Karthik incurs a loss of 40% when he sells an article for Rs. 5,640. At what price should he have sold the article in order to make a profit of 25% on the cost price?

Introduction / Context:
Here we move from a known loss situation to a desired profit situation on the same article. The selling price that produced a loss is given, and we are asked what selling price would produce a specified gain. This problem reinforces the connection between cost price, loss percentage and profit percentage, and how we can move from one scenario to another by recalculating the selling price relative to the cost price.

Given Data / Assumptions:

• Selling price with loss = Rs. 5,640.
• Loss at this selling price = 40% of cost price.
• Required new selling price should give a profit of 25% on the same cost price.
• Cost price is assumed constant in both situations.
• No additional expenses or changes in quantity are involved.

Concept / Approach:
A 40% loss means the selling price equals 60% of cost price. Using SP = 0.60 * CP and SP = 5,640, we can find CP. Then, a 25% profit corresponds to a selling price equal to 125% of cost price. We compute this as SP = 1.25 * CP. The sequence is: derive cost price from a known loss scenario and then apply the desired profit percentage to that cost price to determine the required new selling price.

Step-by-Step Solution:
Step 1: Let cost price = CP. Step 2: Given a loss of 40%, selling price SP1 = 0.60 * CP. Step 3: We know SP1 = Rs. 5,640, so 0.60 * CP = 5,640. Step 4: Therefore, CP = 5,640 / 0.60. Step 5: Compute CP: 5,640 / 0.60 = 5,640 * (10 / 6) = 5,640 * (5 / 3) = 9,400. Step 6: For a profit of 25%, required selling price SP2 = 1.25 * CP. Step 7: SP2 = 1.25 * 9,400 = 9,400 * (5 / 4) = 11,750. Step 8: Thus, the article should be sold at Rs. 11,750 to get a 25% profit.

Verification / Alternative check:
Check both scenarios explicitly. For CP = Rs. 9,400 and SP1 = Rs. 5,640, loss = 9,400 - 5,640 = 3,760. Loss percentage = (3,760 / 9,400) * 100 = 40%. For SP2 = Rs. 11,750, profit = 11,750 - 9,400 = 2,350. Profit percentage = (2,350 / 9,400) * 100 = 25%. Both checks confirm the correctness of the cost price and the required selling price.

Why Other Options Are Wrong:
Selling prices such as Rs. 12,550, Rs. 13,650 or Rs. 11,550 would produce profit percentages different from 25% when calculated with CP = Rs. 9,400. The option Rs. 10,750 results in a profit that is too small. Only Rs. 11,750 corresponds exactly to a 25% profit on the cost price derived from the given loss condition.

Common Pitfalls:
Some learners might incorrectly assume that the increase from the loss selling price to the profit selling price is a simple percentage of the initial SP rather than recalculating from cost price. Others may misinterpret 40% loss as meaning SP is 40% of CP instead of 60% of CP. Carefully writing the equations SP = (1 - loss%) * CP and SP = (1 + profit%) * CP avoids these misunderstandings.

Final Answer:
Karthik should have sold the article for Rs. 11,750 to earn a 25% profit.