DEFINITION
One-Line Definition
Mixtures & Alligations is an algebraic shortcut rule used to find the ratio in which two ingredients of different rates or concentrations must be mixed to produce a mixture of a desired average value.
Key Rule: Cheaper / Dearer = (Dearer - Mean) / (Mean - Cheaper)
CORE INTUITION 🧪
Weighted Average Mental Model
Alligation is simply weighted average in reverse!
- • Instead of solving Aavg = (n1c + n2d) / (n1 + n2)
- • Alligation directly gives the weight ratio n1 : n2 = (d - A) : (A - c)
The closer the mixture price is to one component, the larger the quantity of that component!
💡 WHY THIS CONCEPT MATTERS & REAL-LIFE APPLICATIONS
Alligation reduces 2-minute algebraic equations to 10-second mental subtractions in CAT & MBA CET. Click on any connected topic to jump directly to its Vault page:
Where Are Mixtures & Alligations Used?
📈 Investment Portfolio Blending
🧪 Chemical & Solution Dilution
🥛 Milk & Water Quality Mix
☕ Profit Margin Averaging
📐 THE ALLIGATION CROSS RULE & REPLACEMENT FORMULA
THE ALLIGATION CROSS DIAGRAM
Cheaper (C)
Dearer (D)
\ /
Mean Price (M)
/ \
(D - M)
(M - C)
Quantity of Cheaper : Quantity of Dearer = (Dearer - Mean) : (Mean - Cheaper)
Repeated Replacement / Dilution Formula
PURE LIQUID REMAINING AFTER 'n' REPLACEMENTS
Remaining Amount = V × (1 - x / V)ⁿ
• V = Initial Total Volume of Pure Liquid
• x = Volume withdrawn and replaced with water/other liquid each time
• n = Number of times the operation is performed
• x = Volume withdrawn and replaced with water/other liquid each time
• n = Number of times the operation is performed
📝 SOLVED EXAMPLES (LEVEL 0 TO ADVANCED)
EASY • EXAMPLE 1
In what ratio must wheat at ₹20 per kg be mixed with wheat at ₹30 per kg so that the mixture is worth ₹26 per kg?
Solution:
Cheaper (C) = 20 | Dearer (D) = 30 | Mean (M) = 26
Dearer - Mean = 30 - 26 = 4
Mean - Cheaper = 26 - 20 = 6
Ratio of Cheaper : Dearer = 4 : 6 = 2 : 3
Answer: 2 : 3
MEDIUM • EXAMPLE 2
A 40-liter mixture contains milk and water in the ratio 3 : 1. How much water must be added to make the ratio of milk to water 1 : 1?
Solution:
Initial Milk = (3/4) × 40 = 30 liters | Initial Water = 10 liters
To make ratio 1 : 1, Water volume must equal Milk volume (30 liters).
Water to be added = 30 - 10 = 20 liters
Answer: 20 liters
HARD • EXAMPLE 3 (REPEATED REPLACEMENT)
A vessel contains 80 liters of pure milk. 8 liters of milk are withdrawn and replaced with water. This process is repeated 2 more times (total 3 times). Find the remaining quantity of pure milk.
Solution:
Initial Volume V = 80 liters | Withdrawn x = 8 liters | Repetitions n = 3
Remaining Milk = V × (1 - x / V)ⁿ
Remaining Milk = 80 × (1 - 8 / 80)³ = 80 × (0.9)³ = 80 × 0.729 = 58.32 liters
Answer: 58.32 liters
⚠️ COMMON MISTAKES TO AVOID
❌ Mistake 1: Mixing Different Units in Alligation
All three values in the Alligation Cross (Cheaper, Dearer, Mean) MUST be of the exact same type and unit! For example, do not mix Cost Price with Selling Price.
❌ Mistake 2: Reversing the Output Ratio
Remember: The value under Cheaper is (Dearer - Mean), which gives the quantity of Cheaper ingredient! The value under Dearer gives the quantity of Dearer ingredient.
❌ Mistake 3: Miscounting 'n' in the Replacement Formula
If a question says "withdrawn once and repeated 2 MORE times", total operations n = 1 + 2 = 3, not n = 2!
🚀 CAT & MBA CET SHORTCUTS
⚡ Shortcut 1: Profit & Loss Alligation Trick
When applying Alligation to Profit % and Loss %:
• Put Profit % as positive (+P) and Loss % as negative (-L).
• The output ratio gives the ratio of their COST PRICES!
• Put Profit % as positive (+P) and Loss % as negative (-L).
• The output ratio gives the ratio of their COST PRICES!
⚡ Shortcut 2: Simple Interest Alligation Trick
When a sum is split into two parts at interest rates R1% and R2%:
• Put R1% and R2% at the top, and overall average interest rate R_avg in the center.
• The output ratio gives the ratio of the PRINCIPAL AMOUNTS invested!
• Put R1% and R2% at the top, and overall average interest rate R_avg in the center.
• The output ratio gives the ratio of the PRINCIPAL AMOUNTS invested!
🎯 PRACTICE QUESTIONS
QUESTION 1 • BASIC
In what ratio should tea costing ₹120 per kg be mixed with tea costing ₹180 per kg to obtain a mixture worth ₹150 per kg?
D - M = 180 - 150 = 30.
M - C = 150 - 120 = 30.
Ratio = 30 : 30 = 1 : 1.
M - C = 150 - 120 = 30.
Ratio = 30 : 30 = 1 : 1.
QUESTION 2 • BASIC
In what ratio should a 20% acid solution be mixed with a 50% acid solution to obtain a 30% acid solution?
D - M = 50 - 30 = 20.
M - C = 30 - 20 = 10.
Ratio = 20 : 10 = 2 : 1.
M - C = 30 - 20 = 10.
Ratio = 20 : 10 = 2 : 1.
QUESTION 3 • MODERATE
How many kg of sugar costing ₹45 per kg must be mixed with 30 kg of sugar costing ₹30 per kg so that the mixture is worth ₹40 per kg?
Cheaper (₹30) : Dearer (₹45) = (45 - 40) : (40 - 30) = 5 : 10 = 1 : 2.
Given Cheaper Quantity = 30 kg (1 part).
Dearer Quantity (2 parts) = 30 × 2 = 60 kg.
Given Cheaper Quantity = 30 kg (1 part).
Dearer Quantity (2 parts) = 30 × 2 = 60 kg.
QUESTION 4 • ADVANCED
A container has 100 liters of pure wine. 10 liters are drawn off and replaced with water. 10 liters of the mixture are drawn off again and replaced with water. Find the final quantity of pure wine.
V = 100, x = 10, n = 2.
Remaining Wine = 100 × (1 - 10 / 100)² = 100 × (0.9)² = 100 × 0.81 = 81 liters.
Remaining Wine = 100 × (1 - 10 / 100)² = 100 × (0.9)² = 100 × 0.81 = 81 liters.
❓ FREQUENTLY ASKED QUESTIONS
Q: When can we use the Alligation Rule?
Whenever two ingredients or quantities of different prices, values, or concentrations are mixed together to form a combined mixture of a desired average value.
Q: What does the output ratio of Alligation represent?
It represents the ratio of the physical quantities or weights of the two ingredients being mixed (Quantity of Cheaper : Quantity of Dearer).
Q: Can Alligation be used for Profit & Loss or Simple Interest problems?
Yes! When applied to Profit % and Loss %, the resulting ratio is the ratio of Cost Prices. When applied to Interest Rates %, the resulting ratio is the ratio of Principal amounts invested.