Introduction
Alligation is a shortcut rule for finding the ratio in which two ingredients at different prices (or
concentrations) must be mixed to get a mixture at a given price. It replaces algebra with a simple
cross-subtraction diagram.
The Alligation Cross
For cheaper price C and dearer price D, mixed to give mean price M:
Ratio (Cheaper : Dearer) = (D − M) : (M − C)
Q. In what ratio must tea worth ₹60/kg be mixed with tea worth ₹90/kg so that the mixture is worth ₹75/kg?
C = 60, D = 90, M = 75. Ratio = (D−M):(M−C) = (90−75):(75−60) = 15:15 = 1:1.
Q. A container has 40 litres of milk. 4 litres are withdrawn and replaced with water, and this is done 3 times. Find the final quantity of milk left.
Use the repeated-replacement formula: Final quantity = Initial × (1 − withdrawn/total)ⁿ.
= 40 × (1 − 4/40)³ = 40 × (0.9)³ = 40 × 0.729 = 29.16 litres.
Repeated Dilution Formula
If x units are withdrawn from a vessel of y units (containing pure liquid) and replaced with water, repeated n
times: Final quantity of liquid = y × (1 − x/y)ⁿ
⚠️ Key Insight: Alligation works for any two "concentration" values — price, percentage, speed,
or even marks — not just liquids. Recognise the C-D-M pattern in any two-way mixing scenario.
✅ Practice Focus: Basic alligation cross ratio · Repeated replacement/dilution formula ·
Mixing more than two ingredients using weighted mean.