Percentages, profit and loss, discount, simple and compound interest: previous-year questions
45 real UPSC Prelims questions (2011–2026) with the official answer and a clear explanation. 12 of them came in the last five papers.
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When a runner was crossing the 12 km mark, she was informed that she had completed only 80% of the race. How many kilometres was the runner supposed to run in this event?
Answer (b) & explanation
The race was 15 km long (b). Step 1: 12 km is 80% of the whole race. Step 2: so the full race = 12 ÷ 0·8 = 12 × 10 ÷ 8 = 15 km. Step 3: check: 80% of 15 = 12. It works. Option (a), 14, and option (c), 16, do not give 12 at 80%. Option (d), 16·5, comes from adding 20% of 12 to 12 plus a little, which is the wrong base.
Tip: if a part and its percentage are given, whole = part ÷ percentage. A common mistake is to add 20% of 12 (giving 14·4); the remaining 20% is of the whole race, not of the part already run.
Answer as per the official UPSC answer key.
Raju has ₹ 9000 with him and he wants to buy a mobile handset; but he finds that he has only 75% of the amount required to buy the handset. Therefore, he borrows ₹ 2000 from a friend. Then
Answer (a) & explanation
Raju still does not have enough money (a). Step 1: ₹ 9000 is 75% of the price. So the price = 9000 ÷ 0·75 = ₹ 12,000. Step 2: he borrows ₹ 2000 and now has 9000 + 2000 = ₹ 11,000. Step 3: he is still ₹ 1000 short. So (a) is right. Options (b), (c) and (d) all assume he has enough. They come from working out 25% of 9000 (₹ 2250) instead of 25% of the price.
Tip: the missing 25% is of the price, not of what he has. Find the whole first (part ÷ percentage), then compare.
75% less would be different. Here ₹ 9000 is 75% of the price.
Answer as per the official UPSC answer key.
In an examination, A has scored 20 marks more than B. If B has scored 5% less marks than A, how much has B scored?
Answer (b) & explanation
B scored 380 marks (b). Step 1: B scored 5% less than A. The gap of 20 marks is 5% of A's marks. Step 2: so A = 20 ÷ 0·05 = 400. Step 3: B = 400 − 20 = 380. Check: 5% of 400 is 20. It works. Option (c), 400, is A's score. Option (a), 360, subtracts twice. Option (d) does not fit.
Tip: "5% less than A" means the base is A. So the difference equals 5% of A. Finding A first is the key step. A quick check of each option against both facts also works: only 380 and 400 differ by 20 with a 5% gap based on the larger number.
400 is A's score, not B's.
Answer as per the official UPSC answer key.
Rakesh had money to buy 8 mobile handsets of a specific company. But the retailer offered very good discount on that particular handset. Rakesh could buy 10 mobile handsets with the amount he had. What was the discount the retailer offered?
Answer (b) & explanation
The discount was 20% (b). Step 1: the same money buys 10 handsets instead of 8. So the new price is 8/10 of the old price. Step 2: the price fell by 2/10 = 20%. Step 3: check with numbers: say the money is ₹ 80,000. Old price = 10,000; new price = 8000. Discount = 2000 ÷ 10,000 = 20%. Option (c), 25%, divides by the new price (2000 ÷ 8000). Options (a) and (d) do not fit.
Tip: a discount is always measured on the original price. Buying 25% more items means a 20% price cut, not 25%. Remember: 1 − 8/10 = 20%.
25% is the rise in quantity, not the fall in price.
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Answer as per the official UPSC answer key.
A and B are two heavy steel blocks. If B is placed on the top of A, the weight increases by 60%. How much weight will reduce with respect to the total weight of A and B, if B is removed from the top of A?
Answer (d) & explanation
The weight falls by 37·5% (d). Step 1: take A = 100 for ease. Putting B on top raises the weight by 60%, so B = 60 and the total = 160. Step 2: removing B takes off 60. Step 3: as a share of the total of 160: 60 ÷ 160 = 3/8 = 37·5%. Option (a), 60%, uses A as the base instead of the total. Options (b) and (c) are also wrong.
Tip: the base changes everything. A rise of 60% on 100 is a fall of 37·5% on 160. To reverse an increase of x%, the fall is x ÷ (100 + x) × 100. Here 60 ÷ 160 × 100 = 37·5%. Choosing 100 as the starting value makes such sums easy.
the base changes: 60% of A is not 60% of (A + B).
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Answer as per the official UPSC answer key.
