Ratio, proportion, averages, mixtures and alligation: previous-year questions
44 real UPSC Prelims questions (2011–2026) with the official answer and a clear explanation. 14 of them came in the last five papers.
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Three partners A, B and C entered into a business. A invested one-third of the capital for one-third duration. B invested one-fourth of the capital for one-fourth duration. C invested the remaining capital for the whole duration. Out of a profit of ₹ 17,000, how much profit will C get?
Answer (a) & explanation
C gets ₹12,000 (a). Step 1: profit is shared in the ratio of capital × time. A: 1/3 of capital for 1/3 of the time = 1/9. B: 1/4 for 1/4 = 1/16. C put in the rest: 1 − 1/3 − 1/4 = 5/12, for the whole time = 5/12. Step 2: multiply by 144 to clear fractions: A : B : C = 16 : 9 : 60. Total = 85 parts. Step 3: C's share = 60/85 × 17,000 = ₹12,000. Options (b), (c) and (d) fail.
Tip: in partnerships, multiply capital by time for each partner. Clear fractions with the LCM of the denominators before sharing.
Answer as per the official UPSC answer key.
There are two chemicals which do not react with each other. A container contains 10 litres of the chemical A. One litre of this chemical is removed from it and one litre of the chemical B is poured. Then one litre of the mixture is removed from the container and one litre of B is poured. If this process of replacing one litre of the mixture by one litre of B is performed once more, then what is the volume of B that is present in the container approximately (in percentage)?
Answer (b) & explanation
B is about 27% of the container (b). Step 1: each time, 1 litre of the 10-litre mixture is removed, so 9/10 of chemical A stays. Step 2: after three rounds, A left = 10 × (9/10)³ = 10 × 0.729 = 7.29 litres, that is 72.9%. Step 3: the rest is B: 100 − 72.9 = 27.1%, about 27%. Option (a), 25, and the others come from subtracting 10% three times or similar slips.
Tip: for repeated replacement, use: original left = start × (1 − removed/total)ⁿ. Simple subtraction of 10% each time gives the wrong answer.
it is not 3 × 10% = 30%. Later rounds take out some B as well.
Answer as per the official UPSC answer key.
The class average x in a test increases by 4 when the score of a student is rectified, whose corrected score is 100 instead of 0. Later, the score of another student was found to have been recorded as 81 in place of 56. If there are no other corrections and the final corrected average is y, then y − x is
Answer (b) & explanation
y − x is 3 (b). Step 1: one student's score rose by 100 (from 0 to 100), and the average rose by 4. So the class has 100 ÷ 4 = 25 students. Step 2: the second correction changes 81 to 56: the total falls by 25 marks. Shared by 25 students, the average falls by 1. Step 3: overall, the average went up by 4 and then down by 1: y − x = 3. Options (a), (c) and (d) fail.
Tip: change in average = change in total ÷ number of people. Use the first correction to find the class size, then reuse it.
x is the average before both corrections, so both changes count.
Answer as per the official UPSC answer key.
Suppose x, y and z are variables taking positive real numbers as their possible values. It is given that y is directly proportional to x2 and x is inversely proportional to z. For z = 7/25, the values of x and y are 5 and 50, respectively. If y = 98, what is z equal to?
Answer (b) & explanation
z equals 1/5 (b). Step 1: y is directly proportional to x², so y = k × x². With x = 5 and y = 50: 50 = k × 25, so k = 2. Step 2: y = 98 means 2x² = 98, so x² = 49 and x = 7. Step 3: x is inversely proportional to z, so x × z stays the same. At the start: 5 × 7/25 = 7/5. Now 7 × z = 7/5, so z = 1/5. Options (a), (c) and (d) fail.
Tip: "directly proportional" means the ratio stays fixed; "inversely proportional" means the product stays fixed. Find the constant first, then use it.
Answer as per the official UPSC answer key.
The ratio of male to female workers in two companies A and B is 13 : 10 and 7 : 5, respectively. If both the companies have the same number of female workers, then what is the ratio of the total number of workers in A to those in B?
Answer (b) & explanation
The ratio of total workers is 23 : 24 (b). Step 1: the female numbers must be equal. In A, males : females = 13 : 10. In B, it is 7 : 5, which equals 14 : 10 (doubling both). Step 2: now both have 10 parts of females. A's total = 13 + 10 = 23 parts. B's total = 14 + 10 = 24 parts. Step 3: so A : B = 23 : 24. Option (a) reverses the order. Options (c) and (d) do not equalise the females first.
