1. **State the problem:** Multiply $\frac{1}{2}$ by $7 \frac{1}{2}$. 2. **Convert the mixed number to an improper fraction:** $$7 \frac{1}{2} = \frac{7 \times 2 + 1}{2} = \frac{15}{2}$$ 3. **Write the multiplication of fractions:** $$\frac{1}{2} \times \frac{15}{2}$$ 4. **Multiply the numerators and denominators:** $$\frac{1 \times 15}{2 \times 2} = \frac{15}{4}$$ 5. **Simplify or convert to a mixed number:** $$\frac{15}{4} = 3 \frac{3}{4}$$ **Final answer:** $3 \frac{3}{4}$