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Alligation and Mixtures – Concept, Tricks & Solved Examples

ALLIGATION

Introduction

The technique of alligation is applicable when two extreme values are given and an average value needs to be calculated. This method is commonly used in Percentage, Simple Interest, Ratio & Proportion, and Averages.

Definition

Alligation is a rule used to determine the ratio in which two or more ingredients at different prices must be mixed to achieve a given mean price.

Alligation Rule (Formula)

If two ingredients are mixed, where:

Then the ratio in which these two must be mixed is:

Required Ratio = (D-C)(M-C)

Shortcut Trick:

Draw a simple cross-method diagram:

Cheaper Price(C) Dearer Price(D)
          \       /
        Mean Price (M)

Then, use the alligation rule by subtracting diagonally:

= (D-C)(M-C)

Key Points to Remember:

MIXTURES

Definition

A mixture or alloy contains two or more ingredients of certain quantity mixed together to get a desired proportion.

Example: A 1-liter mixture contains 250 ml water and 750 ml milk.

  • Water = 1/4 of the mixture
  • Milk = 3/4 of the mixture
  • Or in percentage: Water = 25% and Milk = 75%

Concepts & Short Tricks

Concept 1: Finding the Quantity of an Ingredient in a Mixture

Example 1 (SSC, Bank PO, RRB NTPC, UPSC): A mixture contains alcohol and water in the ratio 4:3. If 7 liters of water is added, the ratio becomes 3:4. Find the initial quantity of alcohol in the mixture.

Solution:

Let the quantity of alcohol and water be 4x:3x.

After adding 7 liters of water,

4x/(3x + 7) = 3/4

Solving for x,

4x × 4 = (3x + 7) × 3
16x = 9x + 21
7x = 21
x = 3

Thus, alcohol = 4 × 3 = 12 liters. ✅

Concept 2: Quantity of Ingredient to be Added to Increase Ingredient Percentage

Example 2 (SSC CGL, UPSC, CAT, Banking PO): A mixture of milk and water contains 80% milk. In 50 liters of such a mixture, how much water must be added to reduce milk percentage to 50%?

Solution:

Total mixture = 50 liters

Milk = 80% of 50 = 40 liters

Water = 20% of 50 = 10 liters

Let x liters of water be added.

Now, the new mixture:

  • Milk = 40 liters
  • Water = 10 + x liters
  • Total = 50 + x liters
(10 + x)/(50 + x) = 50%
1/2 (50 + x) = 10 + x
50 + x = 20 + 2x
x = 30 liters

Thus, 30 liters of water must be added. ✅

Concept 3: Changing the Ratio of Ingredients in a Mixture

Example 3 (SSC, RRB, CAT, Banking PO): A 729 ml mixture contains milk and water in the ratio 7:2. How much more water must be added to make the ratio 7:3?

Solution:

Milk in the original mixture = (7/9) × 729 = 567 ml

Water in the original mixture = (2/9) × 729 = 162 ml

Let x ml of water be added. Then,

567/(162 + x) = 7/3
Cross multiply:
567 × 3 = 7 × (162 + x)
1701 = 1134 + 7x
7x = 567
x = 81

Thus, 81 ml of water must be added. ✅

Concept 4: Replacement of a Part of a Solution

If a vessel contains A liters of milk, and B liters is withdrawn and replaced by water n times, then the quantity of milk left after nth operation is:

Final Quantity = A ⁡ ( 1 - B A ) ⁡ n

Example 4 (Bank PO, SSC, RRB NTPC, UPSC): A container has 100 liters of milk. 20 liters is taken out and replaced with water. This process is repeated 3 times. Find the quantity of milk left.

Solution:

Milk left = 100(1 - 20/100)³
= 100(80/100)³
= 100 × 512/1000
= 51.2 liters

Thus, 51.2 liters of milk is left after 3 operations. ✅

Final Short Tricks to Remember 🚀

  1. Mean Price must be between the Cheaper and Dearer prices.
  2. Alligation Rule: Use (D - M)/(M - C) to find the required ratio.
  3. Replacement Formula: Use A(1 - B/A)ⁿ for repeated replacement problems.
  4. Milk-Water Mixture Problems: Convert ratios into fractions and solve.
  5. Quick Calculation: Draw the Alligation Diagram for easy application.