1. **State the problem:** We need to add the fractions $\frac{1}{2}$ and $\frac{1}{3}$ by finding a common denominator. 2. **Formula and rules:** To add fractions, they must have the same denominator. The common denominator is usually the least common multiple (LCM) of the denominators. 3. **Find the LCM of 2 and 3:** The multiples of 2 are 2, 4, 6, 8, ... and the multiples of 3 are 3, 6, 9, 12, ... The smallest common multiple is 6. 4. **Rewrite each fraction with denominator 6:** $$\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$$ $$\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$$ 5. **Add the fractions:** $$\frac{3}{6} + \frac{2}{6} = \frac{3 + 2}{6} = \frac{5}{6}$$ 6. **Final answer:** $$\boxed{\frac{5}{6}}$$