1. **Problem statement:** Given a binary operation $*$ on real numbers defined by $a * b = a + b + \frac{2ab}{5}$, evaluate (i) $3 * 2$, (ii) $2 * (4 * 5)$. 2. **Formula and rules:** The operation is defined as $a * b = a + b + \frac{2ab}{5}$. We will substitute values and simplify. 3. **Step (i):** Calculate $3 * 2$: $$3 * 2 = 3 + 2 + \frac{2 \times 3 \times 2}{5} = 5 + \frac{12}{5} = 5 + 2.4 = 7.4$$ 4. **Step (ii):** Calculate $4 * 5$ first: $$4 * 5 = 4 + 5 + \frac{2 \times 4 \times 5}{5} = 9 + \frac{40}{5} = 9 + 8 = 17$$ Then calculate $2 * (4 * 5) = 2 * 17$: $$2 * 17 = 2 + 17 + \frac{2 \times 2 imes 17}{5} = 19 + \frac{68}{5} = 19 + 13.6 = 32.6$$ **Final answers:** (i) $3 * 2 = 7.4$ (ii) $2 * (4 * 5) = 32.6$