Tip: when one group is the same in two ratios, scale the ratios so that group has the same number of parts. Then compare totals directly.
don't add 13 + 10 and 7 + 5 directly. Equalise the common term first.
Answer as per the official UPSC answer key.
An alloy P contains 20% copper and 80% zinc by weight. Another alloy Q contains 60% copper and 40% zinc by weight. A third alloy R is to be prepared from P and Q so that it contains equal amount of copper and zinc. In what ratio, amounts of P and Q be mixed in order to get R?
Answer (a) & explanation
P and Q must be mixed in the ratio 1 : 3 (a). Step 1: "equal copper and zinc" means R is 50% copper. Step 2: use alligation on copper: P has 20%, Q has 60%, and the mix needs 50%. P : Q = (60 − 50) : (50 − 20) = 10 : 30. Step 3: simplify to 1 : 3. Check: 1 kg P gives 0.2 kg copper; 3 kg Q gives 1.8 kg; total 2 kg copper in 4 kg = 50% ✓. Option (b) reverses the ratio. Options (c) and (d) fail.
Tip: in alligation, the amount of each item is the distance of the other item from the mean. Always check with a quick calculation.
Answer as per the official UPSC answer key.
In a T20 cricket match, three players X, Y and Z scored a total of 37 runs. The ratio of number of runs scored by X to the number of runs scored by Y is equal to ratio of number of runs scored by Y to number of runs scored by Z.
Value-I = Runs scored by X
Value-II = Runs scored by Y
Value-III = Runs scored by Z
Which one of the following is correct?
Answer (d) & explanation
The order cannot be determined (d). Step 1: X : Y = Y : Z means Y² = XZ: the three scores are in continued proportion (a geometric pattern). Step 2: with a total of 37, the whole-number answer is 9, 12, 16 (12² = 9 × 16 = 144). Step 3: but X = 9, Z = 16 and X = 16, Z = 9 both satisfy X : Y = Y : Z. So the scores could rise or fall. Since the order is not fixed, the answer cannot be determined. Options (a), (b) and (c) each pick one order.
Tip: a ratio condition like X : Y = Y : Z is symmetric in X and Z. Check whether swapping values still fits before choosing an order.
Answer as per the official UPSC answer key.
Consider a set of 11 numbers :
Value-I = Minimum value of the average of the numbers of the set when they are consecutive integers ≥ −5.
Value-II = Minimum value of the product of the numbers of the set when they are consecutive non-negative integers.
Which one of the following is correct?
Answer (c) & explanation
The two values are equal (c). Step 1: Value-I: eleven consecutive integers, all at least −5. To make the average smallest, start as low as allowed: −5 to 5. These cancel in pairs, so the average is 0. Step 2: Value-II: eleven consecutive non-negative integers. Start at 0: 0, 1, …, 10. Anything times 0 is 0, so the product is 0, the smallest possible. Step 3: both values are 0, so they are equal. Options (a), (b) and (d) fail.
Tip: for consecutive numbers, the average is the middle term (here 0). Any set that includes 0 has product 0.
Answer as per the official UPSC answer key.
The average of three numbers p, q and r is k. p is as much more than the average as q is less than the average. What is the value of r?
Answer (a) & explanation
r equals k (a). Step 1: "p is as much more than the average as q is less" means p = k + d and q = k − d for some amount d. Step 2: then p + q = 2k. Step 3: the average of all three is k, so their sum is 3k. Hence r = 3k − 2k = k. Options (b) k − 1, (c) k + 1 and (d) k/2 do not follow. For example, p = 7, q = 3, r = 5 gives average 5 = r.
Tip: values that are equally above and below the average cancel out. The remaining value must then sit exactly at the average.
Answer as per the official UPSC answer key.
What percent of water must be mixed with honey so as to gain 20% by selling the mixture at the cost price of honey?
Answer (a) & explanation
Water equal to 20% of the honey must be added (a). Step 1: take 100 litres of honey costing ₹1 a litre, so ₹100 in all. Step 2: a 20% gain means selling for ₹120. Since the mixture is sold at the honey price of ₹1 a litre, we need 120 litres of mixture. Step 3: water added = 120 − 100 = 20 litres, which is 20% of the honey. Options (b), (c) and (d) are too small. (Water is free, so all the extra volume is profit.)
Tip: in "sell at cost price and still gain" questions, the gain comes entirely from the free item. Gain% = water added as a % of the original liquid.
read the base: UPSC asks for water as a percent of honey, not of the mixture.
