1. **Problem:** Find the product of the mixed numbers $1 \frac{2}{4}$ and $3 \frac{5}{6}$.\n\n2. **Convert mixed numbers to improper fractions:**\n$1 \frac{2}{4} = \frac{(1 \times 4) + 2}{4} = \frac{4 + 2}{4} = \frac{6}{4}$\n$3 \frac{5}{6} = \frac{(3 \times 6) + 5}{6} = \frac{18 + 5}{6} = \frac{23}{6}$\n\n3. **Multiply the improper fractions:**\n$$\frac{6}{4} \times \frac{23}{6} = \frac{6 \times 23}{4 \times 6} = \frac{138}{24}$$\n\n4. **Simplify the fraction:**\nCancel common factors: $$\frac{\cancel{6} \times 23}{4 \times \cancel{6}} = \frac{23}{4}$$\n\n5. **Convert back to mixed number:**\nDivide 23 by 4: $23 \div 4 = 5$ remainder $3$, so\n$$5 \frac{3}{4}$$\n\n**Final answer:** $1 \frac{2}{4} \times 3 \frac{5}{6} = 5 \frac{3}{4}$