1. Stating the problem: Simplify the expression $$\frac{1}{6} \times \left(\frac{4}{5} - \frac{1}{3}\right) \div \frac{1}{5}$$. 2. Calculate the subtraction inside the parentheses: $$\frac{4}{5} - \frac{1}{3} = \frac{4 \times 3}{5 \times 3} - \frac{1 \times 5}{3 \times 5} = \frac{12}{15} - \frac{5}{15} = \frac{7}{15}$$ 3. Substitute back into the expression: $$\frac{1}{6} \times \frac{7}{15} \div \frac{1}{5}$$ 4. Multiply the fractions: $$\frac{1}{6} \times \frac{7}{15} = \frac{7}{90}$$ 5. Divide by $$\frac{1}{5}$$ is the same as multiplying by its reciprocal: $$\frac{7}{90} \div \frac{1}{5} = \frac{7}{90} \times \frac{5}{1}$$ 6. Multiply: $$\frac{7}{90} \times \frac{5}{1} = \frac{7 \times 5}{90 \times 1} = \frac{35}{90}$$ 7. Simplify the fraction by dividing numerator and denominator by 5: $$\frac{\cancel{35}^{7}}{\cancel{90}^{18}} = \frac{7}{18}$$ Final answer: $$\frac{7}{18}$$