1. **State the problem:** Add the fractions $\frac{4}{7}$ and $\frac{1}{5}$. 2. **Formula:** To add fractions, use the formula $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$. 3. **Apply the formula:** Here, $a=4$, $b=7$, $c=1$, and $d=5$. So, $$\frac{4}{7} + \frac{1}{5} = \frac{4 \times 5 + 1 \times 7}{7 \times 5} = \frac{20 + 7}{35} = \frac{27}{35}.$$ 4. **Simplify if possible:** $27$ and $35$ have no common factors other than 1, so $\frac{27}{35}$ is in simplest form. 5. **Final answer:** $$\frac{4}{7} + \frac{1}{5} = \frac{27}{35}.$$