AFCAT Arithmetic Master Cheat Sheet: Formulas, Properties & Shortcuts

1. Exam Overview & PYQ Analysis

The Air Force Common Admission Test (AFCAT) Numerical Ability section tests candidates on speed, conceptual clarity, and accuracy. Based on recent NTA and C-DAC exam patterns, Arithmetic forms the backbone of the AFCAT quantitative syllabus. Analyzing past year questions (PYQs) reveals a consistent weightage distributed across fundamental topics like Number Systems, Fractions, Percentages, and Commercial Mathematics.

Chapter / Topic Keyword	Average Questions (AFCAT)	Difficulty Level & Weightage
Number Systems & Divisibility	2 - 3 Questions	Easy to Moderate (High Weightage)
HCF & LCM	1 - 2 Questions	Easy (Direct Formula Based)
Percentages, Profit & Loss	3 - 4 Questions	Moderate (High Speed Required)
Ratio, Proportion & Interest	2 - 3 Questions	Moderate (Shortcut Heavy)

2. Chapter 1: Number Systems & Fundamental Properties

Understanding real numbers and their classifications is critical for quick numerical simplification and conceptual problem-solving in competitive examinations.

Number Sets

• Natural Numbers (\(\mathbb{N}\)): Counting numbers beginning from 1. \[\mathbb{N} = \{1, 2, 3, \dots\}\]
• Whole Numbers (\(\mathbb{W}\)): Natural numbers along with zero. \[\mathbb{W} = \{0, 1, 2, 3, \dots\}\]
• Integers (\(\mathbb{Z}\)): Complete set of positive and negative whole values. \[\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}\]
• Rational Numbers (\(\mathbb{Q}\)): Numbers representable in quotient form. \[\mathbb{Q} = \{\frac{p}{q} \ \Big| \ q eq 0, \ p, q \in \mathbb{Z}\}\]

Fundamental Properties of Arithmetic Operations

• Commutative Property: Addition and multiplication are independent of term ordering. \(a + b = b + a\) and \(ab = ba\)
• Associative Property: Grouping of operands does not alter the evaluation. \((a+b)+c = a+(b+c)\) and \((ab)c = a(bc)\)
• Distributive Property: Multiplication distributes over term summation. \(a(b + c) = ab + ac\)

3. Chapter 2: Fractions and Decimals Operations

Fractions represent proportional values. Mastery over fraction manipulation enables rapid evaluation of algebraic expressions in numerical reasoning.

Addition and Subtraction

To combine fractions with distinct denominators, apply the cross-multiplication or LCM identity:

\[\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}\]

Multiplication and Division

Direct product and reciprocal multiplication rules:

\[\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\] \[\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}\]

4. Chapter 3: Divisibility Rules & Applications

Divisibility shortcuts allow candidates to bypass lengthy division steps when verifying options or simplifying complex numeric factors.

Divisor	Divisibility Condition / Criterion
2	The unit digit of the number must be even (i.e., ending in \(0, 2, 4, 6, 8\)).
3	The sum of all individual digits must be divisible by \(3\).
4	The number formed by the last two digits must be divisible by \(4\).
5	The unit digit must be either \(0\) or \(5\).
9	The sum of all individual digits must be divisible by \(9\).
11	The difference between the sum of digits at odd places and even places is either \(0\) or a multiple of \(11\).

5. Chapter 4: HCF and LCM Relations

The Highest Common Factor (HCF) and Least Common Multiple (LCM) form crucial foundations for modular arithmetic, fraction simplification, and bell/ring timing problems.

Product Formula for Integers

For any two positive integers \(a\) and \(b\):

\[\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b\]

HCF and LCM of Fractional Quantities

When computing factors and multiples for fractional numbers \(\frac{a}{b}\) and \(\frac{c}{d}\):

\[\text{HCF}(\frac{a}{b}, \frac{c}{d}) = \frac{\text{HCF}(a, c)}{\text{LCM}(b, d)}\] \[\text{LCM}(\frac{a}{b}, \frac{c}{d}) = \frac{\text{LCM}(a, c)}{\text{HCF}(b, d)}\]

6. Chapter 5: Percentages & Net Percentage Shifts

Percentages serve as the relative metric across business mathematics, data interpretation, and statistical comparison.

Percentage Change Formula

\[\% \text{ Change} = (\frac{\text{New Value} - \text{Old Value}}{\text{Old Value}}) \times 100\]

Successive Percentage Change Shortcut

If a quantity undergoes two consecutive percentage changes of \(a\%\) and \(b\%\), the effective single percentage change is given by:

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

Note: Use negative values for decreases or discounts (e.g., \(-a\)).

7. Chapter 6: Ratio, Proportion & Theorems

Ratios express relative sizes between two quantities. Equivalence of two ratios \(\frac{a}{b} = \frac{c}{d}\) allows direct algebraic transformations.

Important Ratio Properties

• Invertendo: If \(\frac{a}{b} = \frac{c}{d}\), then: \[\frac{b}{a} = \frac{d}{c}\]
• Alternendo: If \(\frac{a}{b} = \frac{c}{d}\), then: \[\frac{a}{c} = \frac{b}{d}\]
• Componendo & Dividendo: If \(\frac{a}{b} = \frac{c}{d}\), then: \[\frac{a+b}{a-b} = \frac{c+d}{c-d}\]

8. Chapter 7: Profit, Loss & Discount Formulas

Commercial transactions evaluate trade margins using Cost Price (CP), Selling Price (SP), and Marked Price (MP).

Basic Relations

\[\text{Profit} = \text{SP} - \text{CP}, \quad \text{Loss} = \text{CP} - \text{SP}\] \[\text{Profit}\% = (\frac{\text{Profit}}{\text{CP}}) \times 100, \quad \text{Loss}\% = (\frac{\text{Loss}}{\text{CP}}) \times 100\]

Discount Computation

\[\text{Discount} = \text{MP} - \text{SP}\] \[\text{Discount}\% = (\frac{\text{Discount}}{\text{MP}}) \times 100\]

9. Chapter 8: Simple and Compound Interest

Interest calculations measure the time value of money under simple linear accrual and exponential compounding.

Simple Interest (SI)

Calculated uniformly on the initial Principal \(P\) for time \(T\) years at rate \(R\%\) per annum:

\[\text{SI} = \frac{P \times R \times T}{100}\]

Compound Interest (CI)

Calculated on principal plus accumulated interest over \(n\) compounding periods:

\[A = P (1 + \frac{R}{100})^n\] \[\text{CI} = A - P = P [(1 + \frac{R}{100})^n - 1]\]

10. Frequently Asked Questions (FAQ)