Mixture and Alligation MCQs: Rule, Shortcuts and Solved Practice Questions

Mixture and alligation questions appear in almost every SSC, railway and banking exam — usually 1–2 questions worth easy marks if you know two tools: the alligation rule and the removal-replacement formula. This post gives you both, then 10 solved MCQs.

The Alligation Rule

When two ingredients priced (or valued) at a and b are mixed to get a mean value m, the quantities are in the ratio:

Quantity of cheaper : Quantity of dearer = (b − m) : (m − a)

This works for prices, percentages, averages, speeds — any weighted average.

The Removal-Replacement Formula

If a vessel has V litres of pure liquid and x litres are removed and replaced with water n times, the pure liquid left is:

V × (1 − x/V)ⁿ

Solved MCQs

Q1. A container has 30 litres of milk. How much water must be added so that milk : water = 5 : 2?

(a) 10 L (b) 12 L (c) 15 L (d) 8 L

Answer: (b) 12 L. Water needed = 30 × 2/5 = 12 litres.

Q2. A 20-litre solution has milk : water = 3 : 2. How much water must be added to make the ratio 1 : 1?

(a) 2 L (b) 3 L (c) 4 L (d) 5 L

Answer: (c) 4 L. Milk = 12 L, water = 8 L. For 1 : 1 we need water = 12 L, so add 4 L.

Q3. A vessel has 40 litres of mixture with milk : water = 4 : 1. If 10 litres of mixture is removed and replaced with water, the new ratio of milk to water is:

(a) 3 : 1 (b) 3 : 2 (c) 2 : 1 (d) 4 : 3

Answer: (b) 3 : 2. Removing 10 L takes out 8 L milk and 2 L water, leaving milk 24 L, water 6 L; adding 10 L water gives 24 : 16 = 3 : 2.

Q4. Rice at ₹40/kg is mixed with rice at ₹60/kg so that the mixture costs ₹45/kg. The ratio of the two quantities is:

(a) 3 : 1 (b) 1 : 3 (c) 2 : 1 (d) 3 : 2

Answer: (a) 3 : 1. By alligation: (60 − 45) : (45 − 40) = 15 : 5 = 3 : 1 (cheaper : dearer).

Q5. A milkman mixes water and milk in the ratio 1 : 4 and sells the mixture at cost price of pure milk. His profit percentage is:

(a) 20% (b) 25% (c) 30% (d) 15%

Answer: (b) 25%. For every 4 L of milk he sells 5 L at milk price: profit = 1/4 = 25%.

Q6. Vessel A has milk : water = 4 : 3 and vessel B has 2 : 3. Equal volumes from both are mixed. The ratio of milk to water in the result is:

(a) 17 : 18 (b) 6 : 7 (c) 3 : 4 (d) 17 : 20

Answer: (a) 17 : 18. Milk fraction = (4/7 + 2/5)/2 = 34/70 = 17/35, so milk : water = 17 : 18.

Q7. The average weight of one group is 30 kg and another is 40 kg. Combined average is 34 kg. Ratio of members (first : second) is:

(a) 2 : 3 (b) 3 : 2 (c) 4 : 3 (d) 1 : 2

Answer: (b) 3 : 2. Alligation: (40 − 34) : (34 − 30) = 6 : 4 = 3 : 2.

Q8. A vessel has 80 litres of pure milk. 8 litres are removed and replaced with water, and the operation is repeated once more. Milk left is:

(a) 64 L (b) 64.8 L (c) 66 L (d) 62.4 L

Answer: (b) 64.8 L. 80 × (1 − 8/80)² = 80 × 0.81 = 64.8 litres.

Q9. ₹10,000 is split between schemes paying 8% and 12% simple interest per year. Total interest in a year is ₹1,040. Amount at 8% is:

(a) ₹4,000 (b) ₹5,000 (c) ₹6,000 (d) ₹3,500

Answer: (a) ₹4,000. Mean rate = 10.4%. Alligation: (12 − 10.4) : (10.4 − 8) = 2 : 3, so 8% gets 2/5 × 10,000 = ₹4,000. Check: 320 + 720 = ₹1,040.

Q10. Two alloys have copper : zinc = 5 : 3 and 3 : 5. Equal weights of both are melted together. Copper : zinc in the new alloy is:

(a) 1 : 1 (b) 4 : 3 (c) 15 : 16 (d) 5 : 4

Answer: (a) 1 : 1. Copper fraction = (5/8 + 3/8)/2 = 1/2.

How to Practise This Topic

In the exam, decide within seconds whether a question is a direct-ratio case (Q1, Q2), an alligation case (Q4, Q7, Q9) or a replacement case (Q3, Q8) — the setup does most of the work. Practise mixed sets under time on Pareeksha's topic practice and check the quant sections of our full mock tests.

FAQs

Is alligation only for prices? No — it works for any weighted average: percentages, average marks, speeds, interest rates.

How many questions come from this topic? Typically 1–2 in SSC CGL/CHSL and RRB NTPC quant sections, often combined with percentages or profit-loss.