A student has to get 40% marks to pass in an examination. Suppose he gets 30 marks and fails by 30 marks, then what are the maximum marks in the examination?
Answer (c) & explanation
The maximum marks are 150 (c). Step 1: he got 30 marks and failed by 30, so the pass mark is 30 + 30 = 60. Step 2: the pass mark is 40% of the maximum. Step 3: maximum = 60 ÷ 0.4 = 150. Check: 40% of 150 = 60. Option (a), 100, would make the pass mark 40. Option (b), 120, gives 48. Option (d), 300, gives 120.
Tip: "failed by x marks" means he was x marks short of the pass mark. Add it to his score to get the pass mark. Then divide by the pass percentage to find the total.
Answer as per the official UPSC answer key.
A shopkeeper sells an article at ₹ 40 and gets X% profit. However, when he sells it at ₹ 20, he faces same percentage of loss. What is the original cost of the article?
Answer (c) & explanation
The cost price is ₹ 30 (c). Step 1: the profit % and the loss % are equal and are worked out on the same cost price. So the rupee profit equals the rupee loss. Step 2: profit = 40 − C and loss = C − 20. Set them equal: 40 − C = C − 20, so 2C = 60 and C = ₹ 30. Step 3: check: selling at 40 gives a profit of 10, which is 33⅓% of 30. Selling at 20 gives a loss of 10, also 33⅓%. Option (b), ₹ 20, gives zero loss. Options (a) and (d) also fail.
Tip: when profit % and loss % are the same, the cost price is simply the average of the two selling prices: (40 + 20) ÷ 2 = 30.
Answer as per the official UPSC answer key.
A person bought a refrigerator worth ₹ 22,800 with 12·5% interest compounded yearly. At the end of first year he paid ₹ 8,650 and at the end of second year ₹ 9,125. How much will he have to pay at the end of third year to clear the debt?
Answer (d) & explanation
He must pay ₹ 11,250 at the end of the third year (d). Step 1: 12·5% = 1/8. So each year the amount owed is multiplied by 9/8, and then the payment is subtracted. Step 2: year 1: 22,800 × 9/8 = 25,650. Minus 8,650 leaves 17,000. Year 2: 17,000 × 9/8 = 19,125. Minus 9,125 leaves 10,000. Step 3: year 3: 10,000 × 9/8 = ₹ 11,250. This clears the debt. Option (b), ₹ 10,000, forgets the third year's interest. Options (a) and (c) do not fit.
Tip: change common percentages into fractions: 12·5% = 1/8, 20% = 1/5, 25% = 1/4. Multiplying by 9/8 is far quicker than working out 12·5% each time.
₹ 10,000 is the balance before the third year's interest.
Answer as per the official UPSC answer key.
If there is a policy that 1/3rd of a population of a community has migrated every year from one place to some other place, what is the leftover population of that community after the sixth year, if there is no further growth in the population during this period?
Answer (d) & explanation
The leftover population is 64/729 of the original (d). Step 1: each year one-third leave, so two-thirds stay. The cut applies to the population left that year, not the original. Step 2: after one year, 2/3 remains. After two years, 2/3 × 2/3 = 4/9. Keep multiplying. Step 3: after six years: (2/3)⁶ = 64 ÷ 729 = 64/729. Option (b), 32/243, is (2/3)⁵, five years. Option (a), 16/243, and option (c), 32/729, are mixed-up powers.
Tip: a fixed fraction lost each year works like compound interest in reverse. Multiply by the "staying" fraction once for each year. 2⁶ = 64 and 3⁶ = 729.
(a) and (b) are (2/3)4 and (2/3)5 written over 243. Count six years.
Answer as per the official UPSC answer key.
Gopal bought a cell phone and sold it to Ram at 10% profit. Then Ram wanted to sell it back to Gopal at 10% loss. What will be Gopal's position if he agreed?
Answer (c) & explanation
Gopal gains 1% (c). Step 1: suppose Gopal first paid Rs 100 for the phone. He sold it to Ram at 10% profit, for Rs 110. Step 2: Ram sells it back at 10% loss on his own price. 10% of 110 = 11, so the price is Rs 99. Step 3: Gopal now has the same phone for Rs 99 instead of Rs 100. That is Rs 1 less on every Rs 100 he first spent. That is a gain of 1%. Option (a), no loss no gain, is the trap of thinking 10% up and 10% down cancel. Options (b) and (d) do not fit.
Tip: percentages up and down use different bases, so they never simply cancel.