Answer as per the official UPSC answer key.
Two persons P and Q enter into a business. P puts ₹ 14,000 more than Q, but P has invested for 8 months and Q has invested for 10 months. If P’s share is ₹ 400 more than Q’s share out of the total profit of ₹ 2,000, what is the capital contributed by P?
Answer (a) & explanation
P contributed ₹30,000 (a). Step 1: profit shares: P + Q = 2,000 and P − Q = 400. So P gets 1,200 and Q gets 800, a ratio of 3 : 2. Step 2: profit is shared by capital × months: 8p : 10q = 3 : 2. So 16p = 30q, and p : q = 15 : 8. Step 3: the gap of 7 parts is ₹14,000, so one part = ₹2,000 and P's capital = 15 × 2,000 = ₹30,000 (Q's is ₹16,000; check: 30,000 − 16,000 = 14,000). Options (b), (c) and (d) fail the check.
Tip: in partnership questions, profit ratio = capital × time. Always check your answer against every given fact.
Answer as per the official UPSC answer key.
Consider the following :
Weight of 6 boys = Weight of 7 girls = Weight of 3 men = Weight of 4 women
If the average weight of the women is 63 kg, then what is the average weight of the boys?
Answer (b) & explanation
The average weight of the boys is 42 kg (b). Step 1: the average weight of women is 63 kg, so 4 women weigh 4 × 63 = 252 kg. Step 2: 6 boys weigh the same as 4 women: 252 kg. Step 3: one boy, on average, weighs 252 ÷ 6 = 42 kg. The girls and men are extra information and can be ignored. Option (d), 63 kg, just repeats the women's figure. Options (a) and (c) are wrong.
Tip: CSAT often adds data you do not need. Find the shortest link between what is given and what is asked; here women to boys directly.
Answer as per the official UPSC answer key.
The average weight of A, B, C is 40 kg, the average weight of B, D, E is 42 kg and the weight of F is equal to that of B. What is the average weight of A, B, C, D, E and F?
Answer (c) & explanation
The average weight is 41 kg (c). Step 1: A + B + C = 3 × 40 = 120. B + D + E = 3 × 42 = 126. Step 2: adding them gives A + 2B + C + D + E = 246. Since F = B, this is just A + B + C + D + E + F = 246. Step 3: average = 246 ÷ 6 = 41 kg. Option (d) is wrong: F = B fills the gap. Options (a) and (b) are miscalculations.
Tip: when a person is counted twice, look for a clue that replaces the double count (here F = B).
it looks as if B is needed, but F = B makes it cancel out.
One trap spotted is one mark saved. The PYQ Book lines up every trap UPSC has set, so you walk in prepared. Try the free sample
Answer as per the official UPSC answer key.
There are two containers X and Y. X contains 100 ml of milk and Y contains 100 ml of water. 20 ml of milk from X is transferred to Y. After mixing well, 20 ml of the mixture in Y is transferred back to X. If m denotes the proportion of milk in X and n denotes the proportion of water in Y, then which one of the following is correct?
Answer (a) & explanation
m equals n (a). Step 1: both containers end with 100 ml. Whatever water ends up in X must equal the milk left in Y, because they swapped places. Step 2: check: Y has 100 ml water + 20 ml milk. 20 ml of this mix holds 16⅔ ml water and 3⅓ ml milk. Step 3: X now has 80 + 3⅓ = 83⅓ ml milk, and Y keeps 100 − 16⅔ = 83⅓ ml water. So m = n. Options (b), (c) and (d) fail.
Tip: in back-and-forth mixing with equal final volumes, the amount of each "foreign" liquid is always equal. No calculation is needed.
Answer as per the official UPSC answer key.
A student appeared in 6 papers. The maximum marks are the same for each paper. His marks in these papers are in the proportion of 5 : 6 : 7 : 8 : 9 : 10. Overall he scored 60%. In how many number of papers did he score less than 60% of the maximum marks ?
Answer (b) & explanation
He scored below 60% in 3 papers (b). Step 1: the marks are in the ratio 5 : 6 : 7 : 8 : 9 : 10. Their average is 45 ÷ 6 = 7.5. Step 2: his overall score is 60%, and every paper has the same maximum. So the average share, 7.5, stands for 60%. Step 3: any paper with a ratio term below 7.5 is below 60%. Those are 5, 6 and 7, which is 3 papers. Options (a), (c) and (d) count wrongly. The papers with 8, 9 and 10 are above 60%.
