If a merchant offers a discount of 4% on the list price of an article, she incurs a loss of 10% on the cost price. What percentage profit or loss will she make if instead she sells the article at a discount of 20% on the list price?

Introduction / Context:
This problem involves using two different discount scenarios to determine the profit or loss percentage relative to the cost price. The merchant first sells with a small discount and incurs a loss, giving a relationship between list price and cost price. Then the discount is increased, and we must find the new profit or loss percentage. This is a typical question in profit and loss, testing algebraic handling of percentages applied to list price, not directly to cost price.

Given Data / Assumptions:

• Discount of 4% on list price results in a 10% loss on cost price.
• In the second case, discount is 20% on the same list price.
• Let list price be L and cost price be CP.
• We need to compute profit or loss percentage when discount is 20%.

Concept / Approach:
Discount is always calculated on the list price. Profit or loss percentage, however, is computed on the cost price. First, use the 4% discount and 10% loss to derive the ratio between list price and cost price. Then apply a 20% discount to find the new selling price and compare it with the cost price to compute the new loss percentage. This method ensures that all values are consistent and based on the same cost price.

Step-by-Step Solution:
Step 1: Let list price be L.Step 2: With 4% discount, selling price SP1 = 0.96 * L.Step 3: At this selling price, there is a 10% loss, so SP1 = 0.90 * CP.Step 4: Therefore 0.96L = 0.90CP, so CP = (0.96 / 0.90) * L = 1.0666... * L.Step 5: Thus CP is greater than L, meaning the list price is actually below the cost price.Step 6: Now consider a 20% discount. New selling price SP2 = 0.80 * L.Step 7: Loss at SP2 = CP - SP2 = (1.0666... * L) - 0.80L = 0.2666... * L.Step 8: Loss percentage = (Loss / CP) * 100 = (0.2666...L / 1.0666...L) * 100 ≈ 25%.

Verification / Alternative check:
We can set a convenient list price, say L = 100. Then with 4% discount, SP1 = 96. If this is a 10% loss, CP must be 96 / 0.90 ≈ 106.67. With 20% discount, SP2 = 80. Loss at SP2 is 106.67 - 80 = 26.67. Loss percentage is 26.67 / 106.67 ≈ 0.25 or 25%. This numerical confirmation matches the algebraic result and confirms that the second selling situation produces a 25% loss.

Why Other Options Are Wrong:
Option B (4 percent loss) is too small and would correspond to a selling price close to cost price, which is not the case here. Option C (50 percent profit) and option D (26 percent profit) are impossible because both would require the selling price to be above the cost price, but all calculations show that SP2 is well below CP. Only option A accurately reflects a 25 percent loss.

Common Pitfalls:
A frequent mistake is to think that increasing the discount by 16 percentage points from 4% to 20% decreases the profit by exactly the same amount, without recalculating correctly based on list price and cost price. Another pitfall is ignoring the given loss condition to derive CP and instead assuming CP equals list price. Always derive CP first when any profit or loss information is provided, then recompute profit or loss under the new selling condition.

Final Answer:
When the merchant gives a 20% discount on the list price, she makes a 25 percent loss on the cost price.