In-Depth: Mixtures and Alligations

CLAT Application & Relevance

Importance: Low. Direct questions on Mixtures and Alligations are rare in the modern CLAT. However, the underlying principle of combining elements with different properties to achieve a desired average (which is what Alligation is) is essentially a weighted average concept. Understanding this can help in complex DI caselets where averages are involved.

How it's tested: Rarely as a standalone problem. The logic might be implicitly applied within a larger DI set, e.g., combining two groups of students with different average scores or mixing solutions with different concentrations. Replacement problems (where a portion of mixture is removed and replaced with another substance) are a common advanced type.

Section 1: Core Concepts & Formulas

A Mixture is the result of combining two or more distinct ingredients or substances. Alligation (also known as the Rule of Alligation) is a method to determine the ratio in which two ingredients of different prices/concentrations must be mixed to produce a mixture of a desired mean price/concentration.

The Rule of Alligation (Graphical Representation)

This rule helps find the ratio of quantities when two items (or solutions, groups, etc.) with different values are mixed to form a blend with a mean value.

        Quantity of Cheaper (C)     Quantity of Dearer (D)
                   \            /
                    \          /
                      Mean (M)
                     /                            /                      (D - M)            (M - C)

        Ratio of Quantity of Cheaper : Quantity of Dearer = (D - M) : (M - C)

'C' is the value (price, concentration, etc.) of the cheaper/lower quality ingredient.
'D' is the value (price, concentration, etc.) of the dearer/higher quality ingredient.
'M' is the mean value (price, concentration, etc.) of the mixture.
The differences are taken diagonally: (Dearer Value - Mean Value) and (Mean Value - Cheaper Value).

Important Concepts for Mixtures (Beyond Alligation)

Fraction of Component: If a mixture of X and Y is in ratio A:B, then Fraction of X = A / (A+B), Fraction of Y = B / (A+B).
Removal and Replacement: When a certain amount of mixture is removed and replaced by one of the components (e.g., water), the quantity of the other component changes. This often requires careful step-by-step calculation.

Section 2: Solved CLAT-Style Examples

Example 1: Basic Alligation Rule Application (Caselet)

Passage Context: A law firm needs to buy two types of specialized legal books to stock its library. Book A costs ₹400 per copy, and Book B costs ₹650 per copy. The firm wants to maintain an average cost of ₹500 per book in its new purchase batch.

Question: "In what ratio should the firm purchase Book A to Book B to achieve their desired average cost?"

Detailed Solution:
1. Identify Values: Cheaper (C) = ₹400 (Book A) Dearer (D) = ₹650 (Book B) Mean (M) = ₹500
2. Apply the Alligation Rule (D-M : M-C): Ratio of Quantity of Book A : Quantity of Book B = (D - M) : (M - C) = (650 - 500) : (500 - 400) = 150 : 100
3. Simplify the Ratio: Divide both by 50. = 3 : 2.
Answer: The firm should purchase Book A to Book B in the ratio 3:2.

Example 2: Mixture Replacement Problem (Advanced Application)

Passage Context: A large jar in a forensic lab contains a solution of 100 liters with alcohol and water in the ratio 4:1. 20 liters of the mixture are removed from the jar, and then 20 liters of pure water are added to the jar.

Question: "What is the new ratio of alcohol to water in the jar after the process?"

Detailed Solution:
1. Initial Quantities: Total Mixture = 100 liters. Alcohol : Water = 4 : 1. Alcohol = (4/5) * 100 = 80 liters. Water = (1/5) * 100 = 20 liters.
2. After Removing 20 Liters of Mixture: When mixture is removed, components are removed in the same ratio. Amount of Alcohol removed = (4/5) * 20 = 16 liters. Amount of Water removed = (1/5) * 20 = 4 liters. Remaining Alcohol = 80 - 16 = 64 liters. Remaining Water = 20 - 4 = 16 liters. Total Remaining Mixture = 64 + 16 = 80 liters. (This confirms our calculation: 100 - 20 = 80).
3. After Adding 20 Liters of Pure Water: New Alcohol = 64 liters (since only water was added). New Water = Remaining Water + Added Water = 16 + 20 = 36 liters. New Total Mixture = 64 + 36 = 100 liters.
4. New Ratio of Alcohol to Water: New Ratio = 64 : 36. Divide both by their HCF (4). New Ratio = 16 : 9.
Answer: The new ratio of alcohol to water in the jar is 16:9.