1. **State the problem:** Add the fractions $\frac{1}{2}$ and $\frac{4}{5}$. 2. **Formula:** To add fractions, use the formula $$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$ where $a$, $b$, $c$, and $d$ are integers and $b,d \neq 0$. 3. **Apply the formula:** Here, $a=1$, $b=2$, $c=4$, and $d=5$. So, $$\frac{1}{2} + \frac{4}{5} = \frac{1 \times 5 + 4 \times 2}{2 \times 5} = \frac{5 + 8}{10} = \frac{13}{10}$$ 4. **Simplify:** The fraction $\frac{13}{10}$ is an improper fraction and cannot be simplified further. It can also be written as a mixed number: $$\frac{13}{10} = 1 \frac{3}{10}$$ 5. **Final answer:** $$\frac{1}{2} + \frac{4}{5} = \frac{13}{10} = 1 \frac{3}{10}$$