What profit percentage is made by selling an article at a certain price, if selling the same article at two-thirds of that price would result in a loss of 20%?

Introduction / Context:
This question relates two selling prices for the same article: one at which there is a loss and another at which there is a profit. The condition involves a multiplicative relation between the two selling prices. The problem tests algebraic reasoning around the cost price and the connection between loss and profit at different selling levels.

Given Data / Assumptions:

• At two-thirds of some reference selling price S, there is a loss of 20%.
• The cost price remains the same in all cases.
• At the full selling price S, there is a profit that we must find as a percentage.
• The article and all prices are assumed to be consistent for a single transaction.

Concept / Approach:
Let the higher selling price be S. Then the lower selling price where there is a loss is (2 / 3) * S. A loss of 20% at this lower price means that (2 / 3) * S equals 80% of the cost price. From this, we can express the cost price in terms of S. Once the cost price is known symbolically, the profit at selling price S is S minus cost price, and profit percentage is this profit divided by cost price times 100.

Step-by-Step Solution:
Step 1: Let higher selling price = S. Lower selling price = (2 / 3)S.Step 2: At lower selling price, there is a loss of 20%, so (2 / 3)S = 0.8 * CP, where CP is cost price.Step 3: Therefore CP = (2S / 3) / 0.8 = (2S / 3) / (4 / 5) = (2S / 3) * (5 / 4) = 10S / 12 = 5S / 6.Step 4: At price S, profit = S - CP = S - 5S / 6 = S / 6.Step 5: Profit percentage = (Profit / CP) * 100 = (S / 6) / (5S / 6) * 100 = (S / 6) * (6 / 5S) * 100 = (1 / 5) * 100 = 20%.

Verification / Alternative check:
Assume CP = Rs. 100 for an intuitive check.A loss of 20% means lower selling price = 80% of CP = Rs. 80.This lower price corresponds to two-thirds of the higher price, so higher selling price S = 80 * 3 / 2 = Rs. 120.Profit at S: 120 - 100 = Rs. 20, so profit percentage = 20 / 100 * 100 = 20%.

Why Other Options Are Wrong:
Profit percentages like 15%, 25%, 30% or 35% would correspond to different relationships between CP and S that do not match the given condition that two-thirds of S yields a 20% loss. Only 20% is consistent with the algebraic and numerical checks.

Common Pitfalls:
One common mistake is to treat the 20% as a profit instead of a loss when setting up the cost price equation. Another is to assume that the difference between 20% loss and some unknown profit can be added directly without considering the multiplicative relationship between prices. Carefully associating each percentage with its correct base (CP or SP) is essential.

Final Answer:
The profit made by selling at the higher price is 20%.