1. The problem is to simplify the expression $\frac{1}{2} + \left( \frac{3}{5} - \frac{1}{3} \right)$ and verify which option matches the simplified result. 2. Start by simplifying the expression inside the brackets: $$\frac{3}{5} - \frac{1}{3} = \frac{3 \times 3}{5 \times 3} - \frac{1 \times 5}{3 \times 5} = \frac{9}{15} - \frac{5}{15} = \frac{4}{15}$$ 3. Now add $\frac{1}{2}$ to this result: $$\frac{1}{2} + \frac{4}{15} = \frac{1 \times 15}{2 \times 15} + \frac{4 \times 2}{15 \times 2} = \frac{15}{30} + \frac{8}{30} = \frac{23}{30}$$ 4. Comparing $\frac{23}{30}$ to the right-hand side $\frac{2}{3}$: $$\frac{2}{3} = \frac{20}{30}$$ Since $\frac{23}{30} \neq \frac{20}{30}$, the given equation $\frac{1}{2} + \left( \frac{3}{5} - \frac{1}{3} \right) = \frac{2}{3}$ is false. 5. The question asks to find the simplified value of the expression (left-hand side only), which is $\frac{23}{30}$. 6. Now examine the options to find which matches $\frac{23}{30}$ or an equivalent fraction. Options: (1) $\frac{1}{5} = \frac{6}{30}$ (no) (2) $\frac{2}{15} = \frac{4}{30}$ (no) (3) $\frac{1}{10} = \frac{3}{30}$ (no) (4) $-\frac{2}{20} = -\frac{1}{10} = -\frac{3}{30}$ (no) None of these equal $\frac{23}{30}$. Therefore the simplified value is $\frac{23}{30}$, which is not listed among the given options. Final answer: $\boxed{\frac{23}{30}}$