Tip: when all papers carry the same maximum, the overall percentage equals the average paper. Simply compare each term with the average.
Answer as per the official UPSC answer key.
There are two Classes A and B having 25 and 30 students respectively. In Class-A the highest score is 21 and lowest score is 17. In Class-B the highest score is 30 and lowest score is 22. Four students are shifted from Class-A to Class-B. Consider the following statements :
- 1. The average score of Class-B will definitely decrease.
- 2. The average score of Class-A will definitely increase.
Which of the above statements is/are correct ?
Answer (a) & explanation
Only statement 1 is correct (a). Step 1: every student in Class A scores between 17 and 21. Every student in Class B scores 22 or more. Step 2: the four new students in B all score less than anyone already in B. So B's average must fall (statement 1). Step 3: Class A's average depends on who leaves. If the four toppers leave, A's average falls. If the four lowest leave, it rises. So it does not "definitely" increase (statement 2 fails). Options (b), (c) and (d) fail.
Tip: adding values below the average always lowers it. Removing values changes the average only according to which ones are removed; "definitely" needs every case to agree.
the word "definitely". 2 can happen, but it does not have to.
UPSC repeats traps like this year after year. The PYQ Book gathers every trap in one place, so you see them coming. Try the free sample
Answer as per the official UPSC answer key.
An amount of money was distributed among A, B and C in the ratio p : q : r. Consider the following statements :
- 1. A gets the maximum share if p is greater than (q + r).
- 2. C gets the minimum share if r is less than (p + q).
Which of the above statements is/are correct ?
Answer (a) & explanation
Only statement 1 is correct (a). Step 1: if p is greater than q + r, then p is more than half the total. So p is bigger than both q and r, and A gets the maximum share. Statement 1 is true. Step 2: r less than p + q only means r is less than half. That does not make r the smallest. Step 3: try 1 : 5 : 4. Here r = 4 is less than 6, but A gets the least, not C. So statement 2 is false, and options (b), (c) and (d) fail.
Tip: to disprove a "general" statement, find one counter-example with small numbers.
Answer as per the official UPSC answer key.
The average age of a teacher and three students is 20 years. If all the three students are of same age and the difference between the age of the teacher and each student is 20 years, then what is the age of the teacher?
Answer (c) & explanation
The teacher is 35 years old (c). Step 1: the average of 4 people is 20, so their total = 80 years. Step 2: let each student be s years. The teacher is s + 20. So 3s + (s + 20) = 80, which gives 4s = 60 and s = 15. Step 3: teacher = 15 + 20 = 35 years. Check: 15 + 15 + 15 + 35 = 80. Option (d), 45, and options (a) and (b) fail this check.
Tip: turn the average into a total first (average × number). Then use one letter for the unknown and build a simple equation. Always check by adding back.
Answer as per the official UPSC answer key.
A sum of ₹ 2,500 is distributed among X, Y and Z in the ratio 1/2 : 3/4 : 5/6. What is the difference between the maximum share and the minimum share?
Answer (c) & explanation
The difference is ₹ 400 (c). Step 1: clear the fractions by multiplying each by 12 (the LCM of 2, 4 and 6): ½ × 12 = 6, ¾ × 12 = 9, ⅚ × 12 = 10. So the ratio is 6 : 9 : 10. Step 2: total parts = 25. One part = 2500 ÷ 25 = ₹ 100. Step 3: maximum − minimum = (10 − 6) × 100 = ₹ 400. So X gets ₹ 600, Y ₹ 900 and Z ₹ 1000. Options (a), (b) and (d) do not fit.
Tip: to simplify a ratio of fractions, multiply all terms by the LCM of the denominators. Then share the total by parts.
Answer as per the official UPSC answer key.
In a class, there are three groups A, B and C. If one student from group A and two students from group B are shifted to group C, then what happens to the average weight of the students of the class?
Answer (c) & explanation
The class average stays the same (c). Step 1: moving students from groups A and B into group C changes the groups, but not the class. Step 2: the total weight of the class is unchanged, and so is the number of students. Step 3: average = total ÷ number, so the class average stays the same. The group averages may change, but the question asks about the whole class. Options (a), (b) and (d) fail.
Tip: read what the question asks. Shifting people inside a group never changes the overall average. Only adding or removing people from outside can change it.
the groups' averages may change, but the question asks about the class.
Answer as per the official UPSC answer key.