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AFCAT Mixture & Alligation

Study Material — 17 PYQs (2018–2024) · Concept Notes · Shortcuts

AFCAT Mixture & Alligation is a frequently tested subtopic — 17 previous year questions from 2018–2024 papers are included below with concept notes, key rules and shortcut tricks.

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Previous Year Questions

AFCAT Mixture & Alligation — Past Exam Questions

17 questions from actual AFCAT papers · all shown free · click option to reveal solution

Exam Q 12024Previous Year Pattern

Two alloys contain copper in the percentages 30% and 50% respectively. In what ratio should they be mixed to get an alloy with 40% copper?

Exam Q 22024Previous Year Pattern

A shopkeeper mixes two types of rice costing ₹40 per kg and ₹60 per kg in the ratio 3:2. What is the cost price of the mixture per kg?

Exam Q 32018Previous Year Pattern

A container has milk and water in the ratio 3:1. How much water must be added to 40 litres of this mixture to make the ratio 2:1?

Exam Q 42024Previous Year Pattern

A shopkeeper mixes two types of rice costing ₹40 per kg and ₹60 per kg in the ratio 3:2. At what price per kg should he sell the mixture to gain 25% profit?

Exam Q 52020Previous Year Pattern

A shopkeeper mixes two types of tea costing ₹80 per kg and ₹120 per kg in the ratio 3:2. At what price per kg should he sell the mixture to gain 25% profit?

Exam Q 62024Previous Year Pattern

A goldsmith mixes gold and silver in the ratio 2:3 to make an alloy. If he wants to make 500 grams of alloy, how much gold is needed?

Exam Q 72024Previous Year Pattern

Two vessels contain alcohol and water in the ratios 3:1 and 5:3 respectively. If equal quantities from each vessel are mixed together, what is the ratio of alcohol to water in the final mixture?

Exam Q 82024Previous Year Pattern

Two containers have alcohol and water in the ratios 3:1 and 5:3 respectively. If equal quantities from both containers are mixed, what is the ratio of alcohol to water in the final mixture?

Exam Q 92024Previous Year Pattern

In what ratio must sugar costing ₹15 per kg be mixed with sugar costing ₹24 per kg so that the mixture costs ₹18 per kg?

Exam Q 102019Previous Year Pattern

A chemist has two solutions of acid: Solution A contains 30% acid and Solution B contains 70% acid. In what ratio must these two solutions be mixed to obtain 50 litres of a 54% acid solution?

Exam Q 112024Previous Year Pattern

A chemist has two solutions with alcohol concentrations of 20% and 50%. In what ratio should these be mixed to get a solution with 35% alcohol concentration?

Exam Q 122024Previous Year Pattern

A vessel contains 80 litres of a mixture of alcohol and water in the ratio 3:5. How much pure alcohol should be added to make the ratio 1:1?

Exam Q 132024Previous Year Pattern

Two containers have alcohol solutions of 20% and 50% respectively. In what ratio should they be mixed to get a 35% solution?

Exam Q 142024Previous Year Pattern

Two containers have milk and water in ratios 3:1 and 5:3 respectively. If equal volumes are mixed from both containers, what is the ratio of milk to water in the final mixture?

Exam Q 152020Previous Year Pattern

A shopkeeper has two types of tea: Type A costing ₹80 per kg and Type B costing ₹120 per kg. He wants to create a mixture that costs ₹100 per kg. If he uses 15 kg of Type A tea, how many kg of Type B tea should he mix to achieve the desired cost per kg?

Exam Q 162020Previous Year Pattern

A chemist has three containers of alcohol solutions: Container A has 60 litres of 40% alcohol, Container B has 40 litres of 70% alcohol, and Container C has 50 litres of 20% alcohol. The chemist mixes all three containers together, then removes 30 litres of the resulting mixture and replaces it with pure water. What is the percentage concentration of alcohol in the final mixture?

Exam Q 172018Previous Year Pattern

A vessel contains a mixture of milk and water in the ratio 7:3. 20 litres of the mixture is taken out and replaced with pure milk. If the resulting ratio of milk to water becomes 9:1, what is the original volume of the mixture in the vessel?

Concept Notes

Mixture & Alligation— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic

MIXTURE & ALLIGATION — COMPLETE GUIDE FOR SSC CGL ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

CORE CONCEPT — WHAT IS IT? ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

Alligation is a smart method to find the ratio in which two ingredients must be mixed to get a desired mixture at a given price or concentration. Think of it like a see-saw. You are balancing two items around a middle value (the mean price or mean concentration). In plain terms: If you mix a cheap item and an expensive item, alligation tells you EXACTLY how much of each to use.

This topic appears in SSC CGL almost every year — sometimes 2 to 3 questions in one paper. ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

2.Key Rules

Core rules you must know cold

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 1. The mean value (desired value) must ALWAYS lie between the two values being mixed. If it does not, mixing is impossible. 2. The ratio found is of the QUANTITIES of the two ingredients. 3.

