The cash difference between the selling prices of an article at a profit of 4% and at a profit of 6% is Rs. 3; what is the ratio of these two selling prices?

Introduction / Context:
This profit and loss question relates the difference between two selling prices at different profit percentages on the same article. The fixed cash difference in selling prices allows us to determine the cost price and then the actual selling prices. Finally, the ratio of these two selling prices is asked. Such problems strengthen the understanding of how small changes in profit percentage translate into cash differences.

Given Data / Assumptions:

• An article is sold at a profit of 4% in one case.
• The same article is sold at a profit of 6% in another case.
• The difference between these two selling prices is Rs. 3.
• Cost price of the article is the same in both cases.
• We need to find the ratio of the selling price at 4% profit to the selling price at 6% profit.

Concept / Approach:
Let the cost price of the article be CP. Then, selling price at 4% profit is SP1 = 1.04 * CP and selling price at 6% profit is SP2 = 1.06 * CP. The difference SP2 minus SP1 equals 0.02 * CP, which is given as Rs. 3. From this relation, we can determine CP. Once CP is known, SP1 and SP2 can be calculated and their ratio simplified to the lowest terms. This yields the required ratio.

Step-by-Step Solution:
Step 1: Let cost price be CP rupees.Step 2: Selling price at 4% profit, SP1 = 1.04 * CP.Step 3: Selling price at 6% profit, SP2 = 1.06 * CP.Step 4: Difference SP2 - SP1 = (1.06 * CP) - (1.04 * CP) = 0.02 * CP.Step 5: Given that this difference equals Rs. 3, so 0.02 * CP = 3.Step 6: Therefore CP = 3 / 0.02 = 150.Step 7: Compute SP1 = 1.04 * 150 = 156.Step 8: Compute SP2 = 1.06 * 150 = 159.Step 9: Ratio SP1 : SP2 = 156 : 159.Step 10: Simplify this ratio by dividing both numbers by their greatest common divisor which is 3: 156 / 3 = 52 and 159 / 3 = 53.Step 11: Therefore, the ratio of the selling prices is 52 : 53.

Verification / Alternative check:
If the ratio is 52 : 53, assume SP1 = 52k and SP2 = 53k for some constant k. The difference is k. For this to equal Rs. 3, k must be 3. Then SP1 = 156 and SP2 = 159. Check also that these correspond to 4% and 6% profit on the same cost price. If CP is 150, 4% profit gives SP1 = 150 * 1.04 = 156 and 6% profit gives SP2 = 150 * 1.06 = 159. This confirms that our values and the resulting ratio are correct.

Why Other Options Are Wrong:

• 51:52, 53:54, 54:55: None of these ratios are consistent with the known difference of Rs. 3 and the given profits of 4% and 6%. They either lead to a different cost price or a different cash difference.

Common Pitfalls:

• Taking the difference of percentages 6% and 4% but not properly translating that into 0.02 * CP in rupees.
• Incorrectly solving 0.02 * CP = 3 and getting the wrong cost price.
• Forgetting to simplify the final ratio to its lowest terms.

Final Answer:
The ratio of the two selling prices at 4% and 6% profit is 52:53.