1. **State the problem:** We need to divide $\frac{1}{5}$ by the mixed number $2 \frac{6}{7}$. Lucinda's working shows a mistake in converting and multiplying fractions. 2. **Explain the mistake:** Lucinda wrote $1/5 \div 2 \frac{6}{7} = 1/5 \times 2 \frac{7}{6}$, which is incorrect. The reciprocal of $2 \frac{6}{7}$ is not $2 \frac{7}{6}$ but rather the reciprocal of the improper fraction form of $2 \frac{6}{7}$. 3. **Convert the mixed number to an improper fraction:** $$2 \frac{6}{7} = \frac{2 \times 7 + 6}{7} = \frac{14 + 6}{7} = \frac{20}{7}$$ 4. **Find the reciprocal of $\frac{20}{7}$:** $$\text{Reciprocal} = \frac{7}{20}$$ 5. **Rewrite the division as multiplication by the reciprocal:** $$\frac{1}{5} \div \frac{20}{7} = \frac{1}{5} \times \frac{7}{20}$$ 6. **Multiply the fractions:** $$\frac{1 \times 7}{5 \times 20} = \frac{7}{100}$$ 7. **Simplify the fraction if possible:** $\frac{7}{100}$ is already in simplest form. **Final answer:** $$\frac{1}{5} \div 2 \frac{6}{7} = \frac{7}{100}$$