Arithmetic: percentage, ratio, averages, time & work

Answer: (b) 20th May

The two formulas never meet within the first 100 days, after which petrol stays at ₹90. Diesel reaches ₹90 on day 140, and day 140 of 2021 (not a leap year) is 20 May.

1. While both formulas run (days 1–100): 80 + 0.1m = 69 + 0.15m gives 0.05m = 11, so m = 220 — outside 1 to 100, so no match in this period.
2. From day 100 onwards petrol stays at 80 + 0.1 × 100 = ₹90.
3. Diesel reaches ₹90 when 69 + 0.15n = 90, i.e. 0.15n = 21, so n = 140.
4. Days up to 30 April: 31 + 28 + 31 + 30 = 120, so day 140 is the 20th day of May — 20 May.
5. Check: on day 140 diesel = 69 + 0.15 × 140 = 69 + 21 = ₹90, equal to petrol.

Remember · For prices defined piece by piece, test each piece separately; then turn the day number into a date month by month.

Question and answer: UPSC's official GS Paper II (2021, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (d) P = 50 (2)^(n/12)

Doubling every 12 years is exponential growth: after n years the population has doubled n/12 times, so P = 50 × 2^(n/12). Putting n = 0 and n = 12 (giving 50 and 100) confirms it and rules out the other options.

1. Doubling every 12 years means multiplying by 2 once for each 12-year block.
2. In n years there are n/12 such blocks, so P = 50 × 2^(n/12).
3. Check: n = 0 gives 50; n = 12 gives 100; n = 24 gives 200 — doubling every 12 years.
4. Options (a) and (b) add a fixed number each year (linear growth); (c) gives 50 × 2¹² at n = 1, far too fast.

Remember · Test formula options by plugging in simple values — the start (n = 0) and one full period.

Question and answer: UPSC's official GS Paper II (2021, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (d) 20%

Take 100 students: 60 Indian and 40 foreign, with 50 girls. Indian girls are 18, so foreign girls are 32 and foreign boys only 8 — that is 8 out of 40 foreign students, or 20%.

1. Take 100 students: 60 Indian, 40 foreign; 50 girls in all.
2. Indian girls = 30% of 60 = 18.
3. Foreign girls = 50 − 18 = 32, so foreign boys = 40 − 32 = 8.
4. Share of boys among foreign students = 8/40 = 20%.
5. Check: boys = 50; Indian boys 60 − 18 = 42 plus foreign boys 8 = 50.

Remember · Assume a total of 100 and fill a 2 × 2 table; then divide by the base the question names (here, foreign students).

Question and answer: UPSC's official GS Paper II (2021, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (b) 5

Why not the tempting option · UPSC's key is 5. A very literal reading of 'previous height' lets the ball rise once more, to 49.15 cm, and touch the ground a sixth time; but that final contact is the ball coming to rest, not a hit before it stops bouncing, so 5 remains the better count. In the exam, stop at the first height that breaks the threshold and do not add the final landing.

Each bounce reaches 4/5 of the previous height: 150 → 120 → 96 → 76.8 → 61.44 cm. The next rise would be only 49.15 cm, below the 50 cm limit, so the ball does not bounce again; by then it has hit the ground 5 times.

1. Heights in cm: dropped from 150, the ball rises to 4/5 of each previous height — 120, 96, 76.8, 61.44.
2. Every rise is preceded by a hit: hits after falling from 150, 120, 96, 76.8 and 61.44 cm — that is 5 hits.
3. The rise after the 5th hit would be 4/5 × 61.44 = 49.152 cm, below 50 cm, so the ball does not bounce further.
4. Number of times the ball hits the ground before it stops bouncing = 5.

Remember · List the geometric sequence of heights, stop at the first term that breaks the threshold, and count the hits up to that point; do not add an extra landing after the ball has stopped bouncing.

Question and answer: UPSC's official GS Paper II (2021, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 1 Oct 2026 (how we verify). Permalink · Report an error

Answer: (c) Both 1 and 2

Adding the hourly distances, the two close the 15 km gap exactly at the end of the 5th hour. X walks 5 × 1.5 = 7.5 km and Y walks 1 + 1.25 + 1.5 + 1.75 + 2 = 7.5 km, so they meet midway — both statements hold.