Alligation works for price, concentration, speed, percentage — any average-based problem. ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

3.Formula Block

Memorise — at least one formula appears in every paper
Ratio of Cheaper to Dearer = (D - M) : (M - C)
Quantity of Cheaper / Quantity of Dearer = (Dearer - Mean) / (Mean - Cheaper)
Final concentration = (1 - x/n)^t
where t = number of times the operation is repeated.

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

Alligation Rule (Cross Method):

Cheaper Price (C) -------- Dearer Price (D)

\ Mean Price (M) /

(D - M) (M - C)

Simple rule to remember:

For Replacement Problems:

After removing x litres from a vessel of n litres and replacing with water:

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

4.Exam Patterns

What examiners ask — read before attempting PYQs

— HOW IT IS ASKED ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ SSC CGL asks Alligation in these forms: • Mixing two qualities of rice, milk, or tea at different prices to get a target price. • Mixing two solutions of different concentrations to get a target concentration. • Replacement problems — replacing part of a solution with water repeatedly. • Finding average marks, speed, or ratio of students in two groups. ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

5.Worked Example

Solve this step-by-step before moving on
1
Step 1

Write Cheaper = 12, Dearer = 20, Mean = 14.

2
Step 2

Apply alligation cross method. Cheaper side: Dearer - Mean = 20 - 14 = 6 Dearer side: Mean - Cheaper = 14 - 12 = 2

3
Step 3

Ratio of Cheaper to Dearer = 6 : 2 = 3 : 1 Answer: Mix 3 kg of Rs 12 rice with 1 kg of Rs 20 rice. Quick Check: (3 x 12 + 1 x 20) / 4 = (36 + 20) / 4 = 56 / 4 = Rs 14. CORRECT. ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ WORKED EXAMPLE 2 — REPLACEMENT PROBLEM ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Question: A vessel has 40 litres of pure milk. 10 litres is removed and replaced with water. This is done 2 times. Find the amount of milk left.

1
Step 1

Use replacement formula. Final milk = 40 x (1 - 10/40)^2

2
Step 2

1 - 10/40 = 1 - 1/4 = 3/4

3
Step 3

Final milk = 40 x (3/4)^2 = 40 x 9/16 = 360/16 = 22.5 litres Answer: 22.5 litres of milk remain. ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ EXAM SHORTCUTS AND TRICKS ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Shortcut 1 — Direct Ratio Formula: Ratio = (D - M) : (M - C). Just subtract diagonally on the cross diagram. Takes under 10 seconds. Shortcut 2 — For Profit/Loss mixing: If a shopkeeper mixes two items and sells at cost price, use alligation on cost prices directly. No extra steps needed. Shortcut 3 — Mixture of Three Items: Fix one pair using alligation to get a ratio. Then fix the second pair. Combine the results. Always work in pairs. ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

6.Exam Traps

Common mistakes students make — avoid these

— THE NUMBER 1 TRAP ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Most students flip the ratio. They write (Mean - Cheaper) for cheaper quantity and (Dearer - Mean) for dearer quantity. This is WRONG.

Remember: The difference on one side gives the quantity of the OTHER side. Dearer minus Mean gives quantity of CHEAPER. Mean minus Cheaper gives quantity of DEARER.

Think of it as a cross — always swap sides.

Key Points to Remember

  • Alligation finds the ratio of mixing two ingredients to get a desired mean value — price, concentration, or any average.
  • Formula: Ratio of Cheaper to Dearer = (Dearer - Mean) : (Mean - Cheaper).
  • The mean value MUST lie between the cheaper and dearer values — otherwise mixing is impossible.
  • Replacement Formula: Final quantity of original = Initial x (1 - removed/total)^number of times.
  • Alligation works for price problems, concentration problems, speed problems, and percentage problems — same formula.
  • In the cross diagram, always subtract DIAGONALLY and place the result on the OPPOSITE side.
  • For three-ingredient mixing, always break it into two pairs and apply alligation twice.
  • Shortcut: If two groups have known averages and you need the combined average — use alligation in reverse to find ratio.
  • Always do a quick verification: plug ratio back into weighted average formula to confirm mean value is correct.
  • Trap: Do NOT write Mean minus Cheaper on the Cheaper side — the difference always belongs to the opposite ingredient's quantity.

Exam-Specific Tips

  • Alligation Rule Ratio: Quantity of Cheaper / Quantity of Dearer = (Dearer Price - Mean Price) / (Mean Price - Cheaper Price).
  • Replacement Formula for t repetitions: Final pure quantity = Initial quantity x (1 - withdrawal/total volume)^t.
  • If 1/n fraction of a mixture is replaced by water each time, after t times pure liquid left = Initial x ((n-1)/n)^t.
  • In SSC CGL exams, Mixture and Alligation typically contributes 1 to 3 questions per paper in Quantitative Aptitude.
  • Alligation method is also called the Rule of Alligation or Pearls Method in classical mathematics texts.
  • When two solutions of concentration A% and B% are mixed in ratio m:n, resultant concentration = (mA + nB) / (m + n).
  • If the mean price equals the cheaper price, then zero quantity of the dearer ingredient is used — ratio becomes infinite:1.
  • For a mixture problem where profit% is given instead of mean price, convert selling price to effective cost price first before applying alligation.
Practice MCQs

Mixture & Alligation — Practice Questions

25graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 25

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Practice 1easy

A container has 60 litres of a solution with alcohol and water in ratio 7:5. How much pure alcohol must be added to make the ratio 3:1?

