1. **State the problem:** Calculate the sum of $\frac{3}{4}$ and $\frac{1}{2}$.\n\n2. **Formula and rules:** To add fractions, they must have a common denominator. The sum is given by $$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$ where $a,b,c,d$ are integers and $b,d \neq 0$.\n\n3. **Find common denominator:** The denominators are 4 and 2. The least common denominator (LCD) is 4.\n\n4. **Convert fractions:** $\frac{3}{4}$ stays the same. Convert $\frac{1}{2}$ to have denominator 4: $$\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$$\n\n5. **Add fractions:** $$\frac{3}{4} + \frac{2}{4} = \frac{3 + 2}{4} = \frac{5}{4}$$\n\n6. **Simplify and interpret:** $\frac{5}{4}$ is an improper fraction and can be written as a mixed number: $$\frac{5}{4} = 1 \frac{1}{4}$$\n\n7. **Answer:** The sum is $1 \frac{1}{4}$, which corresponds to option C.