In a shop, the cost price of an article is 54% of its marked price. If the shopkeeper allows a discount of 15% on the marked price, what percentage profit does he earn on this article?

Introduction / Context:
This profit and loss question combines the ideas of marked price, cost price, discount and percentage gain. Many competitive exams test whether you can move step by step from marked price to selling price and then compare with cost price. Understanding the chain MP → SP → CP is essential to avoid simple mistakes that lead to the wrong profit percentage.

Given Data / Assumptions:

• Cost price (CP) is 54% of the marked price (MP).
• A discount of 15% is allowed on the marked price.
• We need to find the percentage profit on cost price.
• Assume the article is sold once and there are no other hidden costs or taxes.

Concept / Approach:
The standard approach is to treat the marked price as a convenient base, usually 100 units of money. From that, cost price and selling price are expressed as percentages of this base. Once we know both CP and SP numerically, the profit percentage is calculated using the formula: profit% = (SP - CP) / CP * 100. Using a simple assumed MP keeps the calculations clean and general.

Step-by-Step Solution:
Assume marked price MP = 100 units.Then cost price CP = 54% of 100 = 54 units.Discount = 15% of MP, so selling price SP = 100 - 15 = 85 units.Profit = SP - CP = 85 - 54 = 31 units.Profit percentage = (31 / 54) * 100 ≈ 57.4%.

Verification / Alternative check:
You could also choose MP = 1000 or any other convenient value. For example, if MP = 1000, then CP = 540 and SP = 850. Profit = 310, and profit% = (310 / 540) * 100, which again gives approximately 57.4%. Because ratios are preserved, any consistent base for MP leads to the same percentage result.

Why Other Options Are Wrong:
51.32 %, 49.23 % and 46.8 % are all less than the correct value and typically arise from errors such as taking profit over marked price instead of cost price, or misapplying the discount on cost price instead of marked price. These small numerical slips change the denominator and give a smaller profit percentage than is actually earned.

Common Pitfalls:
A common mistake is to subtract 15 from 54 directly or to treat 54% and 15% on the same base without checking which is applied to the marked price and which is applied to cost price. Another frequent error is to compute profit% as profit divided by selling price instead of cost price. Always remember that percentage profit or loss is defined on cost price unless explicitly stated otherwise in the question.

Final Answer:
The shopkeeper earns a profit of approximately 57.4 % on the article.