GATE CS 2024 Set 2 — Question 7

Let the selling price (SP) of each item be $S$.
For the first item, there is a 10% profit:$$S = CP_1 \times (1 + 0.10) \implies CP_1 = \frac{S}{1.1}$$For the second item, there is a 10% loss:$$S = CP_2 \times (1 - 0.10) \implies CP_2 = \frac{S}{0.9}$$Total Selling Price = $S + S = 2S$
Total Cost Price = $CP_1 + CP_2 = \frac{S}{1.1} + \frac{S}{0.9} = S \left( \frac{1}{1.1} + \frac{1}{0.9} \right) = S \left( \frac{0.9 + 1.1}{0.99} \right) = S \left( \frac{2}{0.99} \right) = \frac{2S}{0.99}$Since Total CP ($\approx 2.02S$) > Total SP ($2S$), there is a loss.
Loss = Total CP - Total SP = $\frac{2S}{0.99} - 2S = 2S \left( \frac{1}{0.99} - 1 \right) = 2S \left( \frac{1 - 0.99}{0.99} \right) = 2S \left( \frac{0.01}{0.99} \right) = \frac{2S}{99}$Loss Percentage = $\frac{\text{Loss}}{\text{Total CP}} \times 100$
$$= \frac{\frac{2S}{99}}{\frac{2S}{0.99}} \times 100 = \frac{0.99}{99} \times 100 = 0.01 \times 100 = 1\%$$Alternatively, for two items sold at the same price with $x\%$ profit and $x\%$ loss, the net result is always a loss of $\frac{x^2}{100}\%$.
Here $x=10$, so Loss $\% = \frac{10^2}{100}\% = 1\%$.