Alligations and Mixture

What are Alligations and Mixture?

Alligation and Mixture is a powerful mathematical technique used to solve problems involving the mixing of two or more ingredients with different prices, concentrations, or properties to create a mixture with a desired average value.
A mixture is created by combining two or more substances together. Alligation helps us determine the ratio in which these ingredients must be mixed to achieve a specific mean value (such as average price, concentration, or strength).
This concept is widely applicable across various quantitative aptitude topics including profit and loss, percentage problems, simple interest calculations, speed and distance problems, and concentration-based questions.
The technique is particularly useful in competitive exams as it provides a faster alternative to traditional algebraic methods for solving mixture problems.

What is Alligation?

Alligation is a mathematical rule that helps us find the ratio in which two or more ingredients at different prices must be mixed to produce a mixture of a desired price (mean price).
When two ingredients with costs 'c' (cheaper) and 'd' (dearer) are mixed to form a mixture with mean price 'm', where c < m < d, the rule of alligation states:

$(Cheaper quantity) : (Dearer quantity) = (d – m) : (m – c)$

The key principle is that the ratio of quantities is inversely proportional to the difference in their prices from the mean price.
This cross-difference method is visualized using the alligation diagram where the mean price is placed in the center, with cheaper price on the left and dearer price on the right.

What is Mixture?

A mixture is a combination of two or more ingredients or substances mixed together in specific proportions to create a new entity with desired properties.
Example: When two varieties of sugar priced at ₹40/kg and ₹50/kg are mixed to create a new variety priced at ₹44/kg, this forms a mixture.
The cost, concentration, or property of a unit quantity of the mixture is called the mean value or mean price.
Mixture problems involve finding the quantities of individual components, the resulting mean value, or the ratio in which ingredients should be combined.
Common real-world applications include mixing milk and water, blending different varieties of tea or coffee, combining metals to form alloys, and diluting acid solutions.

Formulas

Formula 1: Basic Alligation Rule

When two commodities with different prices are mixed together to obtain a mixture with a mean price:

$\frac{\text{Quantity of Cheaper}}{\text{Quantity of Dearer}} = \frac{\text{C.P of Dearer(d) - Mean Price(m)}}{\text{Mean Price(m) – C.P. of Cheaper(c)}}$

This formula calculates the ratio in which two ingredients must be mixed.
Remember: The difference is taken from the opposite ingredient (cross-multiplication pattern).

Formula 2: Repeated Replacement (Successive Operations)

When a container holds 'x' units of liquid A, from which 'y' units are removed and replaced with another liquid (like water), and this operation is repeated 'n' times:

Final quantity of pure liquid A = $x \times (1 - \frac{y}{x})^n$ units

After each operation, the concentration of the original liquid decreases exponentially.
This formula is crucial for solving problems involving successive dilution or replacement.

Formula 3: Alternative Form for Replacement

If a container contains 'x' units of pure liquid and 'y' units are replaced with water in each operation, repeated 'n' times:

Quantity of pure liquid remaining = $x \times (1 - \frac{y}{x})^n$ units

The fraction $(1 - \frac{y}{x})$ represents the remaining proportion after each replacement.
Raising this fraction to power 'n' accounts for the cumulative effect of multiple operations.

Formula 4: Mean Price Calculation

When quantities $q_1$ and $q_2$ of two ingredients with prices $p_1$ and $p_2$ are mixed:

Mean Price = $\frac{q_1 \times p_1 + q_2 \times p_2}{q_1 + q_2}$

This weighted average formula calculates the resultant price per unit of the mixture.

Important Concepts and Tips

Weighted Average Principle: The mean price always lies between the prices of the individual ingredients being mixed. If you get a mean price outside this range, recheck your calculations.
Cross Multiplication: In the alligation diagram, always subtract diagonally (cross-wise). The difference between dearer and mean goes with cheaper quantity, and vice versa.
Ratio Simplification: After finding the ratio using alligation, always simplify it to its lowest terms for the final answer.
Unit Consistency: Ensure all prices are in the same units (per kg, per liter, etc.) before applying alligation formulas.
Multiple Mixtures: For problems involving more than two ingredients, solve by taking pairs of ingredients at a time.
Replacement vs. Addition: Be clear whether the problem involves replacing (removing then adding) or simply adding to the mixture, as the formulas differ.

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: Using alligation rule:

Cheaper price (c) = ₹30, Dearer price (d) = ₹40, Mean price (m) = ₹35

Ratio = (d - m) : (m - c) = (40 - 35) : (35 - 30) = 5 : 5 = 1 : 1
Answer: The two varieties should be mixed in the ratio 1:1

Example 2: Replacement Problem

Problem: A vessel contains 60 liters of milk. 6 liters are removed and replaced with water. This process is repeated once more. What is the final quantity of pure milk?
Solution: Using replacement formula:

x = 60 liters, y = 6 liters, n = 2

Final quantity = $60 \times (1 - \frac{6}{60})^2 = 60 \times (1 - 0.1)^2 = 60 \times 0.9^2 = 60 \times 0.81 = 48.6$ liters
Answer: 48.6 liters of pure milk remains

Example 3: Finding Mean Price

Problem: 20 kg of rice at ₹25/kg is mixed with 30 kg of rice at ₹35/kg. What is the price per kg of the mixture?
Solution: Using mean price formula:

Mean price = $\frac{20 \times 25 + 30 \times 35}{20 + 30} = \frac{500 + 1050}{50} = \frac{1550}{50} = ₹31/kg$
Answer: The mixture costs ₹31 per kg

Common Mistakes to Avoid

Incorrect Cross-Multiplication: Students often subtract in the wrong direction. Remember: difference from dearer goes with cheaper quantity and vice versa.
Forgetting to Simplify Ratios: Always reduce your final ratio to simplest form. A ratio of 10:15 should be written as 2:3.
Confusion in Replacement Formula: The formula $(1 - \frac{y}{x})^n$ must have 'y' (removed quantity) divided by 'x' (total quantity), not the other way around.
Unit Mismatch: Mixing different units (liters with milliliters, kg with grams) without conversion leads to wrong answers.
Misidentifying Cheaper and Dearer: Always identify which ingredient is cheaper and which is dearer before applying the alligation rule.
Applying Wrong Formula: Don't use replacement formula for simple mixing problems and vice versa. Understand the problem type first.

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-liter 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 liters of pure alcohol. 8 liters are removed and replaced with water. If this operation is performed 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 cost price but gains 25%. Find the ratio of water to milk in the mixture.

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