Practice 2easy

A merchant mixes two types of rice costing ₹40 per kg and ₹60 per kg in the ratio 3:2. At what price per kg should the mixture be sold to gain 25% profit?

Practice 3easy

A container has 120 litres of a solution with alcohol and water in the ratio 7:5. How much pure alcohol should be added to make the ratio 3:1?

Practice 4easy

A container has 40 litres of milk. 8 litres of milk is taken out and replaced with water. What is the ratio of milk to water in the final mixture?

Practice 5easy

In what ratio should sugar costing ₹20 per kg be mixed with sugar costing ₹30 per kg to get a mixture costing ₹24 per kg?

Practice 6easy

A goldsmith mixes gold of purity 80% with gold of purity 60% in the ratio 3:2. What is the purity of the resulting mixture?

Practice 7easy

Two containers have milk and water in ratios 3:1 and 5:3 respectively. If equal quantities from each container are mixed, what is the ratio of milk to water in the final mixture?

Practice 8medium

Two alloys A and B contain copper in the ratio 2:3 and 3:4 respectively. Equal weights of both alloys are melted together. What is the ratio of copper to non-copper in the final alloy?

Practice 9medium

A vessel contains 60 litres of a mixture of alcohol and water in the ratio 7:5. How much of the mixture should be removed and replaced with pure alcohol so that the ratio becomes 3:1?

Practice 10medium

A shopkeeper mixes two types of rice costing ₹40 per kg and ₹60 per kg in the ratio 3:2. He then sells the mixture at ₹55 per kg. What is his profit or loss percentage?

Practice 11medium

A shopkeeper mixes two types of rice costing ₹40 per kg and ₹60 per kg. If the mixture costs ₹50 per kg and he uses 15 kg of the cheaper rice, how much of the costlier rice should he use?

Practice 12medium

A mixture contains milk and water in the ratio 5:3. If 16 litres of water is added to the mixture, the ratio becomes 5:7. What is the quantity of milk in the original mixture?

Practice 13medium

A container has 60 litres of a solution with alcohol and water in the ratio 7:5. How much of the solution should be removed and replaced with pure alcohol so that the alcohol content becomes 80% of the total?

Practice 14medium

A shopkeeper mixes two types of rice costing ₹40 per kg and ₹60 per kg in the ratio 3:2. At what price per kg should he sell the mixture to gain 25% profit?

Practice 15medium

A vessel contains 80 litres of a mixture of alcohol and water in the ratio 3:5. How much pure alcohol must be added to make the ratio 1:1?

Practice 16medium

Two containers A and B contain alcohol solutions of 40% and 60% respectively. If 20 litres from A and 30 litres from B are mixed together, what is the percentage of alcohol in the resulting mixture?

Practice 17hard

A merchant blends two types of tea: Premium (₹500/kg) and Standard (₹300/kg). He sells the blend at ₹420/kg and makes a 20% profit. If he uses 15 kg of Premium tea, how many kg of Standard tea must he use?

Practice 18hard

A vessel contains 80 litres of a mixture of alcohol and water in the ratio 3:5. How much of the mixture should be removed and replaced with pure alcohol so that the ratio of alcohol to water becomes 1:1?

Practice 19hard

Two alloys A and B contain copper in the ratio 3:1 and 1:3 respectively. If 20 kg of alloy A is mixed with 30 kg of alloy B, what is the percentage of copper in the resulting mixture?

Practice 20hard

A milkman mixes milk from two sources. Source A has 85% purity and Source B has 60% purity. He wants to create 50 litres of milk with 72% purity. How many litres from Source A must he use?

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60-Second Revision — Mixture & Alligation

  • Formula: Ratio = (Dearer - Mean) : (Mean - Cheaper) — subtract diagonally, place result on OPPOSITE side.
  • Replacement Formula: Final pure liquid = P x (1 - x/n)^t where P = initial amount, x = removed each time, n = total volume, t = times repeated.
  • Trap: Never place (Mean - Cheaper) on the Cheaper side — the value always represents the OTHER ingredient's quantity.
  • Remember: Mean value must ALWAYS be strictly between the two ingredient values — check this first before solving.
  • Shortcut: For three-ingredient questions, pair them up and apply alligation twice — never try to solve all three at once.
  • Verify answer: Multiply each quantity by its value, add totals, divide by total quantity — result must equal the mean value.
  • Pattern Alert: SSC CGL often disguises alligation as profit/loss or speed/distance problems — identify the mean value and apply the same formula.
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