Mixture and Alligation
About 6 min · Quantitative Aptitude · English
Time worked out from the lesson's 398 words at 150 words a minute, plus a minute for each question.
The topic deals with combining two or more components in specific ratios and understanding how their individual properties (price, concentration, strength) blend to create a mixture with averaged characteristics.
Key ideas
- Mixture: A combination of two or more components (liquids, metals, ingredients) mixed together. The resulting mixture has properties that are the weighted average of its components.
- Alligation: A method to determine the ratio in which two or more ingredients at different prices or concentrations must be mixed to produce a mixture of a desired mean value. The alligation rule uses the cross-difference of values.
- Mean Price/Concentration: When components A and B with values a and b are mixed in ratio m:n, the mixture's value = (m×a + n×b)/(m+n). This is the weighted average principle.
- Alligation Rule (Cross-Method): To mix two components with values C₁ and C₂ to get a mixture with mean value M, the ratio of quantities is (M − C₂):(C₁ − M). Visually represented as a cross diagram where you subtract diagonally.
- Replacement: When a portion of a mixture is removed and replaced with one of its pure components, the concentration of that component changes. After n replacements of fraction f each time, final quantity = initial × (1 − f)ⁿ for the non-replaced component.
Quick reference
- Alligation shortcut: Ratio = (M − C₂):(C₁ − M) where C₁ < M < C₂.
- Replacement formula: After n replacements of fraction f, remaining = Initial × (1−f)ⁿ.
- Water addition to reduce concentration: Water needed = (Original volume × Original %) / (Final %) − Original volume.
- Pure substance mixing: Pure has 100% concentration; pure dilutant has 0%.
- Ratio-to-percentage: If A:B = 3:7, then A = 30% and B = 70% of mixture.
Check yourself: 3 questions
1. A milkman wants to sell a milk-water mixture at ₹16 per litre and earn 25% profit. The cost price of milk is ₹16 per litre and water is free. In what ratio must water be mixed with milk?
2. A 60-litre mixture has 40% water. How much water should be added to make it a 50% water mixture?
3. A vessel contains 80 litres of milk. 20 litres of milk is taken out and replaced with water. This process is repeated once more. What is the final quantity of milk in the vessel?
This lesson is cut from the topic's note, not rewritten. Fact-checked on 17 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.
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