A retailer offers a discount of 32 percent on the marked price of goods and therefore sells exactly at the cost price. What is the percentage mark up of the marked price over the cost price?

Introduction / Context:
This question links percentage discount with percentage markup. The retailer first increases the cost price to set a marked price, and then applies a discount such that the final selling price returns to the original cost price. The task is to determine what markup percentage is needed so that a discount of 32 percent exactly cancels the markup.

Given Data / Assumptions:
Let cost price be C. Marked price is some percentage above C. Discount rate on marked price = 32 percent. Selling price after discount equals cost price. We need to find percentage markup of marked price over cost price.

Concept / Approach:
If marked price is C multiplied by (1 plus markup rate), and discount is 32 percent, then selling price is marked price multiplied by (1 minus 32/100). We are told that this selling price equals C, so we can form an equation and solve for markup rate. This is a classic percentage composition problem that combines increase and decrease.

Step-by-Step Solution:
Let markup rate be m percent. Then marked price M = C * (1 + m/100). Discount is 32 percent, so selling price S = M * (1 - 32/100) = M * 0.68. Given S = C, so C = C * (1 + m/100) * 0.68. Divide both sides by C (C is non zero): 1 = (1 + m/100) * 0.68. So 1 / 0.68 = 1 + m/100. Compute 1 / 0.68 = 100 / 68 = 25 / 17 = 1.470588 approximately. Thus 1.470588 = 1 + m/100, so m/100 = 0.470588. Therefore m = 47.0588 percent, which rounds to 47.05 percent.

Verification / Alternative check:
Assume cost price C = 100 units. With markup 47.05 percent, marked price is 147.05. Discount of 32 percent on 147.05 is 0.68 * 147.05 which equals about 100.0, matching the cost price. So the combination of markup and discount returns the selling price to cost price, confirming the result.

Why Other Options Are Wrong:
A markup of 24 percent or 22.34 percent would make the marked price too low, and after 32 percent discount the selling price would fall below cost price. A markup equal to 32 percent would lead to a loss instead of breaking even. Forty percent markup is still insufficient to balance a 32 percent discount. Only 47.05 percent markup precisely compensates for a 32 percent discount.

Common Pitfalls:
Many learners simply subtract 32 from 100 and conclude that 32 percent markup is enough, which is incorrect because increases and decreases are calculated on different bases. Another mistake is to assume that markup must equal discount. Instead, it is necessary to set up the equation linking cost price, marked price, and selling price correctly.

Final Answer:
The retailer must mark up the goods by approximately 47.05 percent over the cost price in order to sell at cost after a 32 percent discount.