The cash difference between the selling price of an article when it is sold at a profit of 4% and when it is sold at a profit of 6% is Rs 3. What is the ratio of the two selling prices corresponding to these profit percentages?

Introduction / Context:
This question is about profits and selling prices. The cost price is not known directly, but we are given the profit percentages and the absolute difference between the two selling prices. Using the definition of profit percentage in terms of cost price and selling price, we can set up an equation for the unknown cost price, find the two selling prices, and then form their ratio.

Given Data / Assumptions:

• Profit in first case = 4% of cost price.
• Profit in second case = 6% of cost price.
• Difference between the two selling prices = Rs 3.
• We assume the cost price is the same in both situations.

Concept / Approach:
Let the cost price be C. A profit of 4% means the selling price is C * (1 + 4 / 100) = 1.04 * C. A profit of 6% means the selling price is 1.06 * C. The difference between these two selling prices is 1.06 * C − 1.04 * C = 0.02 * C. This difference is given as Rs 3, so we can solve for C. Once we know C, we compute both selling prices and then simplify their ratio.

Step-by-Step Solution:
Let cost price = C. Selling price at 4% profit, S1 = 1.04 * C. Selling price at 6% profit, S2 = 1.06 * C. Given S2 − S1 = 3. So 1.06 * C − 1.04 * C = 3. This gives 0.02 * C = 3. Therefore C = 3 / 0.02 = 150. Then S1 = 1.04 * 150 = 156. S2 = 1.06 * 150 = 159. Required ratio S1 : S2 = 156 : 159. Divide both by 3 to simplify: 156 / 3 = 52 and 159 / 3 = 53. So the ratio is 52 : 53.

Verification / Alternative check:
To verify, compute the profits directly. At selling price 156 with cost price 150, profit is 6, which is 6 / 150 * 100 = 4%. At selling price 159, profit is 9, which is 9 / 150 * 100 = 6%. The difference 159 − 156 = 3 matches the given condition. This confirms that our value of C and the final ratio are correct.

Why Other Options Are Wrong:
51 : 53, 55 : 53 and 54 : 53 would correspond to different numerical selling prices which do not yield a difference of exactly Rs 3 when combined with a common cost price. 25 : 26 is unrelated to the specific 4% and 6% profit structure in this problem and does not match the computed ratio from the actual selling prices.

Common Pitfalls:
A common mistake is to treat the 2% difference in profit as directly equal to Rs 3 and forget that this 2% is of the cost price, not of any arbitrary number. Some learners also incorrectly assume that the ratio of selling prices equals the ratio of percentage profits, which is not true because both profits are based on the same cost price. Correct handling of percentages and careful algebraic steps prevent these errors.

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