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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There are some balls of red, green and yellow colour lying on a table. There are as many red balls as there are yellow balls. There are twice as many yellow balls as there are green ones. The number of red balls
Answer (b) & explanation
'Double the number of green balls' is correct (b). Step 1: let the green balls be g. Yellow balls are twice the green, so yellow = 2g. Red balls equal yellow, so red = 2g. Step 2: so red is double the green balls: (b). Step 3: (a) says red = yellow + green = 3g, but red is 2g. (c) says red = yellow − green = g, but red is 2g. (d) is wrong because the relationship can be fixed even without the actual numbers. So (a), (c) and (d) fail.
Tip: when only relations are given, call the smallest group x and express the others in terms of it. 'Cannot be ascertained' is wrong whenever the ratios are fully fixed.
(d) is tempting because no number is given, but a ratio is enough here.
Answer as per the official UPSC answer key.
In a rare coin collection, there is one gold for every three non-gold coins. 10 more gold coins are added to the collection and the ratio of gold coins to non-gold coins would be 1 : 2. Based on the information, the total number of coins in the collection now becomes
Answer (a) & explanation
90 is correct (a). Step 1: gold to non-gold is 1 : 3, so let gold = k and non-gold = 3k. Non-gold coins do not change. Step 2: after adding 10 gold coins, (k + 10) : 3k = 1 : 2. So 2(k + 10) = 3k, which gives 2k + 20 = 3k, and k = 20. Step 3: now gold = 20 + 10 = 30 and non-gold = 60. Total = 90. Check: 30 : 60 = 1 : 2. So (b), (c) and (d) fail.
Tip: in ratio-change questions, fix the quantity that stays the same (non-gold coins here) and build the equation around it. Always check the new ratio with your final numbers.
the question asks for the total now, after adding 10. 80 is the old total.
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Answer as per the official UPSC answer key.
Two glasses of equal volume are respectively half and three-fourths filled with milk. They are then filled to the brim by adding water. Their contents are then poured into another vessel. What will be the ratio of milk to water in this vessel?
Answer (d) & explanation
5 : 3 is correct (d). Step 1: take each glass as 4 units so the fractions are easy. Glass 1 is half milk: 2 milk + 2 water. Glass 2 is three-fourths milk: 3 milk + 1 water. Step 2: pour both into the vessel. Milk = 2 + 3 = 5 units. Water = 2 + 1 = 3 units. Step 3: so milk : water = 5 : 3. (a) 1 : 3 and (b) 2 : 3 put water above milk, and (c) 3 : 2 is the wrong split. So (a), (b) and (c) fail.
Tip: choose a glass size that clears all fractions (the LCM of the denominators, here 4). Then everything becomes whole numbers and simple adding.
Answer as per the official UPSC answer key.
A student on her first 3 tests received an average score of N points. If she exceeds her previous average score by 20 points on her fourth test, then what is the average score for the first 4 tests?
Answer (d) & explanation
N + 5 is correct (d). Step 1: the first three tests total 3N. Step 2: the fourth score is N + 20. The new total is 3N + N + 20 = 4N + 20. Step 3: the new average = (4N + 20) ÷ 4 = N + 5. In simple words, the extra 20 points are shared over four tests, adding 5 to the average. So (a), (b) and (c) fail.
Tip: when one new value is above the old average, the average rises by (excess ÷ new count). Here 20 ÷ 4 = 5.
Answer as per the official UPSC answer key.