Alligation: Formula, Rules and Solved Questions
Alligation is a method for finding the ratio in which two ingredients with different prices or concentrations must be mixed to obtain a required mean value. It uses the differences between the required mean and the two given values. The method applies to mixture price, concentration, purity and similar weighted-average questions.
What Is Alligation?
If the lower value is L, the higher value is H and the required mean is M, then the quantity ratio is lower-value quantity : higher-value quantity = (H − M) : (M − L). The same rule applies to prices, concentrations, efficiencies and other directly comparable quantities. For example, rice costing ₹30 per kg and ₹50 per kg gives a mixture costing ₹38 per kg in the ratio (50 − 38):(38 − 30) = 12:8 = 3:2.
Alligation Formula & Tricks
Important Formulas
Use this when two values are mixed to obtain a specified mean value.
Here, q₁ and q₂ are the quantities and v₁ and v₂ are their corresponding values.
The prices must use the same unit, such as rupees per kilogram.
Use percentages or fractions consistently for all concentrations.
Quick Tricks
Place the lower value and higher value on opposite sides of the required mean. Subtract the mean from the higher value and the lower value from the mean. These cross-differences give the opposite quantities.
The required mean must lie between the two given values. If it equals one given value, only that value is required; if it lies outside the range, ordinary two-value alligation is not possible.
After obtaining the ratio, add its parts and multiply each part by total quantity divided by total ratio parts.
Alligation Concepts
Alligation Rule for Prices
If the cheaper price is C, the dearer price is D and the mean price is M, then cheaper quantity : dearer quantity = (D − M):(M − C). The mean price must be between C and D, and all prices must be expressed in the same unit.
Alligation for Concentration and Purity
For lower concentration L, higher concentration H and required concentration M, the quantity ratio is lower concentration : higher concentration = (H − M):(M − L). Percentages can be used directly because the common factor of 100 cancels.
Finding Actual Quantities from a Ratio
If the ratio is a:b and the total quantity is T, then the quantities are Ta/(a + b) and Tb/(a + b). If one quantity is given, multiply both ratio parts by the factor obtained from that quantity.
Verification Using Weighted Average
For quantities in the ratio a:b and values v₁ and v₂, the resulting mean is (av₁ + bv₂)/(a + b). This calculated mean must equal the required value.
Alligation Video Lessons
Watch short topic-wise lessons for quick revision.
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.
Practice Alligation Questions
Practise published questions related to this topic.
Alligation Quick Quiz
Attempt 5 questions and check your score instantly.
Keep practising
Practice more Alligation questions in the PrepShots app and continue from your current topic.
Practice More Questions - Start ₹1 Trial →Quick Revision Notes
Alligation Revision Points
Use these rules to revise the method and check calculations.
- For lower value L, higher value H and mean M, the ratio is L-value quantity : H-value quantity = (H − M):(M − L).
- The required mean must lie between the two component values.
- Use the same units for all prices, concentrations or other values before applying alligation.
- For a ratio a:b and total T, quantities are Ta/(a+b) and Tb/(a+b).
- Verify the answer with the weighted-average formula: (q₁v₁ + q₂v₂)/(q₁ + q₂).
- A mean closer to the lower value requires more of the lower-value component; a mean closer to the higher value requires more of the higher-value component.
Alligation FAQs
What is the alligation formula?
If L is the lower value, H is the higher value and M is the required mean, then lower-value quantity : higher-value quantity = (H − M):(M − L).
What ratio of ₹40 and ₹60 articles gives a mixture worth ₹45 per kg?
The ratio is (60 − 45):(45 − 40) = 15:5 = 3:1. Thus, ₹40 articles and ₹60 articles are mixed in the ratio 3:1.
How much of each solution is needed to make 60 litres of a 30% solution from 20% and 50% solutions?
The ratio is (50 − 30):(30 − 20) = 2:1. Therefore, use 60 × 2/3 = 40 litres of the 20% solution and 20 litres of the 50% solution.
Can alligation be used when the required mean is outside the two given values?
No. In ordinary two-component alligation, the required mean must be between the lower and higher values. A mean outside this range cannot be obtained by mixing only those two components.
Why are the differences placed opposite the quantities in alligation?
The quantity of one component is inversely related to its distance from the required mean. Therefore, the lower-value quantity receives the difference H − M, while the higher-value quantity receives M − L.
How do you verify an alligation ratio?
Use the weighted average. For a ratio a:b and values v₁ and v₂, calculate (av₁ + bv₂)/(a+b) and check whether it equals the required mean.