1. **State the problem:** Calculate the exact values of the following expressions: (i) $2 \frac{2}{5} - 1 \frac{1}{3} + 3 \frac{1}{2}$ (ii) $(4.14 + 5.75) + (1.62)^2$ 2. **Convert mixed numbers to improper fractions for (i):** $2 \frac{2}{5} = \frac{2 \times 5 + 2}{5} = \frac{12}{5}$ $1 \frac{1}{3} = \frac{1 \times 3 + 1}{3} = \frac{4}{3}$ $3 \frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{7}{2}$ 3. **Calculate (i) using fractions:** $$\frac{12}{5} - \frac{4}{3} + \frac{7}{2}$$ Find the common denominator: $\text{lcm}(5,3,2) = 30$ Rewrite each fraction: $$\frac{12}{5} = \frac{12 \times 6}{30} = \frac{72}{30}$$ $$\frac{4}{3} = \frac{4 \times 10}{30} = \frac{40}{30}$$ $$\frac{7}{2} = \frac{7 \times 15}{30} = \frac{105}{30}$$ Now sum: $$\frac{72}{30} - \frac{40}{30} + \frac{105}{30} = \frac{72 - 40 + 105}{30} = \frac{137}{30}$$ 4. **Simplify (i) if possible:** $137$ and $30$ have no common factors other than 1, so the fraction is in simplest form. 5. **Calculate (ii):** First sum inside parentheses: $$4.14 + 5.75 = 9.89$$ Square $1.62$: $$1.62^2 = 1.62 \times 1.62 = 2.6244$$ Add results: $$9.89 + 2.6244 = 12.5144$$ 6. **Final answers:** (i) $\frac{137}{30}$ (ii) $12.5144$