The marked price of an article is Rs. 1,600 more than its cost price. When the article is sold at a discount of Rs. 500 on the marked price, the shopkeeper earns a profit of 25% on the cost price. For the shopkeeper to earn a profit of 30% on the same cost price, at what price should he sell the article?

Introduction / Context:
This problem combines the ideas of marked price, discount and profit percentage. The marked price of an article is set higher than its cost price by a fixed amount. When a discount is given, the selling price changes and so does the profit. We are told that a certain discount results in a 25% profit and then asked to find a new selling price that would produce a 30% profit on the same cost price. This is a typical profit and loss question that tests algebraic manipulation of prices and percentages.

Given Data / Assumptions:

• Let the cost price (CP) of the article be C.
• The marked price (MP) is C + Rs. 1,600.
• When a discount of Rs. 500 is allowed on MP, the resulting selling price gives a 25% profit.
• We must find the selling price that would yield a 30% profit on the same cost price C.

Concept / Approach:
We use two main profit relationships. First, with the discount of Rs. 500, the selling price is MP − 500, which equals C plus 25% of C. From this, we can solve for C. Once C is known, the required selling price for a 30% profit is simply C * 1.30. The trick is to correctly interpret the relationship between marked price, discount and profit, and express everything in terms of C.

Step-by-Step Solution:
Let cost price = C rupees. Marked price MP = C + 1,600. When a discount of Rs. 500 is given, selling price SP1 = MP − 500 = C + 1,600 − 500. Thus, SP1 = C + 1,100. We are told this selling price yields a 25% profit on C. So SP1 = C * 1.25. Therefore, C + 1,100 = 1.25C. Rearrange: 1.25C − C = 1,100. 0.25C = 1,100. C = 1,100 / 0.25 = 4,400 rupees. Now, to earn a 30% profit, required selling price SP2 = 1.30C = 1.30 * 4,400. SP2 = 4,400 * 1.30 = 5,720 rupees.

Verification / Alternative check:
Check the first scenario. With C = 4,400, MP = 4,400 + 1,600 = 6,000. Discount of Rs. 500 means SP1 = 6,000 − 500 = 5,500. Profit = 5,500 − 4,400 = 1,100. Profit percentage = (1,100 / 4,400) * 100 = 25%, which matches the given condition. For SP2, we tested 5,720, and profit = 5,720 − 4,400 = 1,320. Profit percentage = (1,320 / 4,400) * 100 = 30%, confirming correctness.

Why Other Options Are Wrong:
5,740 and 5,730: These yield profit percentages slightly above or below 30% when compared with 4,400.
5,620: This gives profit = 1,220, so profit percentage is (1,220 / 4,400) * 100 ≈ 27.73%, not 30%.
5,800: Corresponds to a higher profit percentage, greater than 30% and not consistent with the question.

Common Pitfalls:
One frequent mistake is to calculate the 25% profit on the marked price instead of the cost price. Another is to add or subtract profit percentages directly to or from the discount amount. The correct method is always to express selling price in terms of cost price and then apply the given conditions step by step. Also, note that the 1,600 rupees difference is between MP and CP, not between SP and CP.

Final Answer:
To earn a 30% profit on the cost price, the article should be sold for Rs. 5,720.