Quantitative Aptitude
Mixture & Alligation
Mixture & Alligation
Mixture problems involve combining two or more ingredients at different values (price, concentration, strength). Alligation is the shortcut method that finds the mixing ratio directly from the values without setting up equations.
Key Idea
Draw the alligation cross: put C1 top-left, C2 top-right, mean price Cm in centre. The diagonal differences give the mixing ratio: (C2 − Cm) : (Cm − C1).
Core Formulas
Alligation Rule (Mixing Ratio)
Ratio = (C2 − Cm) : (Cm − C1) where Cm is the mean/target value
To find in what ratio two ingredients must be mixed to achieve a target price or concentration.
Repeated Dilution
Pure liquid remaining = V × ((V − x) / V)ⁿ
When x litres are removed and replaced with water n times from a V-litre container.
Weighted Average (Mixed Mixture)
Resultant Avg = (n₁×p₁ + n₂×p₂) / (n₁ + n₂)
When combining two mixtures of different sizes and concentrations.
Milk-Water Problem
Milk% in final mix = Total milk / Total volume × 100
After repeated replacement or combination — track milk and water separately.
Mean Price Constraint
C1 < Cm < C2 (mean price must lie between the two ingredient prices)
To verify the problem setup — if Cm is outside this range, something is wrong.
Relevant Exams
Mixture & Alligation is a consistent topic in SSC CGL Tier 1 and Tier 2, and banking exams. Repeated dilution and profit-based alligation are the hardest and most frequently asked variants.