1. **State the problem:** Add the mixed numbers $1 \frac{1}{2} + \frac{1}{2}$. 2. **Formula and rules:** To add mixed numbers, add the whole numbers and fractions separately. If the fraction sum is improper, convert it to a mixed number. 3. **Convert mixed numbers to improper fractions:** $$1 \frac{1}{2} = \frac{3}{2}$$ $$\frac{1}{2} = \frac{1}{2}$$ 4. **Add the fractions:** $$\frac{3}{2} + \frac{1}{2} = \frac{3+1}{2} = \frac{4}{2}$$ 5. **Simplify the fraction:** $$\frac{4}{2} = \cancel{\frac{4}{2}}^{2} = 2$$ 6. **Convert back to mixed number if needed:** Here, $2$ is a whole number. 7. **Add whole numbers and fractions:** Since the first number had a whole part $1$ and the fraction sum is $2$, total is $1 + 1.5 = 2.5$ or $2 \frac{1}{2}$. **Final answer:** $2 \frac{1}{2}$