1. Stating the problem: Calculate the value of $0.6 \times \frac{2}{7} + \frac{4}{5}$. 2. Convert the decimal to a fraction: $0.6 = \frac{6}{10} = \frac{3}{5}$. 3. Rewrite the expression using fractions: $$\frac{3}{5} \times \frac{2}{7} + \frac{4}{5}$$. 4. Multiply the fractions: $$\frac{3}{5} \times \frac{2}{7} = \frac{3 \times 2}{5 \times 7} = \frac{6}{35}$$. 5. Now the expression is: $$\frac{6}{35} + \frac{4}{5}$$. 6. Find a common denominator for $\frac{6}{35}$ and $\frac{4}{5}$. The least common denominator is $35$. 7. Convert $\frac{4}{5}$ to have denominator $35$: $$\frac{4}{5} = \frac{4 \times 7}{5 \times 7} = \frac{28}{35}$$. 8. Add the fractions: $$\frac{6}{35} + \frac{28}{35} = \frac{6 + 28}{35} = \frac{34}{35}$$. 9. Final answer: $$\frac{34}{35}$$, which is approximately $0.9714$.