1. Stating the problem: Simplify the expression $\left(\frac{6}{8} - \frac{2}{8} \times \frac{1}{3}\right) \times \frac{4}{5}$.\n\n2. Use the order of operations: multiplication and division before addition and subtraction.\n\n3. Calculate the multiplication inside the parentheses first: $\frac{2}{8} \times \frac{1}{3} = \frac{2 \times 1}{8 \times 3} = \frac{2}{24}$.\n\n4. Simplify $\frac{2}{24}$ by canceling common factors: $\frac{\cancel{2}^1}{\cancel{24}^12} = \frac{1}{12}$.\n\n5. Now subtract inside the parentheses: $\frac{6}{8} - \frac{1}{12}$. Find a common denominator, which is 24: $\frac{6}{8} = \frac{6 \times 3}{8 \times 3} = \frac{18}{24}$ and $\frac{1}{12} = \frac{1 \times 2}{12 \times 2} = \frac{2}{24}$.\n\n6. Perform the subtraction: $\frac{18}{24} - \frac{2}{24} = \frac{18 - 2}{24} = \frac{16}{24}$.\n\n7. Simplify $\frac{16}{24}$ by canceling common factors: $\frac{\cancel{16}^4}{\cancel{24}^6} = \frac{4}{6}$.\n\n8. Further simplify $\frac{4}{6}$ by dividing numerator and denominator by 2: $\frac{\cancel{4}^2}{\cancel{6}^3} = \frac{2}{3}$.\n\n9. Multiply the result by $\frac{4}{5}$: $\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}$.\n\nFinal answer: $\frac{8}{15}$.