A trader sells an article at a loss of 5%. When he increases the selling price by Rs. 65, he instead gains 3.33% (that is, one-thirtieth of the cost price). If he sells the same article at Rs. 936, what is his profit percentage?

Introduction / Context:
This question links two different selling prices with two different profit or loss percentages to determine the cost price of an article. Once the cost price is known, a third selling price is used to compute the profit percentage. The problem tests algebraic setup and understanding that small changes in selling price can shift a transaction from loss to profit.

Given Data / Assumptions:

• At the first selling price, there is a loss of 5%.
• When the selling price is increased by Rs. 65, there is a gain of 3.33% (approximately one-thirtieth of cost).
• The same article is then sold at Rs. 936.
• We must find the profit percentage when the selling price is Rs. 936.

Concept / Approach:
Let the cost price be C. A loss of 5% means the first selling price S1 = 0.95C. A profit of 3.33% (which is 1 / 30 of C) means the second selling price S2 = C * (1 + 1 / 30) = 31C / 30. We know S2 = S1 + 65. Solving this equation gives C. Finally, profit at selling price Rs. 936 is (936 - C), so profit percentage is (936 - C) / C * 100.

Step-by-Step Solution:
Step 1: Let cost price be C rupees.Step 2: First selling price with 5% loss: S1 = 0.95C.Step 3: Second selling price with 3.33% gain: S2 = C * (1 + 1 / 30) = 31C / 30.Step 4: Given that S2 = S1 + 65, so 31C / 30 = 0.95C + 65.Step 5: Convert 0.95C to a fraction: 0.95C = 19C / 20, so 31C / 30 = 19C / 20 + 65.Step 6: Bring terms involving C together: 31C / 30 - 19C / 20 = 65.Step 7: Compute left side with common denominator 60: (62C - 57C) / 60 = 5C / 60 = C / 12 = 65, so C = 780.Step 8: When sold at Rs. 936, profit = 936 - 780 = 156.Step 9: Profit percentage = 156 / 780 * 100 = 0.2 * 100 = 20%.

Verification / Alternative check:
First selling price at 5% loss: S1 = 0.95 * 780 = 741 rupees.Second selling price should be S1 + 65 = 741 + 65 = 806 rupees.Check 3.33% gain: 806 - 780 = 26, and 26 / 780 * 100 ≈ 3.33%, confirming C = 780 is correct.Then at Rs. 936, profit percentage is indeed 156 / 780 * 100 = 20%.

Why Other Options Are Wrong:
Profit rates like 15%, 16.67%, or 12.5% correspond to smaller profits at Rs. 936 than the calculated 156 rupees. The option “Data insufficient” is incorrect because the two conditions involving a Rs. 65 change and two percentages uniquely determine the cost price. Only 20% matches the computed value exactly.

Common Pitfalls:
Some learners approximate 3.33% as 3% or 3.5%, which leads to slightly wrong equations and cost prices. Others forget to convert decimals like 0.95 into exact fractions, causing algebraic mistakes. Being precise about 3.33% as 1 / 30 of the cost price and carefully handling the equation ensure the correct result.

Final Answer:
When the article is sold for Rs. 936, the trader's profit percentage is 20%.