1. **State the problem:** Multiply the mixed numbers $1 \frac{2}{3}$ and $1 \frac{1}{6}$ and express the answer as a mixed number in simplest form. 2. **Convert mixed numbers to improper fractions:** $$1 \frac{2}{3} = \frac{3 \times 1 + 2}{3} = \frac{5}{3}$$ $$1 \frac{1}{6} = \frac{6 \times 1 + 1}{6} = \frac{7}{6}$$ 3. **Multiply the improper fractions:** $$\frac{5}{3} \times \frac{7}{6} = \frac{5 \times 7}{3 \times 6} = \frac{35}{18}$$ 4. **Simplify the fraction if possible:** $35$ and $18$ have no common factors other than $1$, so $\frac{35}{18}$ is already in simplest form. 5. **Convert the improper fraction back to a mixed number:** Divide $35$ by $18$: $$35 \div 18 = 1 \text{ remainder } 17$$ So, $$\frac{35}{18} = 1 \frac{17}{18}$$ 6. **Final answer:** $$1 \frac{17}{18}$$ This is the product of $1 \frac{2}{3}$ and $1 \frac{1}{6}$ expressed as a mixed number in simplest form.