An increase of ₹ 3 in the selling price of an article turns a loss of $7\frac{1}{2}\%$ into a gain of $7\frac{1}{2}\%$. The cost price (in ₹) of the article is (C.P.O., 2007)

### Concept & Loss to Gain Transition When an item transitions from being sold at a loss to being sold at a gain, the total absolute difference in selling price is equivalent to the sum of the loss percentage and the gain percentage applied to the Cost Price (CP). $$ \Delta \text{SP} = (\text{Loss \%} + \text{Gain \%}) \times \text{CP} $$ ### Step-by-Step Solution * Given: * Initial state: Loss of $7\frac{1}{2}\% = 7.5\%$ * Final state: Gain of $7\frac{1}{2}\% = 7.5\%$ * Difference in Selling Price ($\Delta \text{SP}$) = ₹ 3 * Calculate the total percentage change on the Cost Price: * The price must cover the initial $7.5\%$ loss just to reach the break-even point (CP), and then increase by another $7.5\%$ to achieve the gain. * Total percentage shift = $7.5\% \text{ (to break even)} + 7.5\% \text{ (new gain)} = 15\%$ * This $15\%$ shift is exactly accounted for by the ₹ 3 price increase. * $15\%$ of CP = 3 * $\frac{15}{100} \times \text{CP} = 3$ * Solve for CP: * $\text{CP} = \frac{3 \times 100}{15}$ * $\text{CP} = \frac{300}{15} = 20$ ### Exam Strategy & Shortcut Visualize the Cost Price as a zero line on a number line. A $7.5\%$ loss is at -7.5, and a $7.5\%$ gain is at +7.5. The total distance covered is 15 units ($15\%$). Since $15\%$ represents ₹ 3, you can scale it down easily: $5\%$ represents ₹ 1. To get $100\%$ (the full CP), multiply by 20. Therefore, $1 \times 20 = 20$. ### Common Pitfall Students sometimes subtract the percentages ($7.5\% - 7.5\% = 0\%$) because one is a loss and the other is a gain, failing to recognize that transitioning from a negative (loss) to a positive (gain) requires adding their absolute values to find the total distance shifted. ### Final Answer Therefore, the correct answer is **20**.