1. **Problem Statement:** We have two clubs: Music and Mathematics. Each originally has $x$ members. Music club registration costs 7000, Mathematics club costs 7200. 5 members move from Music to Mathematics before payment. (a) Find expressions for the amount each member pays after the switch. (b) Given a Music club member pays 40 more than a Mathematics club member, find $x$ and the percentage increase in Music club member's contribution. 2. **Formulas and Rules:** - Cost per member = Total cost / Number of members - After 5 members switch, Music club has $x-5$ members, Mathematics club has $x+5$ members. - Percentage increase = $\frac{\text{new amount} - \text{original amount}}{\text{original amount}} \times 100$% 3. **Step-by-step Solution:** **(a)(i) Music club member contribution after switch:** $$\text{Music contribution} = \frac{7000}{x-5}$$ **(a)(ii) Mathematics club member contribution after switch:** $$\text{Math contribution} = \frac{7200}{x+5}$$ **(b)(i) Form equation and solve for $x$:** Given Music member pays 40 more than Math member: $$\frac{7000}{x-5} - \frac{7200}{x+5} = 40$$ Multiply both sides by $(x-5)(x+5)$: $$7000(x+5) - 7200(x-5) = 40(x-5)(x+5)$$ Expand: $$7000x + 35000 - 7200x + 36000 = 40(x^2 - 25)$$ Simplify left side: $$-200x + 71000 = 40x^2 - 1000$$ Bring all terms to one side: $$40x^2 + 200x - 72000 = 0$$ Divide entire equation by 40: $$x^2 + 5x - 1800 = 0$$ Factor or use quadratic formula: $$x = \frac{-5 \pm \sqrt{5^2 - 4(1)(-1800)}}{2} = \frac{-5 \pm \sqrt{25 + 7200}}{2} = \frac{-5 \pm \sqrt{7225}}{2}$$ $$\sqrt{7225} = 85$$ So, $$x = \frac{-5 + 85}{2} = 40 \quad \text{or} \quad x = \frac{-5 - 85}{2} = -45$$ Since number of members can't be negative, $x=40$. **(b)(ii) Percentage increase in Music club member contribution:** Original contribution: $$\frac{7000}{40} = 175$$ New contribution: $$\frac{7000}{40 - 5} = \frac{7000}{35} = 200$$ Percentage increase: $$\frac{200 - 175}{175} \times 100 = \frac{25}{175} \times 100 = 14.29\%$$ **Final answers:** (a)(i) $\frac{7000}{x-5}$ (a)(ii) $\frac{7200}{x+5}$ (b)(i) $x=40$ (b)(ii) Percentage increase = 14.29%