A man sold two steel chairs for ₹ $500$ each. On one, he gains $20\%$ and on the other, he loses $12\%$. How much does he gain or lose in the whole transaction? (M.A.T., 2006)

### Concept & Formula To find the overall profit or loss percentage, we must first calculate the total Cost Price (CP) and total Selling Price (SP). The formula for CP when SP and Profit/Loss percentages are known is: $$CP = \frac{SP \times 100}{100 \pm \text{Profit/Loss}\%}$$ The overall gain percentage is computed as $\frac{\text{Total SP} - \text{Total CP}}{\text{Total CP}} \times 100$. ### Step-by-Step Solution * **Given:** Selling price of each chair ($SP_1 = SP_2$) = ₹ $500$. * **Total SP:** $500 + 500 = \text{₹ } 1000$. * **CP of first chair (20% gain):** $CP_1 = 500 \times \frac{100}{120} = 500 \times \frac{5}{6} = \frac{2500}{6} = \frac{1250}{3}$. * **CP of second chair (12% loss):** $CP_2 = 500 \times \frac{100}{88} = 500 \times \frac{25}{22} = \frac{12500}{22} = \frac{6250}{11}$. * **Total CP:** $\frac{1250}{3} + \frac{6250}{11} = \frac{13750 + 18750}{33} = \frac{32500}{33}$. * **Overall Gain:** $\text{Total SP} - \text{Total CP} = 1000 - \frac{32500}{33} = \frac{33000 - 32500}{33} = \frac{500}{33}$. * **Gain Percentage:** $\frac{\text{Gain}}{\text{Total CP}} \times 100 = \frac{\frac{500}{33}}{\frac{32500}{33}} \times 100 = \frac{500}{32500} \times 100 = \frac{20}{13}\% \approx 1.538\%$. * Rounding to the nearest half percent gives a $1.5\%$ gain. ### Exam Strategy & Shortcut Unlike problems where the gain and loss percentages are equal (where the net result is always a loss of $\frac{x^2}{100}\%$), different percentages require finding the exact ratio. Using fractions directly instead of decimals prevents rounding errors mid-calculation. ### Common Pitfall A frequent mistake is averaging the percentages: $\frac{20 - 12}{2} = 4\%$ gain, which is entirely incorrect because the cost prices of the two items are different. ### Final Answer Therefore, the correct answer is **1.5% gain**.