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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A bottle contains 20 litres of liquid A. 4 litres of liquid A is taken out of it and replaced by same quantity of liquid B. Again 4 litres of the mixture is taken out and replaced by same quantity of liquid B. What is the ratio of quantity of liquid A to that of liquid B in the final mixture?
Answer (c) & explanation
The final ratio of A to B is 16 : 9 (c). Step 1: each round removes 4 of 20 litres, so it keeps 16/20 = 4/5 of liquid A. Step 2: after two rounds: A = 20 × (4/5) × (4/5) = 20 × 16/25 = 12·8 litres. Step 3: B = 20 − 12·8 = 7·2 litres. Ratio 12·8 : 7·2 = 128 : 72 = 16 : 9. Option (a), 4 : 1, is the ratio after only one round (16 : 4). Options (b) and (d) fail.
Tip: for repeated replacement, use: remaining = original × (1 − removed/total)ⁿ. This formula saves step-by-step work.
Answer as per the official UPSC answer key.
The average score of a batsman after his 50th innings was 46·4. After 60th innings, his average score increases by 2·6. What was his average score in the last ten innings?
Answer (c) & explanation
His average in the last ten innings was 62 (c). Step 1: total after 50 innings = 50 × 46·4 = 2320. Step 2: the new average after 60 innings = 46·4 + 2·6 = 49. Total = 60 × 49 = 2940. Step 3: runs in the last 10 innings = 2940 − 2320 = 620. Average = 620 ÷ 10 = 62. Option (d), 49, is the new overall average. Options (a) and (b) fail.
Tip: always convert averages to totals before subtracting. Shortcut: the last 10 innings must lift 60 innings by 2·6 each and also match the old average: 46·4 + (60 × 2·6) ÷ 10 = 46·4 + 15·6 = 62.
Answer as per the official UPSC answer key.
A family has two children along with their parents. The average of the weights of the children and their mother is 50 kg. The average of the weights of the children and their father is 52 kg. If the weight of the father is 60 kg, then what is the weight of the mother?
Answer (d) & explanation
The mother weighs 54 kg (d). Step 1: the average of the two children and the mother is 50, so their total = 150 kg. The average of the children and the father is 52, so that total = 156 kg. Step 2: both totals include the same two children. So the difference, 156 − 150 = 6 kg, is the father's weight minus the mother's. Step 3: mother = 60 − 6 = 54 kg. Option (b), 50 kg, is just the first average. Options (a) and (c) do not fit.
Tip: when two groups share members, subtract their totals. The shared part cancels, leaving only the difference you need. You never have to find the children's weights.
Answer as per the official UPSC answer key.
The average marks of 100 students are given to be 40. It was found later that marks of one student were 53 which were misread as 83. The corrected mean marks are
Answer (b) & explanation
The corrected mean is 39·7 (b). Step 1: total marks as first counted = 100 × 40 = 4000. Step 2: one mark was read as 83 instead of 53, which is 30 too high. Correct total = 4000 − 30 = 3970. Step 3: correct average = 3970 ÷ 100 = 39·7. Option (d), 40·3, adds the 30 instead of subtracting it. Option (c) ignores the error. Option (a) takes off the whole 1 mark.
Tip: shortcut: the average changes by (error ÷ number of items). Here 30 ÷ 100 = 0·3 too high, so the true average is 40 − 0·3 = 39·7.
Answer as per the official UPSC answer key.
A lift has the capacity of 18 adults or 30 children. How many children can board the lift with 12 adults?
Answer (b) & explanation
10 children can board with 12 adults (b). Step 1: the lift holds 18 adults, so 12 adults use 12/18 = 2/3 of its capacity. Step 2: the remaining space is 1/3 of the lift. Step 3: the full lift holds 30 children, so 1/3 of it holds 30 ÷ 3 = 10 children. Another way: 18 adults = 30 children, so 1 adult = 5/3 child. The 6 missing adults = 6 × 5/3 = 10 children. Option (a), 6, just subtracts 12 from 18. Options (c) and (d) do not fit.
Tip: in capacity questions, turn everything into a fraction of the whole, or into one common unit. Both ways give the answer in seconds.
Answer as per the official UPSC answer key.
