Mixture and Alligation is a high-scoring topic in IBPS PO Prelims that appears both as standalone questions and within Data Interpretation sets. The concept helps you solve problems involving combinations of two or more ingredients at different prices, concentrations, or ratios—without setting up lengthy algebraic equations.
This topic connects directly with Ratio and Proportion, Percentage, and Profit-Loss. Mastering the alligation rule gives you a 30-second shortcut for problems that would otherwise take 2-3 minutes. Expect 1-2 direct questions in Prelims, often disguised as profit/loss or average-based word problems.
Students must be comfortable with: (1) the alligation cross method, (2) repeated replacement formula, and (3) converting between ratio and percentage forms quickly.
Key Concepts
**Mixture**: A combination of two or more ingredients mixed together. Each ingredient has a characteristic value (price, concentration, percentage, etc.).
**Alligation Rule**: When two ingredients at different values are mixed, the ratio of their quantities is inversely proportional to the difference of their values from the mean value.
**Mean Price/Value**: The weighted average characteristic of the final mixture. It always lies between the values of the two ingredients.
**Cheaper and Dearer**: In alligation problems, the ingredient with lower value is "cheaper" and the one with higher value is "dearer." The mean always falls between these two.
**Replacement Problems**: When part of a mixture is removed and replaced with a pure ingredient, the concentration of the other ingredient decreases progressively with each operation.
**Three-Component Mixtures**: Solve by applying alligation twice—first between two components, then combining with the third using the intermediate result.
Formulas / Key Facts
**The Alligation Rule (Cross Method)** ``` Quantity of Cheaper Dearer Value − Mean Value ─────────────────── = ───────────────────────── Quantity of Dearer Mean Value − Cheaper Value ``` Ratio = (d − m) : (m − c), where c = cheaper value, d = dearer value, m = mean value
**Repeated Replacement Formula** When a container has V litres of pure liquid A, and x litres is removed and replaced with liquid B, repeated n times:
Final quantity of A = V × (1 − x/V)ⁿ
Concentration of A after n operations = (1 − x/V)ⁿ × 100%
**Mixing Two Mixtures** If mixture 1 has components in ratio a:b and mixture 2 has components in ratio c:d, and they are mixed in ratio M:N, find the component-wise weighted average.
**Key Facts to Remember**
The alligation diagram always gives ratio by quantity, not by value
For percentage problems, treat percentages as "prices" in the alligation formula
Replacement formula assumes the mixture is homogeneous before each removal
When water is added to a solution, quantity of solute remains constant
Worked Examples
**Example 1: Basic Alligation** In what ratio must rice at ₹42/kg be mixed with rice at ₹60/kg to get a mixture worth ₹48/kg?
**Example 2: Concentration Problem** A vessel contains 80 litres of milk. 8 litres is removed and replaced with water. This process is repeated twice more. What is the final quantity of milk?
*Solution:* Using repeated replacement formula:
V = 80 litres, x = 8 litres, n = 3 operations
Final milk = 80 × (1 − 8/80)³
= 80 × (1 − 1/10)³
= 80 × (9/10)³
= 80 × 729/1000
= 58.32 litres
**Answer: 58.32 litres**
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**Example 3: Finding the Mean** 60 kg of an alloy A contains copper and zinc in ratio 5:3. 100 kg of alloy B contains copper and zinc in ratio 3:2. If both alloys are melted together, find the ratio of copper to zinc in the new alloy.
Alloy B (100 kg): Copper = 100 × 3/5 = 60 kg, Zinc = 100 × 2/5 = 40 kg
Total Copper = 37.5 + 60 = 97.5 kg Total Zinc = 22.5 + 40 = 62.5 kg
Ratio = 97.5 : 62.5 = 975 : 625 = 39 : 25
**Answer: 39 : 25**
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**Example 4: Profit-Loss Based Alligation** A shopkeeper mixes two varieties of tea costing ₹80/kg and ₹120/kg and sells the mixture at ₹110/kg making 10% profit. Find the mixing ratio.
**Reversing the alligation ratio**: Students often write (m − c) : (d − m) instead of (d − m) : (m − c). Remember: the difference from the dearer goes with the cheaper quantity.
**Forgetting to find CP before applying alligation**: In profit/loss problems, students apply alligation using SP directly. Always convert SP to CP first using the profit/loss percentage.
**Confusing number of operations with total removals**: In "process repeated 3 more times," n = 4 total operations, not 3. Read carefully whether it says "3 times" or "3 more times."
**Using wrong base in replacement problems**: The formula (1 − x/V)ⁿ uses the original volume V, not the remaining volume after removal. The volume stays constant because removed liquid is replaced.
**Adding percentages directly in mixture of mixtures**: When mixing two solutions of 20% and 30% concentration in equal parts, the result is 25%, not 50%. Percentages don't add—they average based on quantity.
Quick Reference
Alligation ratio = (Dearer − Mean) : (Mean − Cheaper)
Replacement formula: Final = V × (1 − x/V)ⁿ
Mean value always lies between cheaper and dearer values
For profit problems: Find CP of mixture first, then apply alligation
Water has 0% concentration and ₹0 cost—use these as values in alligation
Check units: ratio is always quantity to quantity, never value to value
👥 Study this together
Invite your prep group — read the same notes, then discuss doubts in this topic's shared room.