AFCAT Arithmetic Master Cheat Sheet: Complete Formula Guide & Study Notes

1. AFCAT Exam Analysis & Overview

The Air Force Common Admission Test (AFCAT) Numerical Ability section tests core speed, structural numerical understanding, and precise problem-solving capabilities. Arithmetic forms the bedrock of this section, contributing to over 70% of the total quantitative questions. Mastering foundational principles such as fraction-to-percentage equivalence, number classifications, remainder properties, and proportional transformations allows candidates to solve complex word problems rapidly without relying on long calculations.

2. Chapter 1: Number Systems & Divisibility Rules

Number systems establish the structural properties of numbers used across competitive examinations. Understanding number classification and divisibility tests is vital for simplifying complex arithmetic equations and determining exact remainders quickly.

Classification of Numbers

• Natural Numbers (\(\mathbb{N}\)): Counting numbers strictly greater than zero, defined as \(\mathbb{N} = \{1, 2, 3, \dots\}\).
• Whole Numbers (\(\mathbb{W}\)): The set of all natural numbers including zero, represented as \(\mathbb{W} = \{0, 1, 2, 3, \dots\}\).
• Integers (\(\mathbb{Z}\)): Complete set of positive, negative, and zero whole numbers, \(\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}\).
• Rational Numbers (\(\mathbb{Q}\)): Numbers that can be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q eq 0\).
• Irrational Numbers: Real numbers that cannot be expressed as a simple fraction; they feature non-terminating, non-repeating decimals (e.g., \(\pi\), \(\sqrt{2}\)).

Core Divisibility Rules

Divisor	Divisibility Criterion
3 or 9	The sum of all individual digits in the number must be divisible by \(3\) or \(9\), respectively.
4	The number formed by the last two digits of the given number must be divisible by \(4\).
8	The number formed by the last three digits of the given number must be divisible by \(8\).
11	The absolute difference between the sum of digits at odd places and the sum of digits at even places is either \(0\) or a multiple of \(11\).

3. Chapter 2: HCF & LCM Concepts & Shortcuts

Highest Common Factor (HCF) and Least Common Multiple (LCM) solve fundamental grouping, timing cycle, and factor-distribution problems in AFCAT arithmetic.

Fundamental Relations

For any two numbers, the product of their HCF and LCM is equal to the product of the two numbers themselves:

\[\text{HCF} \times \text{LCM} = \text{Product of two numbers}\]

When working with rational fractions, apply the following HCF and LCM rules:

\[\text{HCF of Fractions} = \frac{\text{HCF of Numerators}}{\text{LCM of Denominators}}\] \[\text{LCM of Fractions} = \frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}\]

Key Shortcuts & Remainder Theorems

• Representation via Coprimes: If \(H\) represents the HCF of two numbers, the numbers can be written as \(Hx\) and \(Hy\), where \(x\) and \(y\) are co-prime integers such that \(\gcd(x, y) = 1\).
• Greatest Divisor with Remainders: The greatest number that divides numbers \(x\), \(y\), and \(z\) leaving remainders \(a\), \(b\), and \(c\) respectively is given by:

\[\text{Required Number} = \text{HCF}(x-a, y-b, z-c)\]

4. Chapter 3: Percentages & Fast Fractional Conversions

Percentage calculations dictate performance across Profit & Loss, Simple/Compound Interest, and Data Interpretation. Memorizing fractional equivalents eliminates unnecessary division during time-pressured exams.

Core Fraction to Percentage Conversions

Fraction	Percentage	Fraction	Percentage
\(1/2\)	\(50\%\)	\(1/3\)	\(33.33\%\)
\(1/4\)	\(25\%\)	\(1/5\)	\(20\%\)
\(1/8\)	\(12.5\%\)	\(1/16\)	\(6.25\%\)

Percentage Change & Successive Formulas

The general percentage change formula measures relative variation against an initial value:

\[\text{\% Change} = \frac{\text{Final} - \text{Initial}}{\text{Initial}} \times 100\]

If a quantity undergoes two successive percentage changes of \(a\%\) and \(b\%\), the net percentage change is:

\[\text{Net } \% = \bigl( a + b + \frac{ab}{100} \bigr)\%\\]

5. Chapter 4: Ratio, Proportion & Structural Theorems

Ratios express relative magnitudes between quantifies, while proportions define the equality between two ratios. Understanding structural transformations facilitates rapid substitution in algebra and mixture problems.

Properties of Ratios

• Duplicate Ratio: The duplicate ratio of \(a:b\) is given by \(a^2 : b^2\).
• Sub-duplicate Ratio: The sub-duplicate ratio of \(a:b\) is given by \(\sqrt{a} : \sqrt{b}\).
• Triplicate Ratio: The triplicate ratio of \(a:b\) is given by \(a^3 : b^3\).

Componendo & Dividendo Theorem

If two ratios are equal such that \(\frac{a}{b} = \frac{c}{d}\), then applying Componendo & Dividendo yields:

\[\frac{a+b}{a-b} = \frac{c+d}{c-d}\]

6. AFCAT PYQ Weightage & Exam Pattern Analysis

Analysis of Previous Year Questions (PYQs) indicates a stable pattern across AFCAT quantitative aptitude testing. Mastering arithmetic shortcut techniques directly impacts score maximization:

• Chapter Weightage: Percentages, Ratios, and HCF/LCM combined account for 5 to 7 direct questions in every AFCAT attempt.
• Speed & Accuracy: Utilizing fractional conversions and shortcut equations reduces average problem-solving time from 90 seconds to under 30 seconds.
• NTA / Exam Authority Pattern: Questions focus on practical applications rather than purely theoretical proofs. Practicing authentic past questions is essential for familiarization.

7. Frequently Asked Questions (PAA)