1. **State the problem:** Evaluate $\frac{1}{2} \left( \frac{3}{5} + \frac{1}{3} \right)$ and simplify the result. 2. **Use the formula:** To add fractions, find a common denominator and then add the numerators. 3. **Add the fractions inside the parentheses:** $$\frac{3}{5} + \frac{1}{3} = \frac{3 \times 3}{5 \times 3} + \frac{1 \times 5}{3 \times 5} = \frac{9}{15} + \frac{5}{15} = \frac{9 + 5}{15} = \frac{14}{15}$$ 4. **Multiply by $\frac{1}{2}$:** $$\frac{1}{2} \times \frac{14}{15} = \frac{1 \times 14}{2 \times 15} = \frac{14}{30}$$ 5. **Simplify the fraction $\frac{14}{30}$:** $$\frac{14}{30} = \frac{\cancel{14}^{7} \times 2}{\cancel{30}^{15} \times 2} = \frac{7}{15}$$ 6. **Final answer:** $$\boxed{\frac{7}{15}}$$ This is the simplest form of the expression.