Counting all the cash (+110 − 100 − 99), some say Gopal is better off by ₹ 11; the official answer is (c), as printed.
Answer as per the official UPSC answer key.
P = (40% of A) + (65% of B) and Q = (50% of A) + (50% of B), where A is greater than B. In this context, which of the following statements is correct?
Answer (d) & explanation
Nothing can be concluded with certainty (d). Step 1: P − Q = (0.40A + 0.65B) − (0.50A + 0.50B) = 0.15B − 0.10A. Step 2: the sign depends on how much bigger A is than B. If A = 1.2B, then 0.15B − 0.12B is positive, so P > Q. Step 3: if A = 2B, then 0.15B − 0.20B is negative, so P < Q. If A = 1.5B, P = Q. So options (a), (b) and (c) are each possible but not certain.
Tip: when an answer depends on values you are not given, test two very different cases. If they give different results, choose "cannot be concluded".
Answer as per the official UPSC answer key.
In a city, 12% of households earn less than ₹ 30,000 per year, 6% households earn more than ₹ 2,00,000 per year, 22% households earn more than ₹ 1,00,000 per year and 990 households earn between ₹ 30,000 and ₹ 1,00,000 per year. How many households earn between ₹ 1,00,000 and ₹ 2,00,000 per year?
Answer (b) & explanation
240 households earn between Rs 1 lakh and Rs 2 lakh (b). Step 1: 12% earn below Rs 30,000 and 22% earn above Rs 1 lakh. So the middle group, Rs 30,000 to Rs 1 lakh, is 100 − 12 − 22 = 66%. Step 2: 66% = 990 households, so 1% = 15 households. Step 3: those above Rs 1 lakh are 22%, and those above Rs 2 lakh are 6%. So Rs 1 lakh to Rs 2 lakh = 22 − 6 = 16% = 16 × 15 = 240. Options (a), (c) and (d) do not fit.
Tip: "more than Rs 1 lakh" includes "more than Rs 2 lakh". Subtract the inner group to get the band in between.
the 6% earning over ₹ 2 lakh is inside the 22%.
Answer as per the official UPSC answer key.
Anita's mathematics test had 70 problems carrying equal marks i.e., 10 arithmetic, 30 algebra and 30 geometry. Although she answered 70% of the arithmetic, 40% of the algebra and 60% of the geometry problems correctly, she did not pass the test because she got less than 60% marks. The number of more questions she would have to answer correctly to earn a 60% passing marks is :
Answer (b) & explanation
She needed 5 more correct answers (b). Step 1: count her correct answers. Arithmetic: 70% of 10 = 7. Algebra: 40% of 30 = 12. Geometry: 60% of 30 = 18. Total = 37. Step 2: passing needs 60% of 70 = 42 correct answers. Step 3: she needs 42 − 37 = 5 more. Option (a), 1, is too few. Option (c), 7, and option (d), 9, are too many. Since all questions carry equal marks, you can count questions instead of marks.
Tip: work out each part separately and then add. Do not average the percentages (70, 40 and 60), because each section has a different number of questions.
averaging the three percentages ignores the different section sizes.
Answer as per the official UPSC answer key.
Two numbers X and Y are respectively 20% and 28% less than a third number Z. By what percentage is the number Y less than the number X?
Answer (b) & explanation
Y is 10% less than X (b). Step 1: take Z = 100. Then X is 20% less, so X = 80. Y is 28% less, so Y = 72. Step 2: Y is 80 − 72 = 8 less than X. Step 3: the question asks how much less than X, so the base is X. 8 ÷ 80 = 10%. Option (d), 8%, is the trap: it divides by 100, the base of Z. Option (a), 12%, and option (c), 9%, do not fit.
Tip: in percentage comparisons, always divide by the number after "than". Taking 100 as the starting value makes these questions quick.
28 – 20 = 8% uses Z as the base.
Answer as per the official UPSC answer key.
A person allows a 10% discount for cash payment from the marked price of a toy and still he makes a 10% gain. What is the cost price of the toy which is marked Rs. 770?
Answer (c) & explanation
The cost price is Rs 630 (c). Step 1: the marked price is Rs 770. A 10% discount gives a selling price of 90% of 770 = Rs 693. Step 2: he still makes a 10% profit, so 693 is 110% of the cost price. Step 3: cost price = 693 ÷ 1.1 = Rs 630. Check: 630 + 10% = 693. Option (a), Rs 610, and options (b) and (d) do not fit. A common mistake is to think a 10% discount and a 10% gain cancel out, giving 770.
