Profit and Loss – Relating unequal gain and loss, then targeting a 20% profit: On selling an article at ₹ 530, the gain is 20% more than the loss incurred when selling the same article at ₹ 475. Find the cost price first, and then determine the selling price required to earn a 20% profit.

Introduction / Context:
This problem blends two profit–loss scenarios for the same article. We compare a gain at one selling price with a loss at another selling price. Once the cost price (CP) is deduced from the relationship, we compute the selling price (SP) that would yield a 20% gain. The core idea is to express both profit and loss in terms of CP so the equation becomes linear and easy to solve.

Given Data / Assumptions:

• Selling price with gain = ₹ 530
• Selling price with loss = ₹ 475
• Gain at ₹ 530 is 20% more than the loss at ₹ 475
• We want SP for a 20% profit on CP

Concept / Approach:
Let CP = x. Profit at ₹ 530 is (530 − x). Loss at ₹ 475 is (x − 475). The statement “gain is 20% more than the loss” means 530 − x = 1.20 * (x − 475). After finding x, compute the SP for 20% profit as 1.20 * x.

Step-by-Step Solution:
530 − x = 1.20 * (x − 475)530 − x = 1.20x − 5701100 = 2.20x ⇒ x = 500Required SP for 20% gain = 1.20 * 500 = ₹ 600

Verification / Alternative check:
At CP = ₹ 500: loss at ₹ 475 = ₹ 25; gain at ₹ 530 = ₹ 30. Indeed, ₹ 30 is 20% more than ₹ 25. Therefore the relationship is satisfied and the computed CP is consistent.

Why Other Options Are Wrong:
₹ 700 and ₹ 650 overshoot the 20% target on CP 500. ₹ 500 equals CP, implying 0% gain. ₹ 900 is far too high.

Common Pitfalls:
Confusing “20% more than the loss” with “gain minus loss equals 20%.” Always convert the English statement into a precise multiplicative equation.

Final Answer:
₹ 600