A man purchases two articles. He sells one at a loss of 20% and the other at a profit of 25%, and overall he gains 5% on the whole transaction. What is the ratio of their selling prices (first article to second article)?

Introduction / Context:
This is a slightly advanced profit and loss problem involving two articles sold with different loss and profit percentages but yielding an overall gain. The question specifically asks for the ratio of their selling prices rather than cost prices. To solve it, we must express the selling prices in terms of their cost prices and then use the condition on overall profit to derive an equation that links the cost prices. From this, the ratio of selling prices can be computed.

Given Data / Assumptions:

• First article is sold at a loss of 20%.
• Second article is sold at a profit of 25%.
• Overall profit on the two articles together is 5%.
• Cost prices of the two articles may differ.
• We need the ratio of selling prices: SP1 : SP2.

Concept / Approach:
Let the cost prices of the two articles be C1 and C2. Then the corresponding selling prices are SP1 = 0.80C1 (20% loss) and SP2 = 1.25C2 (25% profit). The overall profit of 5% means that total selling price equals 1.05 times total cost price, i.e., SP1 + SP2 = 1.05(C1 + C2). Substituting the expressions for SP1 and SP2 gives a linear relationship between C1 and C2. Once we obtain the ratio C2 : C1, we can then compute SP1 : SP2 in terms of this ratio.

Step-by-Step Solution:
Step 1: Let cost prices be C1 and C2. Step 2: SP1 with 20% loss = 0.80C1. Step 3: SP2 with 25% profit = 1.25C2. Step 4: Overall profit of 5% gives: SP1 + SP2 = 1.05(C1 + C2). Step 5: Substitute SP1 and SP2: 0.80C1 + 1.25C2 = 1.05C1 + 1.05C2. Step 6: Rearrange: 0.80C1 - 1.05C1 + 1.25C2 - 1.05C2 = 0. Step 7: This simplifies to -0.25C1 + 0.20C2 = 0. Step 8: Hence 0.20C2 = 0.25C1, so C2 / C1 = 0.25 / 0.20 = 1.25 = 5/4. Step 9: Take C1 = 4k and C2 = 5k for some k. Step 10: SP1 = 0.80C1 = 0.80 * 4k = 3.2k. Step 11: SP2 = 1.25C2 = 1.25 * 5k = 6.25k. Step 12: Therefore, SP1 : SP2 = 3.2k : 6.25k = 3.2 : 6.25. Step 13: Multiply both terms by 100 to clear decimals: 3.2 : 6.25 = 320 : 625. Step 14: Simplify if possible; 320 and 625 have no common factor other than 5, so divide both by 5: 320/5 = 64, 625/5 = 125, giving 64 : 125.

Verification / Alternative check:
Using C1 = 4k and C2 = 5k with k = 1 for simplicity, total CP = 4 + 5 = 9 units. SP1 = 3.2 units and SP2 = 6.25 units, total SP = 9.45 units. Profit = 9.45 - 9 = 0.45, so profit percent = (0.45 / 9) * 100 = 5%. Thus, the found ratio of selling prices 64 : 125 is fully consistent with the overall 5% profit condition.

Why Other Options Are Wrong:
Ratios such as 4 : 5, 5 : 4, 16 : 25 or 25 : 16 do not satisfy the required balance between individual profit and loss percentages and the overall 5% gain when you compute the corresponding cost price ratios. Only 64 : 125 leads to a combined selling price that is exactly 105% of the total cost price.

Common Pitfalls:
A common mistake is to assume that the cost prices are equal and then attempt to average the profit and loss percentages, which will not give the correct overall 5% gain. Another error is to try and deduce the ratio of selling prices directly from the profit and loss percentages without forming the overall equation SP1 + SP2 = 1.05(C1 + C2). Proper algebraic setup is essential here.

Final Answer:
The ratio of their selling prices is 64 : 125 (first article to second article).