Suppose the average weight of 9 persons is 50 kg. The average weight of the first 5 persons is 45 kg, whereas the average weight of the last 5 persons is 55 kg. Then the weight of the 5th person will be
Answer (c) & explanation
The 5th person weighs 50 kg (c). Step 1: total of all 9 = 9 × 50 = 450 kg. Step 2: total of the first 5 = 5 × 45 = 225 kg. Total of the last 5 = 5 × 55 = 275 kg. Step 3: the first five and the last five together make 10 people. But there are only 9, so the 5th person is counted twice. 225 + 275 − 450 = 50 kg. Option (a), 45 kg, is the first group's average. Options (b) and (d) are guesses between the averages.
Tip: when two groups overlap, add their totals and subtract the whole. What is left is the overlap. This trick saves time in many average questions.
Answer as per the official UPSC answer key.
There are thirteen 2-digit consecutive odd numbers. If 39 is the mean of the first five such numbers, then what is the mean of all the thirteen numbers?
Answer (a) & explanation
The mean of all thirteen numbers is 47 (a). Step 1: for evenly spaced numbers, the mean equals the middle term. The first five have mean 39, so 39 is their middle (3rd) number. The five are 35, 37, 39, 41 and 43. Step 2: the thirteen numbers start at 35 and go up by 2: 35, 37, … up to 59. Step 3: their middle term is the 7th one: 35 + 6 × 2 = 47. So the mean is 47. Options (b) 49 and (c) 51 are too high. Option (d), 45, is too low.
Tip: in any evenly spaced list, mean = middle term = (first + last) ÷ 2. Here (35 + 59) ÷ 2 = 47, which checks out.
Answer as per the official UPSC answer key.
There is a milk sample with 50% water in it. If 1/3rd of this milk is added to equal amount of pure milk, then water in the new mixture will fall down to
Answer (a) & explanation
Water in the new mixture falls to 25% (a). Step 1: take 1 litre of the sample. It has 0.5 litre water and 0.5 litre milk. Step 2: add an equal amount, 1 litre, of pure milk. Now there are 2 litres, with still only 0.5 litre of water. Step 3: water = 0.5 ÷ 2 = 25%. The "1/3rd" only says how much of the sample is used. Since an equal amount of pure milk is added, the share is always halved. Options (b), (c) and (d) do not fit.
Tip: mixing a sample with an equal amount of a pure liquid halves the share of the other part. 50% water becomes 25%, whatever the quantity.
the 1/3rd is a distraction. Equal amounts halve the water share.
Answer as per the official UPSC answer key.
The average rainfall in a city for the first four days was recorded to be 0·40 inch. The rainfall on the last two days was in the ratio of 4 : 3. The average of six days was 0·50 inch. What was the rainfall on the fifth day?
Answer (c) & explanation
Rainfall on the fifth day was 0.80 inch (c). Step 1: total over six days = 6 × 0.50 = 3.00 inches. Total over the first four days = 4 × 0.40 = 1.60 inches. Step 2: so the last two days had 3.00 − 1.60 = 1.40 inches. Step 3: these are in the ratio 4 : 3. 1.40 ÷ 7 = 0.20, so the fifth day = 4 × 0.20 = 0.80 inch and the sixth = 0.60 inch. Option (a), 0.60, is the sixth day. Options (b) and (d) do not fit.
Tip: in average questions, turn every average into a total first. Then subtract to find the missing part, and split it by the ratio.
Answer as per the official UPSC answer key.
The monthly incomes of X and Y are in the ratio of 4 : 3 and their monthly expenses are in the ratio of 3 : 2. However, each saves ₹ 6,000 per month. What is their total monthly income?
Answer (b) & explanation
Their total monthly income is Rs 42,000 (b). Step 1: test the options. Split the total income in the ratio 4 : 3, subtract Rs 6,000 savings from each, and check whether expenses come out as 3 : 2. Step 2: option (b): incomes Rs 24,000 and Rs 18,000. Expenses Rs 18,000 and Rs 12,000. 18 : 12 = 3 : 2. It works. Step 3: option (a), Rs 28,000: incomes 16,000 and 12,000, expenses 10,000 and 6,000 = 5 : 3. It fails. Options (c) and (d) also fail the expense ratio.
Tip: algebra works too: 4x − 6000 : 3x − 6000 = 3 : 2 gives x = 6000. But testing options is often faster in CSAT.
Answer as per the official UPSC answer key.
