(a) Price with 20% gain = 100% = x Selling price = 100% - 10% = 90% i.e. 90% of x = 864 \(\therefore x = 864 \times \frac{100}{90} = N960\) Let cost price = y = 100% x = 20% of y + y = 120% of y. \(y = \frac{100}{120} x = \frac{100}{120} \times N960 = N800\) (b) (i) \((x + 6)^{2} = (x + 5)^{2} + (x - 2)^{2}\) \(x^{2} + 12x + 36 = x^{2} + 10x + 25 + x^{2} - 4x + 4\) \(x^{2} + 12x + 36 = 2x^{2} + 6x + 29\) \(2x^{2} - x^{2} + 6x - 12x + 29 - 36 = 0\) \(x^{2} - 6x - 7 = 0\) \(x^{2} - 7x + x - 7 = 0 \implies x(x - 7) + 1(x - 7) = 0\) \((x - 7)(x + 1) = 0 \implies \text{x = 7 or -1}\) Since measurements cannot be negative, then x = 7 is the suitable answer. (ii) Length of the lawn = (x + 5) metres = (7 + 5) = 12 metres. Width of the lawn = (x - 2) metres = (7 - 2) = 5 metres \(\therefore \text{The area of the lawn} = 12 \times 5 = 60 m^{2}\)