1. **State the problem:** Multiply the mixed numbers $1 \frac{2}{4}$ and $3 \frac{5}{6}$. 2. **Convert mixed numbers to improper fractions:** $1 \frac{2}{4} = 1 + \frac{2}{4} = \frac{4}{4} + \frac{2}{4} = \frac{6}{4}$ $3 \frac{5}{6} = 3 + \frac{5}{6} = \frac{18}{6} + \frac{5}{6} = \frac{23}{6}$ 3. **Multiply the improper fractions:** $$\frac{6}{4} \times \frac{23}{6} = \frac{6 \times 23}{4 \times 6} = \frac{138}{24}$$ 4. **Simplify the fraction by canceling common factors:** $$\frac{\cancel{6} \times 23}{4 \times \cancel{6}} = \frac{23}{4}$$ 5. **Convert the improper fraction back to a mixed number:** Divide 23 by 4: $23 \div 4 = 5$ remainder $3$, so $$\frac{23}{4} = 5 \frac{3}{4}$$ **Final answer:** $$1 \frac{2}{4} \times 3 \frac{5}{6} = 5 \frac{3}{4}$$