A shopkeeper sells a glass for Rs. 1965 at a loss of 25%. If instead he sells the same glass for Rs. 3013, then what percentage profit will he earn on the cost price?

Introduction / Context:
This profit and loss question checks whether you can move comfortably between selling price, cost price, loss percentage and profit percentage. A shopkeeper first sells a glass at a loss and then imagines selling the same glass at a higher price. You are asked to compute the profit percentage in the second case, using the same cost price. Questions of this type are extremely common in aptitude exams because they test basic percentage manipulation and the understanding that the cost price remains fixed for a given item.

Given Data / Assumptions:
- First selling price (with loss) = Rs. 1965. - Loss percentage in the first sale = 25%. - Second selling price (hypothetical) = Rs. 3013. - Cost price of the glass remains the same in both cases. - We need the profit percentage for the second selling price.

Concept / Approach:
To solve this, we first recover the cost price from the loss information. A loss of 25% means the selling price is 75% of the cost price. Once cost price is known, we can compare it with the second selling price to compute the profit percentage. In percentage terms, loss% or profit% is always calculated with respect to the cost price unless clearly stated otherwise. Therefore, the main steps are: find cost price using loss formula, then find profit percentage using the profit formula.

Step-by-Step Solution:
Step 1: Let the cost price be CP. A loss of 25% means that the first selling price SP1 = 75% of CP. Step 2: Mathematically, SP1 = (75 / 100) * CP = 0.75 * CP. Step 3: We are given SP1 = 1965. So, 0.75 * CP = 1965. Step 4: Therefore, CP = 1965 / 0.75. Step 5: CP = 2620 rupees. Step 6: Now, the second selling price SP2 is 3013 rupees. Step 7: Profit in the second scenario = SP2 - CP = 3013 - 2620 = 393 rupees. Step 8: Profit percentage = (Profit / CP) * 100. Step 9: Profit percentage = (393 / 2620) * 100. Step 10: This simplifies exactly to 15% profit.

Verification / Alternative check:
We can verify the calculation quickly. If CP is 2620, then a 25% loss should give SP1 = 2620 - 0.25 * 2620. The loss amount is 655, so SP1 = 2620 - 655 = 1965, which matches the question data. For the second selling price, a 15% profit on 2620 is 2620 * 1.15 = 3013. This exactly matches the given second selling price, confirming that the profit percentage is correctly computed as 15%.

Why Other Options Are Wrong:
10.4% profit is too small because the rise from 1965 to 3013 is significant relative to the cost price. 13% profit is also less than the true value and does not reproduce the given selling price when applied to 2620. 20% profit would produce a selling price of 2620 * 1.20 = 3144, which is greater than 3013 and therefore inconsistent with the question. Only 15% profit correctly matches the data.

Common Pitfalls:
Many learners mistakenly treat the 25% as being on the selling price instead of the cost price. Another common mistake is to directly compare the two selling prices and calculate a percentage difference, which is incorrect because profit and loss percentages must be based on the cost price. It is also easy to mis-handle the division by 0.75, so it is wise to verify the cost price by checking both loss and profit scenarios against the given selling prices.

Final Answer:
The shopkeeper earns a 15% profit on the cost price when he sells the glass for Rs. 3013 instead of Rs. 1965.