A retailer offers a discount of 28% on the marked price of his goods and thereby sells them at cost price.\nWhat is the percentage mark-up on the cost price?

Introduction / Context:
This question tests understanding of the relationship between mark up and discount. The retailer sets a marked price above the cost price and then offers a discount, yet ends up selling at exactly the cost price. From this, you have to back calculate the percentage by which the marked price exceeds the cost price. This is a classic successive percentage and profit loss concept.

Given Data / Assumptions:

• Let the cost price (CP) of the goods be C.
• Let the marked price (MP) be M.
• The retailer offers a discount of 28% on MP.
• After discount, the selling price equals the cost price: SP = CP.
• We must find the percentage mark up, that is how much MP exceeds CP in percentage terms.

Concept / Approach:
A discount of 28% means the selling price is 72% of the marked price, or SP = 0.72M. We are told this SP equals the cost price C. Therefore, C = 0.72M. To find the mark up, we want (M - C) / C * 100. Since C = 0.72M, we can express M in terms of C or C in terms of M and compute the percentage. Working in terms of C keeps the interpretation of markup straightforward.

Step-by-Step Solution:
Step 1: From the discount, we have SP = 72% of MP = 0.72M. Step 2: Given that SP equals CP, we have C = 0.72M. Step 3: Rearrange to express M in terms of C: M = C / 0.72. Step 4: Mark up = M - C = (C / 0.72) - C. Step 5: Factor out C: Mark up = C * (1 / 0.72 - 1). Step 6: Compute 1 / 0.72 = 1.3888... so mark up fraction = 1.3888... - 1 = 0.3888... . Step 7: Convert to percentage: 0.3888... * 100 ≈ 38.88%.

Verification / Alternative check:
Assume the cost price C is Rs. 100 for simplicity. Mark up of 38.88% means MP ≈ 138.88. Now apply a 28% discount on 138.88. Discount = 0.28 * 138.88 ≈ 38.8864. Selling price ≈ 138.88 - 38.8864 ≈ 100, which equals our assumed cost price. This confirms the mark up percentage is correct.

Why Other Options Are Wrong:
18.25% and 22%: These are much smaller than the actual 38.88% needed to offset a large 28% discount and still reach cost price.
28%: This would imply that discount and mark up cancel linearly, which is not the case with successive percentages.
33.33%: This corresponds to a factor of 4/3, which does not match the required 1 / 0.72.

Common Pitfalls:
Many learners incorrectly assume that a 28% discount requires a 28% markup to break even, but markup and discount are applied on different bases (cost vs marked price), so they are not symmetrical. Another pitfall is mismanaging the inversion 1 / 0.72, leading to approximate but incorrect values. Writing clear equations prevents confusion.

Final Answer:
The retailer has marked up the goods by approximately 38.88% over the cost price.