A merchant offers a discount of 10% on the list price of an item and incurs a loss of 25%. If instead the item is sold at a discount of 20% on the same list price, what percentage profit or loss will the merchant make?

Introduction / Context:
Here we again relate discount on list price with profit or loss on cost price. The question describes two different discount scenarios on the same list price and specifies that the merchant makes a loss of 25% in the first case. You are asked to determine the profit or loss percentage in the second case. This type of problem checks whether you can consistently track the relationship between list price, cost price, and selling price.

Given Data / Assumptions:

• Let the list price be L.
• First case: discount = 10%; selling price SP1 = 0.90L.
• Under SP1, there is a loss of 25%.
• Second case: discount = 20%; selling price SP2 = 0.80L.
• We need the profit or loss percentage in the second scenario.
• No extra costs, commissions, or taxes are involved.

Concept / Approach:
We use:

• Loss of 25% implies SP1 = 0.75 * CP.
• We know SP1 in terms of L, so 0.90L = 0.75 * CP.
• From this, we find CP in terms of L.
• Then we compare SP2 = 0.80L with CP to compute the new profit or loss percentage.

The formula remains profit or loss percentage = ((SP - CP) / CP) * 100 (positive for profit, negative for loss).

Step-by-Step Solution:
Step 1: First case SP1 = 0.90L (10% discount). Step 2: Loss of 25% means SP1 = 0.75 * CP. Step 3: Equate 0.90L = 0.75 * CP. Step 4: Solve for CP: CP = 0.90L / 0.75 = 1.20L. Step 5: Second case SP2 = 0.80L (20% discount). Step 6: Profit or loss fraction = (SP2 - CP) / CP = (0.80L - 1.20L) / 1.20L. Step 7: Simplify numerator: 0.80L - 1.20L = -0.40L. Step 8: So fraction = (-0.40L) / 1.20L = -1 / 3. Step 9: Percentage loss = (1 / 3) * 100 ≈ 33.33% loss.

Verification / Alternative check:
Let L = Rs 300 for easier numbers:

• First case: SP1 = 0.90 * 300 = Rs 270.
• Given 25% loss, CP = SP1 / 0.75 = 270 / 0.75 = Rs 360.
• Second case: SP2 = 0.80 * 300 = Rs 240.
• Loss = CP - SP2 = 360 - 240 = Rs 120.
• Loss percentage = 120 / 360 * 100 = 33.33% loss.

This numerical example agrees exactly with the algebraic solution and confirms that the loss is about one third of the cost price.

Why Other Options Are Wrong:
5 percent profit: Incorrect because SP2 is less than CP, so there cannot be any profit.
90 percent profit: This would require SP2 to be almost double CP, which is impossible here because SP2 is lower than CP.
20 percent profit: Again implies SP2 greater than CP, which conflicts with the computed values.
Only an approximate 33.33 percent loss matches the relationship between list price, cost price, and new selling price.

Common Pitfalls:
Learners sometimes assume CP is equal to the list price, which is never stated in the question. Another mistake is adding or subtracting percentages directly (for example, treating 10% discount and 25% loss as related linearly). It is essential to build equations based on how each percentage applies to either the list price or cost price, and then solve logically instead of relying on intuition alone.

Final Answer:
When the item is sold at a 20% discount on the list price, the merchant incurs a 33.33 percent loss.