1. **Problem statement:** Given the prices of pens and the amounts sold in February and April, we need to find expressions for the number of pens sold and calculate quantities based on the given conditions. 2. **Define variables:** Let $x$ be the price of a pen in March. ### (a) Expressions for number of pens sold 1. February price per pen = price in March - 2 = $x - 2$ 2. April price per pen = price in March + 2 = $x + 2$ (i) Number of pens sold in February = Total sales in February / price per pen in February $$\text{Number in February} = \frac{4200}{x-2}$$ (ii) Number of pens sold in April = Total sales in April / price per pen in April $$\text{Number in April} = \frac{4500}{x+2}$$ ### (b) Number of pens sold in February given 50 more pens sold than April We know: $$\frac{4200}{x-2} = \frac{4500}{x+2} + 50$$ Multiply both sides by $(x-2)(x+2)$ to clear denominators: $$4200(x+2) = 4500(x-2) + 50(x-2)(x+2)$$ Expand: $$4200x + 8400 = 4500x - 9000 + 50(x^2 - 4)$$ Simplify right side: $$4200x + 8400 = 4500x - 9000 + 50x^2 - 200$$ Bring all terms to left side: $$4200x + 8400 - 4500x + 9000 - 50x^2 + 200 = 0$$ Simplify: $$-50x^2 - 300x + 17600 = 0$$ Divide entire equation by -50: $$x^2 + 6x - 352 = 0$$ Use quadratic formula: $$x = \frac{-6 \pm \sqrt{6^2 - 4(1)(-352)}}{2} = \frac{-6 \pm \sqrt{36 + 1408}}{2} = \frac{-6 \pm \sqrt{1444}}{2}$$ $$\sqrt{1444} = 38$$ Therefore: $$x = \frac{-6 + 38}{2} = 16 \quad \text{or} \quad x = \frac{-6 - 38}{2} = -22$$ Price cannot be negative, so take $x=16$. Calculate number of pens sold in February: $$\frac{4200}{16 - 2} = \frac{4200}{14} = 300$$ pens. ### (c) Percentage change in pens sold from February to April Number sold in April: $$\frac{4500}{16 + 2} = \frac{4500}{18} = 250$$ pens. Percentage change = $$\frac{\text{February} - \text{April}}{\text{February}} \times 100 = \frac{300 - 250}{300} \times 100 = \frac{50}{300} \times 100 = 16.67\%$$ decrease. **Final answers:** - (a)(i): $\frac{4200}{x-2}$ - (a)(ii): $\frac{4500}{x+2}$ - (b): Number of pens sold in February = 300 - (c): Percentage decrease = 16.67%