A shopkeeper sells an article at $12\frac{1}{2}\%$ loss. If he sells it for ₹ 92.50 more then he gains $6\%$. What is the cost price of the article? (MA.T., 2008)

### Concept & Total Percentage Distance When adjusting the Selling Price (SP) transforms a loss into a profit, the increase in price equals the absolute sum of the initial loss percentage and the final profit percentage, calculated strictly on the Cost Price (CP). $$ \text{Price Increase} = (\text{Loss \%} + \text{Profit \%}) \text{ of CP} $$ ### Step-by-Step Solution * Given: * Initial condition: Loss of $12\frac{1}{2}\% = 12.5\%$ * Final condition: Gain of $6\%$ * Price increment ($\Delta \text{SP}$) = ₹ 92.50 * Determine the total percentage gap bridging the loss and the gain: * Total % difference = $12.5\% \text{ (loss)} + 6\% \text{ (gain)} = 18.5\%$ * Equate this total percentage to the absolute price difference: * $18.5\%$ of CP = 92.50 * $\frac{18.5}{100} \times \text{CP} = 92.50$ * Solve for CP: * $\text{CP} = \frac{92.50 \times 100}{18.5}$ * $\text{CP} = \frac{9250}{18.5}$ * To divide easily, multiply numerator and denominator by 10 to remove decimals: * $\text{CP} = \frac{92500}{185}$ * Note that $185 \times 5 = 925$. * Therefore, $\text{CP} = 500$. ### Exam Strategy & Shortcut Instead of decimals, you can use fractions. $12.5\% = \frac{1}{8}$, and $6\% = \frac{6}{100} = \frac{3}{50}$. Total fraction = $\frac{1}{8} + \frac{3}{50} = \frac{25}{200} + \frac{12}{200} = \frac{37}{200}$. So, $\frac{37}{200}$ of CP = 92.50. $\text{CP} = \frac{92.50 \times 200}{37} = \frac{18500}{37} = 500$. Fractions often prevent decimal arithmetic errors during exams. ### Common Pitfall A typical mistake is treating the $12.5\%$ and $6\%$ as sequential changes and multiplying them, rather than realizing they represent two independent endpoints relative to the same constant Cost Price base. ### Final Answer Therefore, the correct answer is **₹ 500**.