = \frac{5 \times 3 + 2}{3} = \frac{15 + 2}{3} = \frac{17}{3}$ 3. **Multiply the improper fractions:** $$\frac{15}{4} \times \frac{17}{3} = \frac{15 \times 17}{4 \times 3} = \frac{255}{12}$$ 4. **Simplify the fraction:** Find the greatest common divisor (GCD) of 255 and 12, which is 3. $$\frac{\cancel{3} \times 85}{\cancel{3} \times 4} = \frac{85}{4}$$ 5. **Convert the improper fraction back to a mixed number:** Divide 85 by 4: $85 \div 4 = 21$ remainder $1$, so $$\frac{85}{4} = 21\frac{1}{4}$$ **Final answer:** $$3\frac{3}{4} \times 5\frac{2}{3} = 21\frac{1}{4}$$