A shopkeeper offers three successive discounts of 20%, 10% and 5% on the marked price of an article. What single equivalent rate of discount is equal to this series of discounts?

Introduction / Context:
This problem checks understanding of successive discounts and how they combine into one overall or single equivalent discount. In real life, shops frequently announce multiple discounts one after another, and many customers incorrectly add the percentages. For competitive exams, it is important to know that successive discounts are multiplicative on the remaining price, not additive on the original price. The question asks you to convert three successive discounts into one equivalent discount rate that gives the same final selling price.

Given Data / Assumptions:

• Let the marked price of the article be M rupees.
• First discount: 20% on the marked price.
• Second discount: 10% on the reduced price after the first discount.
• Third discount: 5% on the reduced price after the second discount.
• We need one equivalent discount D% such that applying D% once to M gives the same final selling price.

Concept / Approach:
Successive discounts are applied step by step on the reduced price. If a discount of d% is offered, the remaining price factor is (1 − d/100). For multiple discounts, these factors are multiplied. Once we know the final factor multiplying the marked price, we can convert it back into a single equivalent discount by subtracting the factor from 1 and converting the result into a percentage. This approach avoids the common mistake of simply adding the percentage values of discounts, which is always wrong for successive discounts.

Step-by-Step Solution:
Step 1: Let the marked price be M. Step 2: After first discount of 20%, price factor = 1 − 20/100 = 0.8. New price = 0.8M. Step 3: After second discount of 10%, factor = 1 − 10/100 = 0.9. New price = 0.8M * 0.9 = 0.72M. Step 4: After third discount of 5%, factor = 1 − 5/100 = 0.95. Final selling price = 0.72M * 0.95 = 0.684M. Step 5: The final factor 0.684 means the customer pays 68.4% of marked price. Step 6: Therefore equivalent single discount = (1 − 0.684) * 100% = 0.316 * 100% = 31.6%.

Verification / Alternative check:
As a quick check, note that the sum of given discounts is 20% + 10% + 5% = 35%, so the effective discount must be slightly less than 35%, because each later discount is applied to a smaller base. The answer 31.6% is less than 35% and is close to typical exam values, confirming that our calculations are reasonable. Recomputing the multiplication of factors 0.8, 0.9 and 0.95 gives 0.684 again, so the result is consistent.

Why Other Options Are Wrong:

• 31.5% is very close but does not match the exact computed discount of 31.6%.
• 31% is smaller than the true discount and would produce a slightly higher selling price.
• 31.4% is also close but still differs from the exact value 31.6% obtained from the product 0.8 * 0.9 * 0.95.

Common Pitfalls:
The biggest mistake is to simply add the discounts and say 20% + 10% + 5% = 35%, which ignores compounding. Another error is to apply each discount on the original marked price instead of the reduced price. Some students also round intermediate values too early, which can disturb the final answer in close options. Always multiply the remaining fractions step by step and convert the final factor into a single equivalent discount at the end.

Final Answer:
The single equivalent rate of discount is 31.6%.