A shopkeeper fixes the marked price of an item 35% above its cost price; what percentage of discount should be allowed on the marked price so that he still gains 8% on the cost price?

Introduction / Context:
This question explores the relationship between cost price, marked price, discount and final gain. The shopkeeper increases the marked price above cost and later allows a discount but still wants an overall profit. Such questions are common in commercial mathematics because they reflect real retail pricing strategies and test understanding of how multiple percentage changes interact.

Given Data / Assumptions:

• Marked price (MP) is 35% above cost price (CP).
• Desired gain on cost price = 8%.
• A single discount percentage is applied on the marked price.
• We need to find the discount percentage that ensures an 8% profit overall.
• No additional taxes or overheads are mentioned.

Concept / Approach:
We can assume a convenient cost price, for example CP = Rs. 100. Then the marked price becomes 35% more than 100, that is Rs. 135. Let the discount rate on this marked price be d%. After discount, selling price SP will be MP * (1 - d / 100). We want SP to be equal to CP multiplied by 1.08, that is, 108 when CP is 100. Equating SP to 108 allows us to solve for d, the required discount percentage.

Step-by-Step Solution:
Step 1: Assume cost price CP = Rs. 100 for convenience.Step 2: Marked price MP = CP * (1 + 35 / 100) = 100 * 1.35 = Rs. 135.Step 3: Let discount rate be d%. Then selling price SP = MP * (1 - d / 100).Step 4: The desired gain is 8%, so SP should also equal CP * 1.08 = 100 * 1.08 = Rs. 108.Step 5: Set up the equation: 135 * (1 - d / 100) = 108.Step 6: Divide both sides by 135: 1 - d / 100 = 108 / 135.Step 7: Simplify 108 / 135 = 0.8.Step 8: So 1 - d / 100 = 0.8, which implies d / 100 = 0.2.Step 9: Therefore d = 0.2 * 100 = 20%.

Verification / Alternative check:
If CP = 100 and discount is 20%, then MP = 135 and SP = 135 * 0.80 = 108. The profit is SP - CP = 108 - 100 = 8 rupees, which is an 8% gain on the cost price. This matches the desired condition perfectly and confirms that a 20% discount is the correct value.

Why Other Options Are Wrong:

• 18%: This would give SP = 135 * 0.82 = 110.7, resulting in a 10.7% profit, which is higher than required.
• 22%: This would give SP = 135 * 0.78 = 105.3, which corresponds to a 5.3% profit, less than the desired 8%.
• 24%: This would give SP = 135 * 0.76 = 102.6, only a 2.6% profit.

Common Pitfalls:

• Applying the 8% gain on marked price instead of cost price, which changes the target selling price.
• Confusing discount percentage with profit percentage and adding or subtracting them incorrectly.
• Making arithmetic mistakes when simplifying 108 / 135 or solving for the discount.

Final Answer:
The shopkeeper should allow a discount of 20% on the marked price to gain 8% on cost price.