Mixture and Allegation
About 7 min · Numerical and Mathematical Ability · English
Time worked out from the lesson's 460 words at 150 words a minute, plus a minute for each question.
These problems involve combining two or more items with different prices, strengths, or concentrations to form a mixture, or replacing part of a mixture with something else. The alligation rule provides a shortcut formula to find ratios without lengthy equation-solving, making it a time-saver in the exam.
Key ideas
- Mixture: A combination of two or more components (liquids, solids, or abstract quantities like prices) in a certain ratio or quantity.
- Mean or Average Price/Strength: The effective price or concentration of the final mixture, lying between the individual prices or concentrations of the components.
- Alligation Rule: A visual, ratio-based method to find the proportion in which two ingredients are mixed when the mean value is known. It eliminates the need to set up and solve simultaneous equations.
- Replacement: Removing a certain quantity of a mixture and replacing it with a pure component (often water or a cheaper item), which dilutes the original mixture. Repeated replacement follows an exponential decay formula.
- Two-component vs. multi-component mixtures: Multi-component problems, if they appear, can be broken into pairs.
Watch out for
- Inverting the alligation ratio: Wrong: Taking (M − C₁) for the cheaper item. Correct: Cheaper item gets (C₂ − M), dearer gets (M − C₁). Use the cross-multiplication visual to avoid confusion.
- Forgetting to remove the original component in replacement: Wrong: Thinking that adding water increases milk quantity. Correct: When x litres are removed from a mixture, the milk removed = (milk fraction) × x. Only then is water added.
- Using percentage instead of absolute values in alligation: Wrong: Directly using percentages like 40% and 60% without converting to actual concentrations if needed. Correct: Ensure C₁, C₂, and M are in the same unit (rupees per kg, percentage concentration, grams per litre, etc.).
Quick reference
- Alligation ratio formula: (Dearer − Mean) : (Mean − Cheaper).
- Single replacement: Final = Initial × (1 − removed/total).
- n replacements: Final = Initial × (1 − removed/total)ⁿ.
- Mean price/concentration: Sum of (price × quantity) divided by total quantity.
- Draw the alligation cross: Visually connect cheaper, mean, dearer to avoid ratio inversion.
Check yourself: 3 questions
1. A vessel contains 60 litres of milk. 12 litres of milk is taken out and replaced with water. What is the ratio of milk to water in the resulting mixture?
2. In what ratio must rice at Rs. 12 per kg be mixed with rice at Rs. 18 per kg so that the mixture is worth Rs. 15 per kg?
3. A container has 80 litres of milk. 20 litres of milk is removed and replaced with water. This process is repeated once more. What is the final quantity of milk in the container?
This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 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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