Tip: discount is always on the marked price, and profit is always on the cost price. Keep the two bases separate.
Answer as per the official UPSC answer key.
There is an order of 19000 quantity of a particular product from a customer. The firm produces 1000 quantity of that product per day out of which 5% are unfit for sale. In how many days will the order be completed?
Answer (c) & explanation
The order takes 20 days (c). Step 1: the firm makes 1000 pieces a day, but 5% are unfit. Good pieces per day = 95% of 1000 = 950. Step 2: the customer needs 19,000 good pieces. Days = 19,000 ÷ 950 = 20. Step 3: option (b), 19 days, is the trap: it divides 19,000 by 1000 and forgets the faulty pieces. Option (a), 18, is even less. Option (d), 22, is too many.
Tip: always work with the useful output, after removing waste or defects. A quick check: 950 × 20 = 19,000, which matches the order exactly.
Answer as per the official UPSC answer key.
In a test, a candidate attempted only 8 questions and secured 50% marks in each of the questions. If he obtained a total of 40% in the test and all questions in the test carried equal marks, how many questions were there in the test?
Answer (b) & explanation
There were 10 questions (b). Step 1: he got half marks on each of 8 questions. That equals the full marks of 4 questions. Step 2: those marks are 40% of the total. So 4 questions = 40% of all questions. Step 3: all questions = 4 ÷ 0.4 = 10. Check: 50% of 8 questions = 4 questions' marks; 4 out of 10 = 40%. Options (a), (c) and (d) fail the check.
Tip: turn percentages into "question-equivalents". Half marks on 8 questions is the same as full marks on 4. This avoids having to know the marks per question.
Answer as per the official UPSC answer key.
Candidates in a competitive examination consisted of 60% men and 40% women. 70% men and 75% women cleared the qualifying test and entered the final test where 80% men and 70% women were successful. Which of the following statements is correct?
Answer (c) & explanation
More men cleared the exam than women (c). Step 1: take 100 candidates: 60 men and 40 women. Step 2: men who passed both tests: 60 × 0.7 × 0.8 = 33.6. Women: 40 × 0.75 × 0.7 = 21. So more men cleared (c). Step 3: test (a). Men's success rate = 33.6 ÷ 60 = 56%. Women's = 21 ÷ 40 = 52.5%. Women's rate is lower, so (a) is wrong. Step 4: test (b). Overall = 33.6 + 21 = 54.6%, above 50%, so (b) is wrong. Option (d) is wrong too.
Tip: for multi-stage pass rates, multiply the stages. Taking 100 as the total turns every answer into a percentage directly.
Answer as per the official UPSC answer key.
If Sohan, while selling two goats at the same price, makes a profit of 10% on one goat and suffers a loss of 10% on the other
Answer (c) & explanation
He loses 1% (c). Step 1: take an easy selling price that works for both goats, say ₹99 each. Step 2: 10% profit means cost × 1.1 = 99, so cost = ₹90. 10% loss means cost × 0.9 = 99, so cost = ₹110. Step 3: total cost = 90 + 110 = ₹200. Total sale = 99 + 99 = ₹198. Loss = ₹2 on ₹200 = 1%. Option (a) is the trap: the profit and loss are on different costs, so they do not cancel. Options (b) and (d) are wrong.
Tip: whenever two things sell at the same price, one at x% profit and one at x% loss, there is always a loss of (x × x ÷ 100)%. Here 10 × 10 ÷ 100 = 1%.
"+10 and –10 cancel" is wrong. The cost prices differ.
Answer as per the official UPSC answer key.
A and B decide to travel from place X to place Y by bus. A has ₹ 10 with him and he finds that it is 80% of the bus fare for two persons. B finds that he has ₹ 3 with him and hands it over to A. In this context, which one of the following statements is correct?
Answer (c) & explanation
A is left with 50 paise (c). Step 1: A's ₹10 is 80% of the fare for two people. So full fare = 10 ÷ 0.8 = ₹12.50. Step 2: B gives ₹3, so A now has 10 + 3 = ₹13. Step 3: after buying two tickets: 13 − 12.50 = ₹0.50 left. Option (a) is wrong: ₹13 is more than "just enough". Option (b) is wrong: A needs nothing extra. Option (d) is wrong: the money is enough.
Tip: to find the whole from a percentage, divide by the percentage as a decimal: whole = part ÷ 0.8. Quick check: 80% of 12.50 = 10.
Answer as per the official UPSC answer key.