Alligations and Mixture

What are Alligations and Mixture?

Alligation and mixture is a set of techniques used to solve problems about combining two or more ingredients with different prices, concentrations or strengths to obtain a mixture with a desired average (mean) value.
A mixture is formed by combining substances in specified proportions. Alligation helps quickly determine the ratio in which ingredients must be mixed to get a required mean value (for example, mean price or concentration).
These techniques are commonly used in quantitative aptitude problems covering concentrations, price mixing, replacement, and practical examination questions.
Alligation provides a faster and more straightforward alternative to algebraic methods for many mixture problems encountered in competitive exams and real-life situations.

What is Alligation?

Alligation is a rule to find the ratio in which two (or more) ingredients at different prices or strengths should be mixed to obtain a mixture of a given mean price or strength.
When two ingredients with prices \(c\) (cheaper) and \(d\) (dearer) are mixed to obtain a mean price \(m\), where \(c < m < d\), the alligation rule gives:

(Cheaper quantity) : (Dearer quantity) = (d − m) : (m − c)

The ratio is inversely proportional to the difference of each component from the mean. This is often shown using the alligation diagram with the mean in the centre and the two prices at the ends.

What is a Mixture?

A mixture is a combination of two or more substances combined in certain proportions to form a new product with an aggregate property (such as average price or concentration).
Example: Mixing sugar at ₹40/kg with sugar at ₹50/kg to obtain a mixture priced at ₹44/kg.
The property (cost, concentration, etc.) per unit of the mixture is the weighted average (mean) of the components.
Mixture problems typically ask for the ratio of components, the resulting mean value, or quantities required to achieve a target mean.
Practical examples include blending milk and water, mixing different grades of tea/coffee, alloy formation, and solution dilutions.

Formulas

Formula 1: Basic Alligation Rule

For two commodities with costs \(c\) (cheaper) and \(d\) (dearer) mixed to get mean price \(m\):

\(\dfrac{\text{Quantity of Cheaper}}{\text{Quantity of Dearer}} = \dfrac{d - m}{m - c}\)

Use diagonal differences (cross subtraction) to set up the ratio and then simplify to lowest terms.

Formula 2: Replacement (Repeated Operations)

If a container has \(x\) units of a liquid A and \(y\) units are removed and replaced with another liquid (e.g., water), repeated \(n\) times, the remaining quantity of pure A is:

\(\text{Remaining pure A} = x\left(1 - \dfrac{y}{x}\right)^n\)

After each replacement, the proportion of the original liquid decreases multiplicatively by the factor \(\left(1 - \dfrac{y}{x}\right)\).

Formula 3: Mean (Weighted Average)

If \(q_1\) units at price \(p_1\) and \(q_2\) units at price \(p_2\) are mixed, the mean price is:

\(\text{Mean Price} = \dfrac{q_1 p_1 + q_2 p_2}{q_1 + q_2}\)

More generally, the weighted average formula applies for any number of ingredients.

Important Concepts and Tips

Weighted Average Principle: The mean always lies between the minimum and maximum component values. If the mean is outside that range, check your inputs.
Cross Differences: In alligation, subtract diagonally: (dearer − mean) pairs with cheaper quantity and (mean − cheaper) pairs with dearer quantity.
Simplify Ratios: Always reduce the final ratio to its simplest form before presenting the answer.
Unit Consistency: Make sure all rates are expressed in the same unit (per kg, per litre, etc.) before applying formulas.
Multiple Ingredients: For more than two components, combine ingredients pairwise or use the weighted average formula directly.
Replacement vs Addition: Distinguish between removing-and-replacing (replacement) and simple addition; they use different formulas.

Solved Examples

Example 1: Basic Alligation Problem

Problem: A shopkeeper mixes two varieties of rice costing ₹30/kg and ₹40/kg. In what ratio should they be mixed to get a mixture costing ₹35/kg?
Solution: Cheaper \(c=30\), Dearer \(d=40\), Mean \(m=35\). Ratio = (d − m) : (m − c) = (40 − 35) : (35 − 30) = 5 : 5 = 1 : 1.
Answer: Mix in the ratio 1 : 1.

Example 2: Replacement Problem

Problem: A vessel contains 60 litres of milk. 6 litres are removed and replaced with water. The operation is repeated once more. How much pure milk remains?
Solution: Here \(x=60\), \(y=6\), \(n=2\). Remaining milk \(=60\left(1-\dfrac{6}{60}\right)^2=60\times0.9^2=60\times0.81=48.6\) litres.
Answer: 48.6 litres of pure milk remain.

Example 3: Finding Mean Price

Problem: 20 kg of rice at ₹25/kg is mixed with 30 kg at ₹35/kg. What is the price per kg of the mixture?
Solution: Mean price \(=\dfrac{20\times25 + 30\times35}{20+30}=\dfrac{500+1050}{50}=\dfrac{1550}{50}=₹31/kg\).
Answer: ₹31 per kg.

Common Mistakes to Avoid

Wrong subtraction direction: Do not subtract the mean from the wrong term. Follow the diagonal subtraction rule in alligation.
Not simplifying ratios: Always reduce to lowest terms — for example 10:15 → 2:3.
Replacement formula confusion: Ensure the fraction is \(\dfrac{y}{x}\) where \(y\) is the removed quantity and \(x\) is the total, i.e. \((1 - \dfrac{y}{x})^n\).
Unit mismatches: Convert units (kg, g, litre, ml) before mixing; inconsistent units lead to incorrect answers.
Mislabeling cheaper/dearer: Identify which price is cheaper and which is dearer before applying alligation.
Using wrong formula: Check whether the problem requires mixing, replacement, or weighted averaging and select the proper formula.

Practice Problems

Problem 1: In what ratio must tea at ₹62/kg be mixed with tea at ₹72/kg so that the mixture costs ₹65/kg?
Problem 2: A 40-litre mixture contains milk and water in the ratio 3:1. How much water should be added to make the ratio 2:1?
Problem 3: A container has 80 litres of pure alcohol. 8 litres are removed and replaced with water. If this is done 3 times, how much pure alcohol remains?
Problem 4: Two varieties of pulses costing ₹15/kg and ₹20/kg are mixed and the mixture is sold at ₹18/kg. If the profit is 20%, in what ratio were the pulses mixed?
Problem 5: A dishonest milkman mixes water with milk and sells it at the cost price but gains 25%. Find the ratio of water to milk in the mixture.

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