1. **State the problem:** Calculate the value of the expression $\frac{1}{2} + \frac{3}{5} - \frac{2}{3}$.\n\n2. **Find a common denominator:** The denominators are 2, 5, and 3. The least common denominator (LCD) is 30.\n\n3. **Rewrite each fraction with denominator 30:**\n$$\frac{1}{2} = \frac{1 \times 15}{2 \times 15} = \frac{15}{30}$$\n$$\frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30}$$\n$$\frac{2}{3} = \frac{2 \times 10}{3 \times 10} = \frac{20}{30}$$\n\n4. **Perform the addition and subtraction:**\n$$\frac{15}{30} + \frac{18}{30} - \frac{20}{30} = \frac{15 + 18 - 20}{30} = \frac{13}{30}$$\n\n5. **Final answer:**\n$$\frac{13}{30}$$