If the cost price of 48 articles is equal to the selling price of 32 articles, what is the profit percentage on each article?

Introduction / Context:
This question compares the total cost price of a larger batch of articles with the total selling price of a smaller batch. From this relationship, you must derive the profit percentage on each article. It is a standard type of problem that tests your understanding of how cost and selling prices scale with the number of articles, and how to convert that into a percentage gain.

Given Data / Assumptions:
- Cost price of 48 identical articles equals the selling price of 32 identical articles. - Each article has the same cost price and selling price. - We must find the profit percentage on each article.

Concept / Approach:
Let cost price of one article be C and selling price of one article be S. Then, total cost price of 48 articles is 48C. Total selling price of 32 articles is 32S. The problem states that 48C = 32S. From this equation we can find the ratio S / C, which indicates how many times the selling price exceeds the cost price. Once we have S in terms of C, profit per article is S - C and profit percentage is (S - C) / C * 100.

Step-by-Step Solution:
Step 1: Let cost price per article be C and selling price per article be S. Step 2: Given that cost of 48 articles = selling price of 32 articles. Step 3: So, 48C = 32S. Step 4: Divide both sides by 16 to simplify: 3C = 2S. Step 5: Express S in terms of C: S = (3 / 2) * C = 1.5 * C. Step 6: Profit per article = S - C = 1.5 * C - C = 0.5 * C. Step 7: Profit percentage = (Profit / C) * 100 = (0.5 * C / C) * 100. Step 8: Profit percentage = 0.5 * 100 = 50%.

Verification / Alternative check:
Assume a convenient cost price, for example C = Rs. 2. Then S = 1.5 * 2 = Rs. 3. Cost of 48 articles = 48 * 2 = Rs. 96. Selling price of 32 articles = 32 * 3 = Rs. 96. This satisfies the given condition. Profit per article is 3 - 2 = Rs. 1, and profit percentage is 1 / 2 * 100 = 50%. This numerical check supports the algebraic result.

Why Other Options Are Wrong:
If the profit were 20%, the selling price would be 1.2 * C, making 48C not equal to 32 * 1.2C. Similarly, 25% profit corresponds to S = 1.25 * C and 75% profit corresponds to S = 1.75 * C. None of these values would make 48C equal to 32S. Only a 50% profit, with S = 1.5 * C, satisfies the given equality between the total cost and total selling price.

Common Pitfalls:
Some learners may wrongly assume that the ratio 48:32 immediately yields the profit percentage without doing the algebra. Others might confuse whether the relationship is between cost total and selling total or between numbers of articles. Carefully setting up the equation 48C = 32S and methodically solving for S / C is the best way to avoid mistakes.

Final Answer:
The trader earns a 50% profit on each article.