Arithmetic: percentage, ratio, averages, time & work

Answer: (c) 20%

For a fixed job, the number of men is inversely proportional to the number of days. Cutting the days from 6k to 5k needs 6/5 times the men, which is a 20% increase.

1. Work = men × days, so M × 6k = M′ × 5k.
2. M′ = M × 6k/5k = (6/5)M; the k cancels.
3. Increase = (6/5 − 1) × 100% = (1/5) × 100% = 20%.
4. Check: with k = 1, 5 men for 6 days = 30 man-days = 6 men for 5 days, and 6 is 20% more than 5.

Remember · For fixed work, new men = old men × old days ÷ new days. Common factors like k cancel out.

Question and answer: UPSC's official GS Paper II (2024, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (c) 6 hours 45 minutes

X is the fastest, so X should take the first hour and every alternate hour after it. In six hours 21 of the 24 units are done, and X finishes the last 3 units in 45 minutes of the seventh hour.

1. Take the work as 24 units (LCM of 6 and 8): X does 4 units per hour, Y 3 and Z 3.
2. Nobody may work two hours in a row, so X can work at most every other hour; start with X so that he gets as many hours as possible.
3. Hours 1 to 6: X, Y, X, Z, X, Y gives 3 × 4 + 3 × 3 = 21 units, and all three have worked.
4. Hour 7 is X’s again: the remaining 3 units at 4 units per hour take 3/4 hour = 45 minutes.
5. Total = 6 hours 45 minutes.
6. Check: if Y or Z starts, X still gets only three of the first six hours (21 units), but hour 7 then goes to Y or Z at 3 units per hour, finishing at exactly 7 hours.

Remember · In alternate-hour work problems, give the fastest worker the first slot and every second slot after it.

Question and answer: UPSC's official GS Paper II (2024, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (a) 20%

Selling at the honey’s cost price and still gaining 20% means 120 litres of mixture must be sold for what 100 litres of honey cost. So 20 litres of water go into every 100 litres of honey, i.e. 20% of the honey.

1. Take 100 litres of honey at ₹1 per litre: cost = ₹100.
2. A 20% gain means sales must bring ₹120; at ₹1 per litre that is 120 litres of mixture.
3. Water added = 120 − 100 = 20 litres, which is 20% of the honey.
4. Check: water is 20/120 = 16.67% of the mixture, which is not an option, so the question means water as a percentage of honey.

Remember · When free water is sold at the price of the pure item, the gain percent equals water as a percent of the pure item.

Question and answer: UPSC's official GS Paper II (2024, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (a) ₹30,000

Out of ₹2,000, P gets ₹1,200 and Q ₹800, a ratio of 3 : 2. Profit is shared in the ratio of capital × time, so 8(Q + 14,000) : 10Q = 3 : 2, which gives Q = ₹16,000 and P = ₹30,000.

1. Shares add to 2,000 and differ by 400, so P gets ₹1,200 and Q ₹800; ratio 3 : 2.
2. Let Q’s capital be x; then P’s capital is x + 14,000.
3. 8(x + 14,000) : 10x = 3 : 2 → 16(x + 14,000) = 30x → 14x = 2,24,000 → x = 16,000.
4. P’s capital = 16,000 + 14,000 = ₹30,000.
5. Check: 30,000 × 8 = 2,40,000 and 16,000 × 10 = 1,60,000, a ratio of 3 : 2.

Remember · Profit share is proportional to capital × time. Turn the profit clue into a ratio first, then solve one equation.

Question and answer: UPSC's official GS Paper II (2024, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (b) 56.25%

Take R’s salary as 100: Q gets 80 and P gets 64. R’s salary is 36 more than P’s, and 36 out of 64 is 56.25%.

1. Let R’s salary be 100. Q’s is 20% lower: 80. P’s is 20% lower than Q’s: 64.
2. R − P = 100 − 64 = 36.
3. R is more than P by 36/64 × 100 = 56.25%.
4. Check: 64 × 1.5625 = 100.

Remember · Take 100 as the base. ‘More than P’ means divide the difference by P’s value, not by 100.

Question and answer: UPSC's official GS Paper II (2024, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (d) 93.75%

The correct result is 4x and the mistaken one x/4. The result falls by 4x − x/4 = 15x/4, which is 15/16 of the correct result, i.e. 93.75%.

