(a) A shop owner marked a shirt at a price to enable him to make a gain of 20%. During a special sales period, the shirt was sold at 10% reduction to a customer at N864.00. What was the original cost to the shop owner?
(b) A rectangular lawn of length (x + 5) metres is (x – 2) metres wide. If the diagonal is (x + 6) metres, find ;
(i) the value of x ; (ii) the area of lawn.
Explanation
(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
\(\therefore \text{The area of the lawn} = 12 \times 5 = 60 m^{2}\)