1. **State the problem:** Multiply the fractions $\frac{1}{6}$ and $\frac{2}{3}$. 2. **Formula:** To multiply fractions, multiply the numerators together and the denominators together: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$ 3. **Apply the formula:** $$\frac{1}{6} \times \frac{2}{3} = \frac{1 \times 2}{6 \times 3} = \frac{2}{18}$$ 4. **Simplify the fraction:** Both numerator and denominator can be divided by 2: $$\frac{\cancel{2}}{\cancel{18}} = \frac{1}{9}$$ 5. **Final answer:** $$\frac{1}{9}$$ This means one-sixth times two-thirds equals one-ninth.