A shopkeeper wants to earn 33% profit on an article even after offering a 30% discount to customers. By what percentage above the cost price should he mark the article?

Introduction / Context:
In this question, a shopkeeper plans pricing in such a way that, even after giving a substantial discount, he still earns a desired profit on the cost price. This is a classic combination of markup and discount where you must work backward to find the required marked price. The core idea is that the final selling price after discount must correspond to the desired profit percentage on the cost price, and the marked price is adjusted accordingly.

Given Data / Assumptions:
- Desired profit percentage on cost price = 33%. - Discount percentage offered to customers = 30% on marked price. - The article has a definite cost price, denoted by CP. - We must find by what percentage above CP the article should be marked. - There are no extra taxes, and discount is applied only once.

Concept / Approach:
Let the cost price be CP. To achieve 33% profit, the required selling price after discount must be SP = 1.33 * CP. The shopkeeper will mark the price at some level MP such that, after giving a 30% discount, the customer pays exactly SP. A discount of 30% means the customer pays 70% of the marked price, so SP = 0.70 * MP. Combine these ideas: 0.70 * MP = 1.33 * CP. Solve this equation to express MP as a multiple of CP, then convert that multiple into a markup percentage above cost price.

Step-by-Step Solution:
Step 1: Let cost price be CP. Step 2: Desired profit = 33% of CP, so selling price after discount SP = CP * 1.33. Step 3: Let marked price be MP. Step 4: Discount is 30%, so the buyer actually pays 70% of MP. Step 5: Hence SP = 0.70 * MP. Step 6: Equate the expressions for SP: 0.70 * MP = 1.33 * CP. Step 7: MP = (1.33 / 0.70) * CP. Step 8: Compute the factor: 1.33 / 0.70 = 1.9. Step 9: Therefore MP = 1.9 * CP. Step 10: This means the marked price is 90% above the cost price, because 1.9 * CP = CP + 0.9 * CP.

Verification / Alternative check:
Assume CP = Rs. 100 for a quick check. To earn 33% profit, SP must be Rs. 133. If the shopkeeper marks at 90% above CP, MP = 100 * 1.9 = Rs. 190. Discount of 30% on 190 is 0.30 * 190 = 57, so the buyer pays 190 - 57 = Rs. 133. This matches the required selling price for 33% profit, confirming that a 90% markup is correct.

Why Other Options Are Wrong:
If the markup were only 63%, 69%, or 72%, the effective selling price after 30% discount would be less than 1.33 times the cost price, resulting in a lower profit percentage. Each smaller markup produces a smaller final selling price and thus fails to achieve the target 33% profit. Only a 90% markup creates a situation where a 30% discount still leaves the shopkeeper with the desired gain.

Common Pitfalls:
A frequent error is to subtract the discount percentage directly from the desired profit percentage, for example 33% + 30% or 33% - 30%, which is not logically correct. Another mistake is to apply discount on the cost price instead of the marked price. Always remember that discount is applied on the marked price and profit is calculated on the cost price. Setting up proper equations connecting marked price, selling price, and cost price is the safest and most systematic method.

Final Answer:
The shopkeeper should mark the article at 90% above the cost price to earn 33% profit after giving a 30% discount.