Averages, Mixtures, and Alligations: A Complete Guide

Understanding averages, mixtures, and alligations is vital for anyone preparing for entrance tests or aiming to strengthen their quantitative aptitude. These concepts not only feature prominently in exams like CAT, SSC, Railways, and GMAT, but they also mirror many real-life problem-solving scenarios, from cooking recipes to financial planning.

In this detailed blog, we’ll break down each of these concepts, show you why they matter for your exams, walk through problem types and solutions, and reveal the shortcuts and tips to boost your speed and accuracy.

What are Averages, Mixtures, and Alligations?

Averages

The average simply tells you the “central value” of a set of numbers. It’s calculated as:

$$\text{Average (A)}=\frac{\text{Sum of all values (S)}}{\text{Number of items (n)}}$$

For instance, if you scored 60, 70, and 80 in three tests, your average score is:

$$A=\frac{60+70+80}{3}=\frac{210}{3}=70$$

Averages crop up everywhere—finding class averages, average speed while traveling, or splitting expenses among friends.

Mixtures

A mixture is formed by combining two or more substances (called components), like blending different varieties of rice, milk and water, or alloys. In exams, mixtures often ask you to calculate the resulting concentration, price, or ratio of ingredients after mixing.

Alligations

Alligation is a specific rule that helps calculate the ratio in which two or more components should be mixed to achieve a desired average value (like price, concentration, etc.). It’s a shortcut, often faster than using algebraic equations, and becomes especially handy under time pressure.

Why Are These Topics Important for Entrance Exams?

• Weightage: Almost every major aptitude-based exam features these topics in the quantitative section.

• Real-World Application: Problems require logical reasoning, calculation speed, and understanding of practical situations.

• Time-Saving: Mastering alligation methods can halve your solution time for many mixture-based questions.

• Repeating Patterns: The types of questions from these areas are consistent every year.

Dive Deeper: Types of Problems and Detailed Solutions

1. Simple Average Problems

Problem: Find the average of 20, 30, and 50.

Solution: $(20+30+50)\mathrm{/}3=100\mathrm{/}3=33.33$

Weighted Average

Suppose a class has 10 boys (average age 15 years) and 20 girls (average age 13 years). What’s the average age of the class?

$$\text{Weighted Average}=\frac{10\times 15+20\times 13}{10+20}=\frac{150+260}{30}=\frac{410}{30}=13.67$$

2. Mixture Problems

Example 1: Finding the Mean Price

Two types of tea costing $30/kg and $50/kg are mixed in the ratio 2:3. What is the mean price?

Solution:

$$\text{Mean price}=\frac{2\times 30+3\times 50}{2+3}=\frac{60+150}{5}=\frac{210}{5}=\mathrm{\$}42\mathrm{/}kg$$

Example 2: Changing Concentration

You have 10L of milk with 20% water. How much water to add to make it 40%?

Let $x$ be the amount of water added.

$$\frac{0.2\times 10+x}{10+x}=0.4⟹\frac{2+x}{10+x}=0.4$$

$$\begin{gathered} \text{Solve for }x:2+x=0.4(10+x) \\ 2+x=4+0.4x \\ x-0.4x=4-2 \\ 0.6x=2⟹x=\frac{2}{0.6}\approx 3.33L \end{gathered}$$

3. Alligation Problems

The Alligation Rule

When mixing two ingredients of different values (say, cost or concentration):

$$\frac{\text{Quantity of Cheaper}}{\text{Quantity of Dearer}}=\frac{\text{Dearer Value}-\text{Mean Value}}{\text{Mean Value}-\text{Cheaper Value}}$$

Example: Mix sugar at ₹30/kg (cheaper) with ₹50/kg (dearer) to obtain a mixture at ₹40/kg.

$$\frac{\text{Cheaper}}{\text{Dearer}}=\frac{50-40}{40-30}=\frac{10}{10}=1:1$$

So, mix in equal proportions!

4. Complex Mixture Types

• Simple Mixture: Mixing two components directly.

• Compound Mixture: Mixing already prepared mixtures; needs repeated alligation or algebra.

Example: Alloy A (zinc:tin 5:2) and B (zinc:tin 3:4). Mix to get equal zinc and tin in the final mix.

Let alloy A’s zinc = 5/7; B’s zinc = 3/7. Final required: 1/2.

Alligation gives required ratio:

$$\frac{3\mathrm{/}7-1\mathrm{/}2}{1\mathrm{/}2-5\mathrm{/}7}=\frac{(6-7)\mathrm{/}14}{(7-10)\mathrm{/}14}=\frac{-1\mathrm{/}14}{-3\mathrm{/}14}=\frac{1}{3}$$

So, alloy A:alloy B = 1:3.

Tips and Tricks for Faster Solutions

• Learn the Cross Alligation Diagram: Visually subtract diagonally for ratios; saves time.

• Remember the Weighted Average Formula: Use for mean price/concentration or group averages.

• Practice Back Substitution: If unsure, plug answer options back into the formula for speed.

• Stay Alert for Variants: Some questions use time, speed, or profit/loss in alligation guise.

• Check Units and Terms: Always match units—price per kg, concentration per L, etc.

• Repeated Dilution Formula: For repeated replacement problems,

  $$\text{Final Quantity}=\text{Initial}\times {(1-\frac{\text{Taken out}}{\text{Total}})}^n$$
  

Common Problem Patterns to Practice

• Mixing commodities to achieve a target price/concentration.

• Replacing a portion of a mixture (e.g., salt water replaced by pure water).

• Finding average speed for a round trip at different speeds.

• Dealing with groups (teachers/students, men/women) and finding averages.

Why Consistent Practice Works

Practicing mixture and alligation problems for 30 minutes daily can improve your score by 15–20 points in quantitative sections. With mixture problems making up about 20% of many aptitude tests, mastering this topic gives you a real edge.

Final Thoughts: Mastering Averages, Mixtures, and Alligations

To ace exams and everyday math:

• Practice a variety of problems.

• Memorize the alligation rule and weighted average formula.

• Use cross-method diagrams for speed.

• Apply these techniques in mock tests to improve accuracy.

With these strategies, averages, mixtures, and alligations will become some of your strongest topics for any entrance exam or real-world application!

Happy practicing, and may your mixing and averaging always give you the perfect result!

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