CAT Averages & Weighted Averages: The Balance Method

Averages in CAT are rarely about simple calculation. They are about 'balance' and 'deviation'. This guide teaches you the 'Assumed Mean' method and the powerful 'Alligation' technique to solve weighted average problems in seconds.

Section 1: The Core Toolkit

The Foundation: Averages as Balance Points

Stop calculating sums. Learn to view the Average as a fulcrum or balance point, allowing you to solve problems by observing deviations.

Interactive Data Set Analyzer

The #1 Skill: Working with Sums

Learn the fundamental strategy for solving replacement and removal problems: converting averages to sums to track changes efficiently.

Step 1The Core Inference

Whenever you see an Average, immediately calculate the Sum.

$$ \text{Sum} = \text{Avg} \times \text{Count} $$

Step 2Example 1

Given: Class of 20 students, Avg Age 14.
Infer: $$ \text{Total Age} = 20 \times 14 = 280 $$

Step 3Example 2

Given: Avg daily visits in June is 120.
Infer: $$ \text{Total Visits} = 120 \times 30 = 3600 $$ (June has 30 days).

Section 2: Advanced CAT Concepts

Application: Weighted Averages (WAvg)

Master the concept of Weighted Averages for scenarios where groups have different sizes. Essential for Data Interpretation sets.

Step 1The Formula

When values have different weights (frequencies):
$$ \text{Avg} = \frac{w_1 x_1 + w_2 x_2}{w_1 + w_2} $$

Step 2Scenario: Class Scores

24 students get 60, 16 get 70, 10 get 80.
$$ \frac{(24 \times 60) + (16 \times 70) + (10 \times 80)}{50} = 67.2 $$

Step 3Scenario: Stock Price

10 shares at $50, 40 shares at $60.
$$ \frac{(10 \times 50) + (40 \times 60)}{50} = \$58 $$

Shortcut: The Alligation Method (Mixtures)

Learn the 'Rule of Alligation', a visual shortcut for solving weighted average and mixture problems without complex equations.

The Allegation Method

A visual shortcut for mixture problems. Solves $$ \frac{w_1 C_1 + w_2 C_2}{w_1 + w_2} = \text{Mean} $$ without complex algebra.

Step 1Scenario

Vessel A: 75% Spirit.
Vessel B: 40% Spirit.
Target Mixture: 60% Spirit.
Find Ratio A:B

Step 2The Allegation X

75
40
↘
↙
60
↙
↘
(60-40)=20
(75-60)=15

Step 3Simplify

$$ A : B = 20 : 15 $$

Ratio = 4 : 3

Special Properties: Arithmetic Progressions

Explore the powerful property for evenly spaced sets (AP): Mean = Median. This is a frequent shortcut in CAT Number System questions.

Multiply entire set by 'm'

Change the multiplier 'm' to see how the Average scales perfectly with it.

Interactive Practice

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