The monthly average salary paid to all the employees of a company was Rs. 5000. The monthly average salary paid to male and female employees was Rs. 5200 and Rs. 4200 respectively. Then the percentage of males employed in the company is :
Answer (b) & explanation
Males are 80% of the staff (b). Step 1: use alligation. The overall average is Rs 5000. Males earn Rs 5200, which is 200 above it. Females earn Rs 4200, which is 800 below it. Step 2: the counts are in the reverse ratio of these gaps. Males : females = 800 : 200 = 4 : 1. Step 3: males = 4 out of 5 = 80%. Option (a), 75%, would give an average of 4950. Options (c) and (d) give averages above 5000.
Tip: in mixture averages, the bigger group sits closer to the overall average. Since 5000 is much closer to 5200, males must be the large majority.
Answer as per the official UPSC answer key.
The sum of the ages of 5 members comprising a family, 3 years ago was 80 years. The average age of the family today is the same as it was 3 years ago, because of an addition of a baby during the intervening period. How old is the baby?
Answer (b) & explanation
The baby is 1 year old (b). Step 1: three years ago, the 5 members' ages added to 80. So the average was 80 ÷ 5 = 16. Step 2: today each of the five is 3 years older. Their total = 80 + 5 × 3 = 95. Step 3: with the baby there are 6 members, and the average is still 16. So the total = 6 × 16 = 96. Baby = 96 − 95 = 1 year. Option (a), 6 months, and options (c) and (d) do not fit.
Tip: in age-average questions, work with totals, not averages. Every year adds 1 year to each person, so a group of n people gains n years in total each year.
Answer as per the official UPSC answer key.
The total emoluments of two persons are the same, but one gets allowances to the extent of 65% of his basic pay and the other gets allowances to the extent of 80% of his basic pay. The ratio of the basic pay of the former to the basic pay of the latter is :
Answer (d) & explanation
The ratio of their basic pays is 12 : 11 (d). Step 1: the first person's total is basic + 65% of basic = 1.65 × basic. The second person's total is 1.80 × basic. Step 2: the totals are equal, so 1.65 × B1 = 1.80 × B2. Step 3: B1 : B2 = 1.80 : 1.65. Multiply by 100 to get 180 : 165, then divide by 15 to get 12 : 11. The person with the smaller allowance must have the bigger basic pay. Option (a) 16 : 13 and option (b) 5 : 4 are traps from the 80 and 65 figures. Option (c) does not fit.
Tip: if a × X = b × Y, then X : Y = b : a. The ratio flips.
80 : 65 = 16 : 13 forgets that the total is basic + allowance.
Answer as per the official UPSC answer key.
The average monthly income of a person in a certain family of 5 is Rs. 10,000. What will be the average monthly income of a person in the same family if the income of one person increased by Rs. 1,20,000 per year?
Answer (a) & explanation
The new average is Rs 12,000 (a). Step 1: the family's total monthly income = 5 × 10,000 = Rs 50,000. Step 2: one person's income rises by Rs 1,20,000 a year. That is 1,20,000 ÷ 12 = Rs 10,000 a month. Step 3: new total = 50,000 + 10,000 = 60,000. New average = 60,000 ÷ 5 = Rs 12,000. Option (d), Rs 34,000, wrongly adds the yearly figure to the monthly one. Options (b) and (c) do not fit.
Tip: watch the time units. Convert yearly to monthly (divide by 12) before mixing figures. An increase shared over 5 people raises the average by one-fifth of it.
Rs. 34,000 adds the yearly figure to a monthly average.
Answer as per the official UPSC answer key.
30 g of sugar was mixed in 180 ml water in a vessel A, 40 g of sugar was mixed in 280 ml of water in vessel B and 20 g of sugar was mixed in 100 ml of water in vessel C. The solution in vessel B is :
Answer (d) & explanation
Vessel B is less sweet than vessel C (d). Step 1: sweetness = sugar ÷ water. A = 30/180 = 1/6. B = 40/280 = 1/7. C = 20/100 = 1/5. Step 2: compare the fractions. 1/5 is bigger than 1/6, which is bigger than 1/7. So C is sweetest and B is least sweet. Step 3: so B is less sweet than C, option (d). Option (a) says the opposite. Option (b) is wrong because B is less sweet than A too. Option (c) is wrong because B and C are not equal.
Tip: to compare mixtures, turn each into "per unit" form and simplify. With the same numerator 1, a smaller denominator means a stronger mixture.
Answer as per the official UPSC answer key.
In a class, there are 18 very tall boys. If these constitute three-fourths of the boys and the total number of boys is two-thirds of the total number of students in the class, what is the number of girls in the class?
