Rule of Alligation: Formula, Method and Solved Examples
Rule of Alligation is a method used to find the ratio in which two ingredients with different values or prices must be mixed to obtain a desired mean value. It uses the differences between the two given values and the mean value. The method is commonly applied to mixture, price, concentration and average-based questions.
What Is the Rule of Alligation?
Let the cheaper value be C, the dearer value be D, and the required mean value be M. Then, quantity of cheaper item : quantity of dearer item = (D − M) : (M − C). The mean value must lie between C and D. For example, if rice costing ₹40 per kg and ₹60 per kg is mixed to obtain rice worth ₹48 per kg, the ratio is (60 − 48):(48 − 40) = 12:8 = 3:2.
Rule of Alligation Formula & Tricks
Important Formulas
C is the cheaper value, D is the dearer value and M is the required mean value.
q₁ and q₂ are the quantities of the cheaper and dearer items respectively.
The total ratio parts are divided in proportion to the two quantities.
Quick Tricks
Write the cheaper value, mean value and dearer value in increasing order. Subtract the mean from the dearer value for the cheaper quantity, and subtract the cheaper value from the mean for the dearer quantity.
The required mean must be greater than the cheaper value and less than the dearer value. If it is outside this range, a positive mixture of both items is not possible.
After finding the ratio, multiply each value by its corresponding ratio part and divide by the total parts.
Rule of Alligation Concepts
Cross-Difference Method
Place C, M and D in increasing order. The difference D−M gives the ratio part of the cheaper item, while M−C gives the ratio part of the dearer item. Thus, cheaper quantity:dearer quantity = (D−M):(M−C).
Alligation with Prices
The mixture price must lie between the two individual prices. If ₹80 and ₹120 goods are mixed to obtain a mixture priced at ₹95, the ratio of ₹80 goods to ₹120 goods is (120−95):(95−80) = 25:15 = 5:3.
Alligation with Concentrations
Use the lower concentration as C, the higher concentration as D and the required concentration as M. The ratio of lower-strength solution to higher-strength solution is (D−M):(M−C).
Finding Quantities from the Ratio
If the ratio of two components is a:b and the total mixture is T, their quantities are Ta/(a+b) and Tb/(a+b). For a 2:1 ratio in eighteen litres, the quantities are 18×2/3 = 12 litres and 18×1/3 = 6 litres.
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Alligation Rule: Cross Method
Learn how to apply the rule of alligation using the cross method to compare mixture values and determine the required ratio of ingredients.
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Rule of Alligation: Quick Revision
Use the cross-difference rule to find the ratio of two components producing a given mean value.
- For cheaper value C, dearer value D and mean value M: cheaper quantity:dearer quantity = (D−M):(M−C).
- The mean value must lie strictly between the two component values for positive quantities of both.
- The difference opposite a component gives that component's quantity ratio.
- Always simplify the ratio before calculating actual quantities.
- For ratio a:b and total T, quantities are Ta/(a+b) and Tb/(a+b).
- Verify the result using the weighted mean formula M = (Cq₁ + Dq₂)/(q₁ + q₂).
Rule of Alligation FAQs
What is the rule of alligation formula?
If C is the cheaper value, D is the dearer value and M is the mean value, then cheaper quantity:dearer quantity = (D−M):(M−C).
What ratio is obtained when ₹30 and ₹50 items are mixed to get a mean price of ₹38?
The ratio is (50−38):(38−30) = 12:8 = 3:2. Therefore, ₹30 items and ₹50 items are mixed in the ratio 3:2.
How much of each solution is needed to make 24 litres of a 30% solution from 20% and 50% solutions?
The ratio is (50−30):(30−20) = 2:1. Hence, the quantities are 24×2/3 = 16 litres of 20% solution and 24×1/3 = 8 litres of 50% solution.
Can alligation be used when the required mean is equal to one component value?
Only a zero quantity of the other component would be required. For a positive mixture of both components, the mean must lie strictly between their values.
Why does the difference from the dearer value give the cheaper quantity?
The cheaper item needs a larger quantity when the mean is closer to the cheaper value. Therefore, its ratio part is the opposite difference, D−M.
How can an alligation ratio be verified?
Multiply each component value by its ratio quantity, add the products and divide by the sum of the ratio parts. For values 40 and 60 in ratio 3:2, the mean is (40×3+60×2)/5 = 48.