1. **State the problem:** Add the fractions $\frac{1}{2}$ and $\frac{5}{4}$. 2. **Formula and rules:** To add fractions, they must have a common denominator. The sum is given by $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$. 3. **Find a common denominator:** The denominators are 2 and 4. The least common denominator (LCD) is 4. 4. **Convert fractions:** Convert $\frac{1}{2}$ to have denominator 4: $\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$. 5. **Add the fractions:** $\frac{2}{4} + \frac{5}{4} = \frac{2 + 5}{4} = \frac{7}{4}$. 6. **Final answer:** The sum is $\frac{7}{4}$ or $1 \frac{3}{4}$.