Mixture and Alligation — SSC CGL Study Notes
Overview
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.
Mastery of this topic is essential because questions appear in both direct ratio-calculation form and as word problems involving replacement, repeated mixing, or finding optimal combinations. The alligation rule provides a shortcut method that eliminates lengthy algebraic manipulation, making it invaluable under time pressure. Students must develop comfort with the visual cross-method diagram and recognize when to apply the weighted average concept versus the alligation shortcut.
The core competency tested is your ability to set up ratio relationships correctly, handle percentage-to-ratio conversions, and work backward from a mixture's properties to find constituent quantities or their proportions.
Key Concepts
- 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.
- Percentage to Ratio: A mixture with 40% A and 60% B has A:B = 40:60 = 2:3. Always simplify ratios to lowest terms for easier calculation.
- Successive Mixing: When mixtures are combined in stages, apply the weighted average principle at each stage or use alligation iteratively.
Formulas / Key Facts
1. Alligation Formula: To obtain mean M from C₁ and C₂, mix in ratio = (M − C₂):(C₁ − M).
2. Weighted Average: If quantities Q₁ and Q₂ with values V₁ and V₂ are mixed, mean value = (Q₁V₁ + Q₂V₂)/(Q₁ + Q₂).
3. Single Replacement Formula: Final quantity of a component after one replacement = Initial quantity × (1 − fraction removed).
4. Multiple Replacements: After n identical replacements of fraction f, remaining quantity of non-replaced component = Initial × (1 − f)ⁿ.
5. Milk-Water Ratio Conversion: If milk:water = a:b, then milk% = [a/(a+b)]×100 and water% = [b/(a+b)]×100.
6. Cost Price in Alligation: When mixing items at different costs to achieve a target selling price or profit margin, use cost prices as C₁ and C₂.
7. Concentration Problems: Pure substance has 100% concentration; pure dilutant (water, oil) has 0% of the active ingredient.
8. Ratio Addition: When mixing A:B = m:n with C:D = p:q in quantities x and y, the combined mixture requires finding the total of each component.
Worked Examples
Example 1: Basic Alligation Problem: A merchant mixes rice at ₹30/kg with rice at ₹40/kg to sell the mixture at ₹35/kg. In what ratio should he mix them?
Solution: Using alligation rule: C₁ = 30 (cheaper), C₂ = 40 (dearer), M = 35 (mean) Ratio = (M − C₂):(C₁ − M) = (35 − 40):(30 − 35) = (−5):(−5) Taking absolute values: 5:5 = 1:1
Answer: Mix in ratio 1:1.
Example 2: Percentage Concentration Problem: How much water must be added to 60 litres of milk containing 8% water to make it 20% water?
Solution: Initial: 8% water means 92% milk. Final: 20% water means 80% milk. Milk quantity remains constant = 60 × 0.92 = 55.2 litres. If final mixture has 80% milk: Total mixture = 55.2/0.80 = 69 litres. Water to be added = 69 − 60 = 9 litres.
Answer: 9 litres.
Example 3: Replacement Problem Problem: A container has 40 litres of milk. 4 litres are removed and replaced with water. This is done twice. What is the final quantity of milk?
Solution: Fraction removed each time = 4/40 = 1/10. After first replacement: Milk = 40 × (1 − 1/10) = 40 × 9/10 = 36 litres. After second replacement: Milk = 36 × 9/10 = 32.4 litres. Or directly: Milk = 40 × (9/10)² = 40 × 81/100 = 32.4 litres.
Answer: 32.4 litres.
Example 4: Three-Component Mixture (Iterative Alligation) Problem: Mix liquids at ₹20, ₹30, and ₹40 per litre to get a mixture at ₹28 per litre. If the first two are in ratio 2:3, find the ratio of all three.
Solution: Step 1: Mix ₹20 and ₹30 in ratio 2:3. Mean of this mix = (2×20 + 3×30)/5 = (40+90)/5 = 26. Step 2: Now mix this ₹26 mixture with ₹40 to get ₹28. Using alligation: (28−40):(26−28) = (−12):(−2) = 12:2 = 6:1. So ₹26 mixture : ₹40 = 6:1. Since ₹26 mixture contains ₹20:₹30 = 2:3, scale this by 6 parts. Final ratio ₹20:₹30:₹40 = 12:18:5.
Answer: 12:18:5.
Common Mistakes
Mistake 1: Confusing which value to subtract in alligation → Fix: Always subtract diagonally in the cross-method: (Mean − Lower value):(Higher value − Mean). Both differences should be positive.
Mistake 2: Not converting percentages to actual quantities → Fix: When dealing with percentages, first find the actual quantity of each component. A 25% solution of 100L has 25L solute, not 25% of mixture.
Mistake 3: Using simple average instead of weighted average → Fix: If 2L of ₹30 liquid mixes with 3L of ₹40 liquid, mean ≠ (30+40)/2. Use (2×30 + 3×40)/(2+3) = 36.
Mistake 4: Incorrect replacement formula application → Fix: The formula Final = Initial × (1−f)ⁿ applies only to the component NOT being added. The added component follows a different pattern.
Mistake 5: Forgetting to simplify ratios → Fix: Always reduce ratios to simplest form. The ratio 10:15:20 should be written as 2:3:4 to avoid calculation errors and match answer options.
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.
- Cross-verify with weighted average: Your answer ratio should satisfy (Q₁V₁ + Q₂V₂)/(Q₁+Q₂) = Mean value.