1. Take the number as 16. Correct result = 16 × 4 = 64; mistaken result = 16 ÷ 4 = 4.
2. Change = 64 − 4 = 60.
3. Percentage change = 60/64 × 100 = 93.75% (a fall).
4. Check in general: (4 − 1/4)/4 = 15/16 = 0.9375.

Remember · Pick a convenient number and measure the change against the correct result, which is the base.

Question and answer: UPSC's official GS Paper II (2024, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (b) 20%

20% failed in English and 30% in Hindi, with 15% failing both. So 5% failed only English and 15% failed only Hindi: 20% failed in exactly one subject.

1. Failed in English = 100 − 80 = 20%; failed in Hindi = 100 − 70 = 30%.
2. Failed in both = 15%.
3. Failed only in English = 20 − 15 = 5%; failed only in Hindi = 30 − 15 = 15%.
4. Failed in only one subject = 5 + 15 = 20%.
5. Check: passed in both = 100 − (20 + 30 − 15) = 65%, and 65 + 20 + 15 = 100.

Remember · Work with the failure sets: ‘only one’ = (A − both) + (B − both).

Question and answer: UPSC's official GS Paper II (2024, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (a) 30 years

n years ago the father was the son’s present age, so n is the gap between their ages. Using ‘father now = 4 × son n years ago’ gives father : son = 8 : 5, so with a total of 130 they are 80 and 50, a difference of 30 years.

1. Let the father’s age be F and the son’s S. ‘n years back I was as old as you are now’: F − n = S, so n = F − S.
2. Son’s age n years back = S − n = S − (F − S) = 2S − F.
3. F = 4(2S − F) → 5F = 8S → F : S = 8 : 5.
4. F + S = 130 → 13 parts = 130 → 1 part = 10, so F = 80 and S = 50.
5. Difference = 80 − 50 = 30 years.
6. Check: n = 30; the son was 20 thirty years ago, and 4 × 20 = 80.

Remember · ‘I was your present age n years ago’ means n equals the age gap; turn the rest into a ratio.

Question and answer: UPSC's official GS Paper II (2024, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (b) 42 kg

Four women weigh 4 × 63 = 252 kg, and six boys weigh the same. So the average boy weighs 252 ÷ 6 = 42 kg.

1. Weight of 4 women = 4 × 63 = 252 kg.
2. Weight of 6 boys = weight of 4 women = 252 kg.
3. Average weight of a boy = 252 ÷ 6 = 42 kg.
4. Check: 6 × 42 = 252 = 4 × 63.

Remember · In a chain of equal totals, use only the two links the question needs and ignore the rest.

Question and answer: UPSC's official GS Paper II (2024, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (d) ₹1,350

Take Q’s cost as 100: R costs 120 and P costs 150, a total of 370 units. As 370 units = ₹3,330, one unit = ₹9, so P costs 150 × 9 = ₹1,350.

1. Let Q = 100. R is 20% more: 120. P is 25% more than R: 150.
2. Total = 100 + 120 + 150 = 370 units = ₹3,330, so 1 unit = ₹9.
3. P = 150 × 9 = ₹1,350.
4. Check: Q = ₹900, R = ₹1,080, P = ₹1,350, and 900 + 1,080 + 1,350 = 3,330.

Remember · Chain the percentages from the cheapest item taken as 100, then scale to the given total.

Question and answer: UPSC's official GS Paper II (2024, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error

Answer: (c) Both 1 and 2

4O + 6M + 8A = 2(O + 2M + 5A) simplifies to O + M = A, which is statement 2. Statement 1 then follows, because 3O + 5M + 9A differs from 4O + 6M + 8A by A − O − M = 0.

1. Let the prices be O, M and A. Then 4O + 6M + 8A = 2O + 4M + 10A.
2. So 2O + 2M = 2A, i.e. O + M = A. Statement 2 is correct.
3. (3O + 5M + 9A) − (4O + 6M + 8A) = A − O − M = 0. Statement 1 is correct.
4. Check with O = 1, M = 2, A = 3: 4 + 12 + 24 = 40 = 2 × (1 + 4 + 15), and 3 + 10 + 27 = 40 as well.

• ✓ 1. The two totals differ by A − O − M, which is 0 because O + M = A.
• ✓ 2. Simplifying the given condition gives exactly O + M = A.

Remember · Simplify the given condition first; then test each statement by subtracting one side from the other.

Question and answer: UPSC's official GS Paper II (2024, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink · Report an error