Mixtures and alligation is one of those topics that looks scary but has ONE trick that solves almost everything — the alligation (cross) method. Once we get this, we can solve mixing problems, average problems, percentage problems, and even profit/loss problems in seconds.
What Is Alligation?
Alligation is a method to find the ratio in which two ingredients at different prices (or concentrations, or averages) should be mixed to get a mixture at a desired price (or concentration, or average).
In simple language, if we’re mixing cheap and expensive stuff to get something in between, alligation tells us the exact ratio.
The Alligation (Cross) Method
This is the most important diagram in this entire topic.
The trick is: we take the cross differences. The cheaper quantity’s ratio part = (Dearer - Mean), and the dearer quantity’s ratio part = (Mean - Cheaper).
Simple Mixing Example
Example: In what ratio should Rs 40/kg tea be mixed with Rs 60/kg tea to get a mixture worth Rs 45/kg?
Using alligation:
- Cheaper (C) = 40, Dearer (D) = 60, Mean (M) = 45
- Ratio = (D - M) : (M - C) = (60 - 45) : (45 - 40) = 15 : 5 = 3 : 1
So we mix 3 parts of the Rs 40 tea with 1 part of the Rs 60 tea. Done in 10 seconds.
Alligation Beyond Mixing
Here’s the cool part — alligation works for ANY weighted average situation, not just physical mixing.
For Averages
Example: Section A (30 students) has average marks 70. Section B (20 students) has average marks 80. We already know the combined average is 74 (from the averages chapter). But we can also verify the ratio:
Ratio of A:B students = (80 - 74) : (74 - 70) = 6 : 4 = 3 : 2. And indeed, 30:20 = 3:2 ✓
For Percentages
Example: A shopkeeper mixes two types of sugar — one at Rs 20/kg (with 60% purity) and another at Rs 30/kg (with 90% purity). In what ratio should they be mixed to get 70% pure sugar?
Using alligation on purity: Ratio = (90 - 70) : (70 - 60) = 20 : 10 = 2 : 1
For Profit/Loss
Example: A shopkeeper sells some goods at 10% profit and the rest at 20% profit. If overall profit is 14%, find the ratio of cost prices.
Using alligation on profit%: Ratio = (20 - 14) : (14 - 10) = 6 : 4 = 3 : 2
Milk and Water Problems
These are the most classic mixture problems.
Type 1: Finding the Ratio
Example: A container has 40 liters of milk. How much water should be added to make a 4:1 milk-to-water ratio?
Milk = 40L. We want milk:water = 4:1, so water = 40/4 = 10L. Add 10 liters of water.
Type 2: Mixing Two Mixtures
Example: Vessel A contains milk and water in 3:1 ratio. Vessel B contains milk and water in 5:3 ratio. In what ratio should we mix from A and B to get a 7:3 mixture?
Milk concentration in A = 3/4 = 75% Milk concentration in B = 5/8 = 62.5% Desired milk concentration = 7/10 = 70%
Using alligation: Ratio A:B = (70 - 62.5) : (75 - 70) = 7.5 : 5 = 3 : 2
Repeated Dilution (Replacement Problems)
This is a favorite exam type. We start with a full container of substance X. Each time, we remove some and replace with water (or another substance).
Example: A container has 80 liters of milk. 8 liters are drawn out and replaced with water. This is done 3 times. How much milk remains?
Milk remaining = 80 × (1 - 8/80)³ = 80 × (1 - 1/10)³ = 80 × (9/10)³
= 80 × 729/1000 = 58.32 liters
Key insight: Each replacement reduces the milk by the same fraction. After n replacements, the fraction remaining is (1 - x/V)ⁿ.
Worked Examples
Example 1: A milkman mixes water with milk. The cost of pure milk is Rs 24/liter. He sells the mixture at Rs 26/liter and makes a 30% profit. Find the ratio of water to milk.
His effective CP = 26/1.3 = Rs 20 per liter. But pure milk costs Rs 24/liter and water costs Rs 0/liter. Using alligation: Water:Milk = (24 - 20) : (20 - 0) = 4 : 20 = 1 : 5
Example 2: Two vessels contain milk-water mixtures. Vessel 1 has milk:water = 7:3, Vessel 2 has milk:water = 1:2. They are mixed in ratio 2:1. Find the milk:water ratio in the new mixture.
Milk fraction in V1 = 7/10, in V2 = 1/3. Mixed in 2:1 ratio: Average milk fraction = (2 × 7/10 + 1 × 1/3) / (2 + 1) = (14/10 + 1/3) / 3 = (42/30 + 10/30) / 3 = (52/30) / 3 = 52/90 = 26/45
Milk:Water = 26 : (45-26) = 26 : 19
Example 3: 20 liters of a mixture contains milk and water in the ratio 3:1. How much mixture should be removed and replaced with water so that milk and water become equal (1:1)?
Current: Milk = 15L, Water = 5L. After removing x liters and replacing with water:
- Milk removed = x × 3/4 = 3x/4
- Remaining milk = 15 - 3x/4
- Total water = 5 - x/4 + x = 5 + 3x/4
For equal ratio: 15 - 3x/4 = 5 + 3x/4 10 = 6x/4 = 3x/2 x = 20/3 = 6.67 liters
Example 4: A jar contains 100 liters of pure acid. 10 liters are removed and replaced with water. This is done twice more. What percentage of acid remains?
Acid remaining = 100 × (1 - 10/100)³ = 100 × (0.9)³ = 100 × 0.729 = 72.9 liters. Percentage = 72.9%
Example 5: In what ratio must a person mix two kinds of coffee costing Rs 300/kg and Rs 400/kg so that by selling the mix at Rs 420/kg, they make a 20% profit?
SP = Rs 420, Profit = 20%. So CP of mix = 420/1.2 = Rs 350/kg. Alligation: Ratio = (400 - 350) : (350 - 300) = 50 : 50 = 1 : 1
Common Exam Patterns
- “In what ratio to mix two things” → Alligation cross method
- “Repeated removal and replacement” → V × (1 - x/V)ⁿ
- “Mixing two mixtures” → Find concentration of each, then apply alligation
- “Selling mixture at profit” → Find effective CP of mixture, then apply alligation
- “How much to add to change ratio from a:b to c:d” → Set up equation using current quantities
- “Alligation for average/profit%” → Same cross method, just use averages or percentages instead of prices
Practice Problems
Q1: A shopkeeper mixes rice at Rs 50/kg with rice at Rs 65/kg in the ratio 2:3. At what price per kg should he sell the mixture to make a 20% profit?
Q2: A container has 120 liters of pure milk. 30 liters are taken out and replaced with water. This process is repeated twice more. How much milk remains?
Q3: Two solutions have acid concentrations of 30% and 70%. In what ratio should they be mixed to get a 50% solution?
Answers:
A1: CP of mix = (2×50 + 3×65)/(2+3) = (100+195)/5 = 295/5 = Rs 59/kg. SP for 20% profit = 59 × 1.2 = Rs 70.80/kg
A2: Milk = 120 × (1 - 30/120)³ = 120 × (3/4)³ = 120 × 27/64 = 50.625 liters
A3: Alligation: Ratio = (70-50) : (50-30) = 20 : 20 = 1 : 1