Selling an article at a profit of 20%, Aman receives Rs. 400 more than if he sells it at a loss of 20%. What is the cost price of the article?

Introduction / Context:
This question compares two different selling scenarios of the same article: one with a 20% profit and the other with a 20% loss. The difference between the two selling prices is given in rupees. This is a classic type of problem where we can exploit the symmetry in percentages above and below cost price to derive the cost price using a simple linear equation.

Given Data / Assumptions:

• Profit scenario: article is sold at 20% profit.
• Loss scenario: article is sold at 20% loss.
• Difference in selling prices between the two scenarios = Rs. 400.
• The underlying cost price is the same in both scenarios.
• We must find the cost price (CP) of the article.

Concept / Approach:
Let CP be C. When sold at 20% profit, the selling price is 1.20C. When sold at 20% loss, the selling price is 0.80C. The difference between these two selling prices is given as Rs. 400. Therefore, 1.20C - 0.80C = 0.40C must equal 400. Solving this simple equation gives us the cost price. Because the percentages are symmetrically placed around the cost price, the difference comes out as a neat fraction of CP.

Step-by-Step Solution:
Step 1: Let cost price be C rupees. Step 2: Selling price at 20% profit, SP1 = C * (1 + 20/100) = 1.20C. Step 3: Selling price at 20% loss, SP2 = C * (1 - 20/100) = 0.80C. Step 4: Given that SP1 - SP2 = Rs. 400. Step 5: Substitute SP1 and SP2: 1.20C - 0.80C = 400. Step 6: Simplify left side: (1.20 - 0.80)C = 0.40C. Step 7: So 0.40C = 400. Step 8: Therefore, C = 400 / 0.40 = 400 * (10 / 4) = 400 * 2.5 = 1,000.

Verification / Alternative check:
If CP = Rs. 1,000, then SP1 at 20% profit = 1.20 * 1,000 = Rs. 1,200. SP2 at 20% loss = 0.80 * 1,000 = Rs. 800. Difference between the two selling prices = 1,200 - 800 = Rs. 400, which matches the given information exactly and confirms the cost price.

Why Other Options Are Wrong:
If CP were Rs. 900, for example, the difference in selling prices would be 0.40 * 900 = Rs. 360, not 400. Similarly, cost prices of Rs. 1,020, Rs. 1,140 or Rs. 1,210 would lead to differences of 0.40C that are not equal to Rs. 400. Only Rs. 1,000 makes 0.40C exactly equal to 400.

Common Pitfalls:
Some learners wrongly take 20% of Rs. 400 or try to treat 400 as a percentage directly. Others confuse the base for percentages and do not realise that both 20% profit and 20% loss are taken on the same cost price. Writing out both selling prices in terms of cost price and then subtracting them is the cleanest and most reliable method.

Final Answer:
The cost price of the article is Rs. 1000.