1. Distance closed in each hour: 1.5 + 1 = 2.5, then 1.5 + 1.25 = 2.75, then 3, 3.25, 3.5 km.
2. Running total: 2.5, 5.25, 8.25, 11.5, 15 km — they meet at the end of the 5th hour.
3. X covers 5 × 1.5 = 7.5 km; Y covers 1 + 1.25 + 1.5 + 1.75 + 2 = 7.5 km.
4. Each covers half of 15 km, so they meet midway.
5. Check: Y's average speed over 5 hours is 7.5 ÷ 5 = 1.5 km/hr, equal to X's speed — hence midway.

• ✓ 1. The combined distance reaches 15 km exactly after 5 hours.
• ✓ 2. Both walk 7.5 km in those 5 hours, so the meeting point is the midpoint.

Remember · With speeds changing hour by hour, tabulate the hours; for an arithmetic progression of speeds, the average speed gives a quick check.

Question and answer: UPSC's official GS Paper II (2021, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (b) 3

With equal maximum marks, the overall 60% equals the average of the six papers' shares, and the average of 5 to 10 is 7.5. Only the papers with shares 5, 6 and 7 fall below that level — 3 papers.

1. Let the marks be 5k, 6k, 7k, 8k, 9k and 10k; the total is 45k.
2. Since every paper has the same maximum, the overall 60% equals the average paper score: 45k ÷ 6 = 7.5k is 60% of the maximum.
3. Papers below 7.5k: 5k, 6k and 7k — 3 papers.
4. Check: if the maximum is 12.5k, the marks are 40%, 48%, 56%, 64%, 72%, 80% — three below 60%.

Remember · When maximum marks are equal, the overall percentage is the average of the paper percentages; compare each paper with the mean.

Question and answer: UPSC's official GS Paper II (2021, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (d) 30 or more

There are only 40 foreigners, so at most 40 of the 70 English speakers can be foreigners. At least 30 must be Indians, and the number could be higher, so the answer is '30 or more'.

1. Foreigners = 120 − 80 = 40.
2. Even if all 40 foreigners speak English, at least 70 − 40 = 30 English speakers are Indians.
3. If all 70 English speakers were Indians, the number would be 70; the data do not fix it.
4. So the number of Indians who speak English is 30 or more.

Remember · Minimum overlap of two groups = (size of A + size of B) − total; give a range when the data do not fix an exact value.

Question and answer: UPSC's official GS Paper II (2021, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (a) 1 only

Every shifted student scored at most 21, below Class-B's lowest score of 22, so Class-B's average must fall. Class-A's average can rise or fall depending on which four students leave, so statement 2 is not certain.

1. Each shifted student scored at most 21; every Class-B student scored at least 22.
2. Adding scores lower than all existing scores pulls Class-B's average down — statement 1 is certain.
3. Class-A's new average depends on whether the four who left were above or below Class-A's average.
4. If the four scored 21 each, Class-A's average falls; if they scored 17 each, it rises — so statement 2 is not definite.

• ✓ 1. New members all score below Class-B's minimum, so its average must decrease.
• ✗ 2. Removing students raises the average only if they were below it; the data do not say which four moved.

Remember · Adding values below every existing value lowers the mean; removing values changes it only relative to the current mean.

Question and answer: UPSC's official GS Paper II (2021, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (c) The price of a pen is two times the price of a pencil

Both spent the same, so 3 pens + 5 pencils cost as much as 2 pens + 7 pencils. Cancelling the common items leaves 1 pen = 2 pencils.

1. Equal spending: 3 pens + 5 pencils = 2 pens + 7 pencils.
2. Remove 2 pens and 5 pencils from both sides: 1 pen = 2 pencils.
3. So a pen costs twice as much as a pencil.

Remember · With equal totals, cancel what both sides share — the leftovers must balance.

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Answer: (a) 360

Q's score is 10% below P's, so the 40-mark gap is 10% of P's score. P scored 400 and Q scored 360.

1. Q scored 10% less than P, so the gap of 40 marks is 10% of P's marks.
2. P = 40 ÷ 0.10 = 400.
3. Q = 400 − 40 = 360.
4. Check: 360 is 90% of 400.

Remember · 'x% less than P' means the difference is x% of P — use it to find P first.

Question and answer: UPSC's official GS Paper II (2021, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (b) ₹ 360

The second condition gives X + 40 = Y = Z − 40, and the first gives Y − 40 = Z/2. Solving, Z = 160, Y = 120 and X = 80, a total of ₹360.

