IIM CAT 2026 Arithmetic Cheat Sheet - Essential Formulas & PYQ Guide

1. Number Systems & Fundamentals

Number systems form the backbone of Quantitative Aptitude for CAT 2026. A clear understanding of core set definitions ensures precision when tackling algebra and higher-level arithmetic problem sets.

• Natural Numbers (\(\mathbb{N}\)): Positive counting integers: \(\{1, 2, 3, \dots\}\).
• Whole Numbers (\(\mathbb{W}\)): Non-negative integers including zero: \(\{0, 1, 2, 3, \dots\}\).
• Integers (\(\mathbb{Z}\)): Complete set of whole numbers and their negatives: \(\{\dots, -2, -1, 0, 1, 2, \dots\}\).
• Rational Numbers (\(\mathbb{Q}\)): Any number expressible as \(\frac{p}{q}\) where \(p, q \in \mathbb{Z}\) and \(q eq 0\).
• Irrational Numbers: Numbers that cannot be expressed as simple ratios, featuring non-terminating, non-repeating decimals (e.g., \(\pi\), \(\sqrt{2}\)).

2. Divisibility Rules & Applications

Divisibility shortcuts allow CAT aspirants to quickly factorize large numbers and solve remainder theorem questions efficiently.

• Divisibility by 2: The unit digit is even (\(0, 2, 4, 6, 8\)).
• Divisibility by 3: The sum of all digits is a multiple of 3.
• Divisibility by 4: The final two digits form a number divisible by 4.
• Divisibility by 5: The unit digit is either \(0\) or \(5\).
• Divisibility by 6: The number satisfies criteria for both 2 and 3 simultaneously.
• Divisibility by 9: The sum of all digits is a multiple of 9.
• Divisibility by 11: The absolute difference between the sum of digits at odd places and even places is either \(0\) or divisible by 11.

3. HCF and LCM Theorems

Highest Common Factor (HCF) and Least Common Multiple (LCM) rules are crucial for solving timing, grouping, and modular arithmetic questions in the IIM CAT exam pattern.

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

HCF and LCM of Fractional Quantities

To calculate HCF and LCM when inputs are expressed as rational fractions \(\frac{a}{b}\) and \(\frac{c}{d}\):

4. Fractions and Decimals Operations

A rigorous command over basic fractional operations prevents algebraic errors in word problems.

• Addition/Subtraction: \(\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}\)
• Multiplication: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)
• Division: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}\)

5. Percentages & Successive Changes

Percentages serve as the foundational tool across arithmetic topics including Data Interpretation, Profit & Loss, and Simple/Compound Interest.

Basic Percentage Equations

Successive Percentage Formula

When a quantity is altered sequentially by \(a\%\) and \(b\%\), the net change is given by:

6. Ratio, Proportion & Componendo-Dividendo

Ratios measure relative scale, providing clean alternatives to full equations when solving mixture and partnership problems.

If \(a:b = c:d\), then \(\frac{a}{b} = \frac{c}{d}\), which implies:

Componendo and Dividendo Rule

Useful for simplifying complex rational equations in algebra and arithmetic:

7. Profit, Loss, and Discount Mechanics

Commercial mathematics problems in CAT heavily feature Cost Price (\(CP\)), Selling Price (\(SP\)), and Marked Price (\(MP\)).

• \(\text{Profit} = \text{SP} - \text{CP}\)
• \(\text{Loss} = \text{CP} - \text{SP}\)
• \(\text{Profit}\% = ( \frac{\text{Profit}}{\text{CP}} ) \times 100\)
• \(\text{Loss}\% = ( \frac{\text{Loss}}{\text{CP}} ) \times 100\)
• \(\text{Discount} = \text{MP} - \text{SP}\)
• \(\text{Discount}\% = ( \frac{\text{Discount}}{\text{MP}} ) \times 100\)

8. Simple and Compound Interest

Interest models evaluate financial growth across time horizons using linear or exponential scales.

Simple Interest (SI)

Where \(P = \text{Principal}\), \(R = \text{Annual Rate}\), and \(T = \text{Time in Years}\).

Compound Interest (CI)

Where \(A = \text{Total Accumulated Amount}\) and \(n = \text{Number of Compounding Periods}\).

9. Speed, Time, and Distance Essentials

Motion mechanics demand unit conversions and speed-time relationship modeling.

Unit Conversion Factor

10. PYQ Analysis & Chapter Weightage for CAT 2026

Analyzing recent CAT patterns reveals that Arithmetic consistently accounts for 40% to 50% of the Quantitative Aptitude section (typically 8 to 11 questions out of 22). Key priority areas include:

• High Weightage Topics: Percentages, Ratios, Profit & Loss, and Time, Speed & Distance.
• Moderate Weightage Topics: Simple & Compound Interest, Averages, Mixtures, and Divisibility Rules.
• NTA / CAT Exam Pattern Insight: CAT emphasizes conceptual depth, variable-based word problems, and multi-concept fusion rather than direct formula application.

11. Frequently Asked Questions (FAQ)