If the cost price of an article is 75% of its selling price, then by how many percent does the selling price exceed the cost price, that is, what is the profit percentage on the cost price?

Introduction / Context:
This question is about the relationship between cost price and selling price when one is given as a percentage of the other. It checks if you can correctly convert a statement like “cost price is 75% of selling price” into a profit percentage. Such problems are very common in profit and loss chapters and they test whether you understand which quantity is the base for the percentage being described. Careful algebra quickly leads to the correct profit percentage without confusion.

Given Data / Assumptions:
- Cost price (CP) is 75% of selling price (SP). - That means CP = 0.75 * SP. - We must find the profit percentage on the cost price. - All values are conceptual; no actual rupee amount is needed.

Concept / Approach:
Profit percentage is always calculated using cost price as the base: Profit% = (Profit / CP) * 100, where Profit = SP - CP. The statement CP = 75% of SP can be rewritten in algebraic form and used to express SP and CP in terms of a convenient variable. Typically, we assume a simple numerical value for SP (for example, Rs. 100) to simplify the arithmetic. Once we have CP and SP in that form, profit and profit percentage become straightforward to compute. This approach is particularly useful for mental math in exams.

Step-by-Step Solution:
Let selling price (SP) be Rs. 100 for convenience. Given that cost price is 75% of selling price, CP = 75% of 100 = Rs. 75. Profit = SP - CP = 100 - 75 = Rs. 25. Profit percentage is calculated on CP: Profit% = (Profit / CP) * 100. So, Profit% = (25 / 75) * 100. Compute 25 / 75 = 1 / 3. Thus, Profit% = (1 / 3) * 100 ≈ 33.33%.

Verification / Alternative check:
We can work symbolically: Let SP = S, then CP = 0.75S. Profit = S - 0.75S = 0.25S. Profit% = (0.25S / 0.75S) * 100 = (0.25 / 0.75) * 100. Simplify 0.25 / 0.75 = 1 / 3, so profit% = (1 / 3) * 100 ≈ 33.33%. This symbolic method perfectly matches the earlier numeric approach.

Why Other Options Are Wrong:
- 23.47% and 25% are lower than the correct value; they arise from incorrect division or using selling price as the base. - 20% assumes a smaller difference between SP and CP than actually exists in the given relationship. - 30% is close but still incorrect; it results from approximate mental math that ignores exact fractions.

Common Pitfalls:
A common mistake is to treat 75% as the profit percentage instead of a relationship between CP and SP. Another error is to calculate profit percentage using selling price as the base, giving (25 / 100) * 100 = 25%, which is incorrect. Students sometimes forget that profit is always calculated relative to cost price, not relative to selling price.

Final Answer:
The selling price exceeds the cost price by a profit of approximately 33.33%.