1. Let X, Y and Z have x, y and z rupees.
2. If Z gives X ₹40, all three are equal: x + 40 = y = z − 40.
3. If Y gives X ₹40, Y has half of Z: y − 40 = z/2.
4. Put y = z − 40: z − 80 = z/2, so z = 160, y = 120, x = 80.
5. Total = 80 + 120 + 160 = ₹360.
6. Check: after Y gives ₹40, Y has 80, half of Z's 160; after Z gives ₹40, everyone has 120.

Remember · Turn 'then all will be equal' into a chain of equalities; it reduces everything to one unknown.

Question and answer: UPSC's official GS Paper II (2021, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (a) 9

Full marks would be 90 × 5 = 450, and each wrong answer costs 5 + 2 = 7 marks compared with a right one. The shortfall of 63 marks means 9 wrong answers.

1. If all 90 were correct: 90 × 5 = 450 marks.
2. Each wrong answer loses 5 + 2 = 7 marks compared with a correct one.
3. Shortfall = 450 − 387 = 63, so wrong answers = 63 ÷ 7 = 9.
4. Check: 81 × 5 − 9 × 2 = 405 − 18 = 387.

Remember · Start from full marks and divide the shortfall by (marks for a correct answer + penalty for a wrong one).

Question and answer: UPSC's official GS Paper II (2021, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (a) 1 only

If one part exceeds the other two combined, it certainly exceeds each of them, so A gets the most. But being less than the sum of the other two says nothing about being the smallest, as 1 : 3 : 2 shows.

1. Statement 1: shares are positive, so p > q + r means p > q and p > r — A gets the maximum. Correct.
2. Statement 2: take p : q : r = 1 : 3 : 2. Here r = 2 < p + q = 4, yet A, not C, gets the least. Incorrect.
3. So only Statement 1 is correct.

• ✓ 1. p > q + r forces p above both q and r.
• ✗ 2. r < p + q can hold even when r is not the smallest, e.g. 1 : 3 : 2.

Remember · Test 'if … then' claims with a quick counterexample: 'greater than the sum' implies 'greatest', but 'less than the sum' does not imply 'least'.

Question and answer: UPSC's official GS Paper II (2021, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (b) 2 only

X's age at Y's birth is the age gap X − Y, which equals twice Y's present age; so X = 3Y and Y = 14. Eight years ago they were 34 and 6 (not five times), but in 14 years they will be 56 and 28 (exactly twice).

1. At Y's birth X's age was X − Y, and this equals 2Y, so X = 3Y.
2. X = 42, so Y = 14.
3. Statement 1: 8 years ago X was 34 and Y was 6; 5 × 6 = 30, not 34. Incorrect.
4. Statement 2: after 14 years X will be 56 and Y 28; 56 = 2 × 28. Correct.

• ✗ 1. 34 is not five times 6.
• ✓ 2. 56 is exactly twice 28.

Remember · 'My age at your birth' is simply the age difference, which never changes.

Question and answer: UPSC's official GS Paper II (2021, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (a) 0%

A 20% cut takes ₹100 to ₹80, and 25% of 80 is 20, bringing it back to ₹100. The multipliers 0.8 × 1.25 = 1, so there is no net change.

1. Let the price be ₹100. After a 20% decrease it is ₹80.
2. A 25% increase on ₹80 is ₹20, giving ₹100.
3. Net change = 0%.
4. Check: 0.8 × 1.25 = 1.

Remember · Successive changes multiply: a fall of 1/5 is exactly undone by a rise of 1/4.

Question and answer: UPSC's official GS Paper II (2021, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (d) 15

He does 1/8 of the job every 3 days, so the whole job takes 24 days. When the job grows to 1.5 times its size, 5/8 of the original job remains, which takes 15 more days.

1. 7/8 of the job in 21 days, so the whole job takes 21 ÷ 7/8 = 24 days (1/24 of the job per day).
2. The work is increased by 50%, so the total becomes 1.5 jobs.
3. Work left = 1.5 − 7/8 = 12/8 − 7/8 = 5/8 of the original job.
4. Days needed = 5/8 × 24 = 15.
5. Check: 7/8 + 15/24 = 7/8 + 5/8 = 12/8 = 1.5 jobs.

Remember · Turn work done into a daily rate first; then express the remaining work as a fraction of the original job.

Question and answer: UPSC's official GS Paper II (2021, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error