A loss of 20% calculated on the selling price of an article is equal to a loss of x% calculated on its cost price. What is the value of x?

Introduction / Context:
This question explores the difference between calculating percentage loss on selling price and on cost price. The same absolute loss amount can represent different percentages depending on which base value is used. It is a slightly conceptual profit and loss question that requires algebraic manipulation and understanding of definitions.

Given Data / Assumptions:
- Loss is 20% of the selling price. - The same loss is x% of the cost price. - We assume one single article with cost price C and selling price S. - Loss is positive and both prices are greater than zero.

Concept / Approach:
Let cost price be C and selling price be S. Loss = C - S. Given that this loss is 20% of selling price, we have:
C - S = 0.20 * S We also know that the same loss equals x% of cost price:
C - S = (x / 100) * C We solve these equations to express x in terms of known constants. The key idea is to first find the relationship between C and S and then compute x from that relationship.

Step-by-Step Solution:
Step 1: From C - S = 0.20S, we get C = S + 0.20S = 1.20S. Step 2: Therefore S = C / 1.20. Step 3: Loss L = C - S = C - C / 1.20. Step 4: L = C * (1 - 1 / 1.20) = C * (1 - 0.8333...) = C * 0.1666..., which is C / 6. Step 5: Since L = (x / 100) * C, we have C / 6 = (x / 100) * C. Step 6: Cancel C on both sides to get 1 / 6 = x / 100, so x = 100 / 6 = 16.666..., which is 16 2/3%.

Verification / Alternative check:
Take an example. Let selling price S = Rs 120. Then loss is 20% of 120, that is Rs 24. So cost price C = S + loss = 120 + 24 = Rs 144. Now loss as a percentage of cost price is 24 / 144 * 100 = 16.666..., which is 16 2/3%. This matches our algebraic solution.

Why Other Options Are Wrong:
Option 20% repeats the given percentage on selling price, not on cost price, and does not satisfy the equations. Option 20 with no percent sign is incomplete. Option 16 understates the loss relative to cost price and would not give equal loss amounts in both definitions.

Common Pitfalls:
Many students incorrectly assume that the same percentage applies to both cost price and selling price, which is not true. Another mistake is to treat x as 20 directly without doing the algebra. Always derive the relationship between cost price and selling price when the loss is given as a fraction of one of them.

Final Answer:
The required value of x is 16 2/3%.