Answer (b) & explanation
There are 12 girls (b). Step 1: 18 tall boys are three-fourths of the boys. So boys = 18 ÷ 3/4 = 24. Step 2: boys are two-thirds of all students. So students = 24 ÷ 2/3 = 36. Step 3: girls = 36 − 24 = 12. Option (a), 6, is a quarter of the boys, the boys who are not tall. Option (c), 18, is the number of tall boys. Option (d), 21, does not fit.
Tip: "x is a fraction of y" means y = x ÷ fraction. Work backwards one step at a time from the number you know. Check: girls are 12 of 36, which is one-third, the remaining part.
Answer as per the official UPSC answer key.
An automobile owner reduced his monthly petrol consumption when the prices went up. The price-consumption relationship is as follows :
| Price (in ₹ per litre) | 40 | 50 | 60 | 75 |
|---|---|---|---|---|
| Monthly consumption (in litres) | 60 | 48 | 40 | 32 |
If the price goes up to ₹ 80 per litre, his expected consumption (in litres) will be
Answer (a) & explanation
He will use 30 litres at ₹80 (a). Step 1: look at the table. Multiply price by consumption in each column: in every case the answer is 2,400. So he always spends ₹2,400 a month on petrol, whatever the price. Step 2: at ₹80 a litre: 2,400 ÷ 80 = 30 litres. Options (b), (c) and (d) would mean a different monthly spend.
Tip: in price-quantity tables, check whether price × quantity stays the same. If it does, the person has a fixed budget, and the new quantity is simply budget ÷ new price. This is quicker than looking for a pattern in the differences.
the drops (12, 8, 8) do not follow a steady pattern. Look for the constant product.
Answer as per the official UPSC answer key.
Two equal glasses of same type are respectively 1/3 and 1/4 full of milk. They are then filled up with water and the contents are mixed in a pot. What is the ratio of milk and water in the pot?
Answer (a) & explanation
The milk to water ratio is 7 : 17 (a). Step 1: pick a glass size that both fractions divide into easily. 12 units works for 1/3 and 1/4. Step 2: glass 1 is 1/3 milk: 4 units of milk, 8 of water. Glass 2 is 1/4 milk: 3 units of milk, 9 of water. Step 3: in the pot: milk = 4 + 3 = 7, water = 8 + 9 = 17. Total 24. Ratio 7 : 17. Options (b), (c) and (d) do not match.
Tip: choose a size equal to the LCM of the denominators (here 3 and 4 give 12). This turns fractions into whole numbers and makes mixing easy.
Answer as per the official UPSC answer key.
A person ordered 5 pairs of black socks and some pairs of brown socks. The price of a black pair was thrice that of a brown pair. While preparing the bill, the bill clerk interchanged the number of black and brown pairs by mistake which increased the bill by 100%. What was the number of pairs of brown socks in the original order ?
Answer (d) & explanation
The original order had 25 brown pairs (d). Step 1: let a brown pair cost ₹1, so a black pair costs ₹3. Step 2: test the options. Try 25 brown pairs. Correct bill: 5 black × 3 + 25 brown × 1 = 15 + 25 = ₹40. Step 3: wrong bill, with the numbers swapped: 25 black × 3 + 5 brown × 1 = 75 + 5 = ₹80. Step 4: ₹80 is double ₹40, a 100% increase. This matches. Options (a), (b) and (c) do not double the bill: for example, 10 brown gives ₹25 and ₹35.
Tip: when an algebra setup seems long, test the options. Give prices simple values like 1 and 3 to keep the arithmetic easy.
Answer as per the official UPSC answer key.
The monthly incomes of Peter and Paul are in the ratio of 4 : 3. Their expenses are in the ratio of 3 : 2. If each saves ₹ 6,000 at the end of the month, their monthly incomes respectively are (in ₹)
Answer (a) & explanation
Their incomes are ₹24,000 and ₹18,000 (a). Step 1: every option keeps incomes in the ratio 4 : 3, so test each one using savings of ₹6,000. Step 2: option (a): expenses = 24,000 − 6,000 = 18,000 and 18,000 − 6,000 = 12,000. 18,000 : 12,000 = 3 : 2. This matches. Step 3: check option (b): expenses 22,000 and 15,000, which is not 3 : 2. Options (c) and (d) also fail.
Tip: when all options satisfy one condition, use the other condition to test them. Testing options is often faster than setting up two equations.
Answer as per the official UPSC answer key.