Introduction
The Alligation rule (mean proportion) gives a fast way to find the ratio in which two solutions of different concentrations must be mixed to obtain a desired concentration. This pattern is useful because it avoids algebra for many two-component mixture problems and is widely used in aptitude tests and quick estimations.
You'll learn a simple difference-based method and a short worked example that is easy to follow.
Pattern: Alligation Rule (Mean Proportion Method)
Pattern
Key concept: The ratio of the quantities of two mixtures (lower concentration : higher concentration) = (higher - mean) : (mean - lower).
Steps to apply:
1. Identify the lower concentration, higher concentration and the required mean (target) concentration.
2. Subtract: (higher - mean) and (mean - lower).
3. The two differences give the ratio of quantities of lower : higher respectively.
4. Simplify the ratio to smallest whole numbers and use parts to find actual amounts.
Step-by-Step Example
Question
You have solutions at 10% and 30%. In what ratio should they be mixed to get an 18% solution? If you want 50 litres of the final 18% solution, how many litres of each are needed?
Solution
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Step 1: Identify concentrations
Lower concentration = 10%; Higher concentration = 30%; Required mean = 18%.
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Step 2: Compute the differences
Difference for lower (against mean) = (higher - mean) = 30 - 18 = 12.
Difference for higher (against mean) = (mean - lower) = 18 - 10 = 8.
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Step 3: Form the ratio
Ratio (lower : higher) = (higher - mean) : (mean - lower) = 12 : 8.
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Step 4: Simplify the ratio
12 : 8 = 3 : 2 (divide both by 4). So mix 3 parts of 10% with 2 parts of 30%.
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Step 5: Apply to required final volume
Total parts = 3 + 2 = 5 parts. For 50 litres → one part = 50 ÷ 5 = 10 L.
Amount of 10% = 3 × 10 = 30 L. Amount of 30% = 2 × 10 = 20 L.
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Final Answer:
Mix in ratio 3 : 2 (10% : 30%). For 50 L → 30 L of 10% and 20 L of 30%.
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Quick Check:
Pure substance = 30×0.10 + 20×0.30 = 3 + 6 = 9 L. Final concentration = 9 ÷ 50 = 0.18 = 18% ✅
Quick Variations
1. If asked for ratio of higher : lower, simply reverse the order of the differences.
2. When volumes are given instead of desired mean, use alligation to check if the mean matches the weighted average.
3. For cost-based mixing (value alligation), replace concentrations with prices per unit - the same difference method applies.
Trick to Always Use
- Step 1 → Always label lower, mean (target), and higher concentrations clearly.
- Step 2 → Compute (higher - mean) and (mean - lower) exactly; these map to parts of lower and higher respectively.
- Step 3 → Simplify the ratio and convert parts to actual quantities using total parts.
Summary
Summary
In the Alligation Rule (Mean Proportion Method) pattern:
- The rule converts a mixing problem into two simple differences: (higher - mean) and (mean - lower).
- These differences give the ratio of quantities for lower and higher concentrations.
- The ratio (lower : higher) = (higher - mean) : (mean - lower).
- Use ratio parts to calculate actual amounts when total volume is known.
- Quick check: The final concentration must lie between the two given concentrations.
Hint: Differences (higher - mean) and (mean - lower) give parts for lower and higher respectively.
Common Mistakes: Swapping which difference corresponds to which component.