Consider the following Assertion and Reason : Assertion : 15% discount on the marked price of an item is more than the discount of 10% on the marked price of the same item and a subsequent (successive) discount of 5%. Reason : The subsequent (successive) discount is on the discounted amount. Which one of the following is correct with regard to the above Assertion and Reason ?
- (a)Both the assertion and the reason are individually true and the reason is a correct explanation of the assertion.
- (b)Both the assertion and the reason are individually true, but the reason is not a correct explanation of the assertion.
- (c)Assertion is true, but Reason is false.
- (d)Assertion is false, but Reason is true.
Answer
Why
Correct — A, (a) Both the assertion and the reason are individually true and the reason is a correct explanation of the assertion. Test each part separately, then test whether the second accounts for the first.
THE ASSERTION IS TRUE. Take the marked price as ₹ 100, which is what makes discount arithmetic easy. Single discount of 15 per cent: the customer pays 100 − 15 = ₹ 85, and the discount is ₹ 15. Successive discounts of 10 per cent and then 5 per cent: the first takes the price to ₹ 90; the second is 5 per cent OF 90, that is ₹ 4·50, so the customer pays ₹ 85·50 and the total discount is ₹ 14·50. Since ₹ 15 is more than ₹ 14·50, a straight 15 per cent discount is indeed larger than 10 per cent followed by 5 per cent. The assertion holds.
THE REASON IS TRUE. The second discount in a successive scheme is applied to the price already reduced, not to the marked price. That is what 'on the discounted amount' says, and it is how successive discounts work everywhere.
THE REASON EXPLAINS THE ASSERTION. This is the part that decides between options (a) and (b), and here the link is exact. Because the 5 per cent is charged on ₹ 90 instead of ₹ 100, it yields ₹ 4·50 instead of ₹ 5 — and that shortfall of 50 paise is precisely the gap between ₹ 15 and ₹ 14·50. The reason does not merely happen to be true alongside the assertion; it is the mechanism that produces it.
The general formula makes the same point once and for all. Successive discounts of a per cent and b per cent are equivalent to a single discount of a + b − ab/100. With a = 10 and b = 5 this is 10 + 5 − 0·5 = 14·5 per cent. The subtracted term ab/100 is exactly the part of the second discount that is lost by charging it on a smaller base, and since it is always positive, successive discounts ALWAYS come to less than their sum. That is worth carrying as a rule: a shop offering '10% + 5% off' is offering less than 15 per cent off.
Why the others are wrong
- (b)Both the assertion and the reason are individually true, but the reason is not a correct explanation of the assertion. — The two statements are indeed both true, so this option gets that far; where it fails is on the link. The reason is not an unrelated fact standing beside the assertion — it is the whole cause of it. Charging the second discount on the reduced price is what makes 5 per cent worth ₹ 4·50 rather than ₹ 5, and that is the entire difference the assertion asserts. In this option set the only word separating (a) from (b) is 'not', so read the two to the end before choosing.
- (c)Assertion is true, but Reason is false. — The reason is a correct statement of how successive discounts operate: the second is calculated on the amount left after the first. Nothing in commercial practice or in the arithmetic makes it false. A candidate choosing this has usually verified the assertion, been satisfied, and not troubled to test the reason on its own — which is the one thing an assertion-reason item requires.
- (d)Assertion is false, but Reason is true. — This would require the successive discounts to be worth at least as much as the single 15 per cent. They are worth 14·5 per cent, and by the formula a + b − ab/100 they must always be worth less than a + b. The temptation is to reason that 10 and 5 add to 15 and so the two schemes must be equal — which is exactly the misconception the item is built to expose.
Concept
Discounts are percentages of the marked price, and successive discounts are percentages applied one after another to a shrinking base. If the marked price is M and discounts of a and b per cent are applied in turn, the final price is M(1 − a/100)(1 − b/100), which expands to M[1 − (a + b − ab/100)/100]. Hence the equivalent single discount is a + b − ab/100, always less than a + b. The order of two successive discounts does not matter, since multiplication is commutative. The same structure appears with successive percentage INCREASES, where the equivalent single increase is a + b + ab/100, and with an increase followed by a decrease of the same percentage, which always leaves the amount lower than it started.
Assertion-Reason is a distinct format and needs a distinct routine: judge the assertion alone, judge the reason alone, and only if both are true ask whether the reason explains the assertion. Skipping to the third question is what makes candidates choose between (a) and (b) by feel. This paper carries exactly one item in the format, standing on its own rather than heading a run, and it is worded with the labels Assertion and Reason rather than Statement I and Statement II.
Assertion-Reason is a format with its own routine, and the routine has three steps in a fixed order: judge the assertion alone; judge the reason alone; and only if both stand, ask whether the reason explains the assertion. Candidates who collapse the second and third steps end up choosing between the first two options by feel, and those two options differ by a single word. This paper carries exactly one item in the format, standing on its own rather than heading a run of four, so the routine has to be recalled cold rather than warmed up on a neighbouring question.
Key facts
- A single discount of 15 per cent on ₹ 100 leaves ₹ 85.
- Successive discounts of 10 per cent and 5 per cent on ₹ 100 leave ₹ 85·50, a total discount of 14·5 per cent.
- Successive discounts of a and b per cent equal a single discount of a + b − ab/100.
- The subtracted term ab/100 is the amount lost because the second discount is charged on a reduced base.
- Successive discounts therefore always amount to less than the sum of the two rates.
- The order in which two successive discounts are applied does not change the final price.
- Successive increases of a and b per cent are equivalent to a single increase of a + b + ab/100.
- An increase of p per cent followed by a decrease of p per cent leaves the amount below its original value.
Study next
Common traps
- Adding successive discount rates as though they applied to the same base.
- Choosing between the first two options without asking whether the reason explains the assertion.
- Accepting the assertion and neglecting to test the reason independently.
- Overlooking a single negating word that distinguishes two nearly identical options.
Assertion-Reason items appear in most papers in this family and are usually built on a common misconception — here, that percentages applied in succession can be added. Work the numbers on a base of one hundred, which turns every percentage question of this kind into simple subtraction.
Related PYQs
EPFO_EOAO_2017_Q107The price of an article is increased by 20%. Further, there is a tax of 5% on the increment. If the article costs ₹ 1,331 to the customer, then what was the price of the article before the increase in price?
- (a) ₹ 1,000
- (b) ₹ 1,064
- (c) ₹ 1,100
- (d) ₹ 1,200
Answer(c) ₹ 1,100
The same arithmetic in a commercial setting from the earlier paper — a price increased by a percentage with a tax charged on the increment, where each step is applied to a different base.
Practice
- practice — not a real PYQ
Successive discounts of 20 percent and 10 percent are equivalent to a single discount of which one among the following ?
- (a)28 percent
- (b)30 percent
- (c)25 percent
- (d)32 percent
Answer(a) 28 percent
- practice — not a real PYQ
The price of an article is increased by 10 percent and the new price is then decreased by 10 percent. The final price, compared with the original, is which one among the following ?
- (a)The same
- (b)1 percent higher
- (c)1 percent lower
- (d)10 percent lower
Answer(c) 1 percent lower