Arithmetic Cheat Sheet - IIM CAT 2026

1. Number Systems & Classification

Understanding the fundamental structure of real numbers is essential for solving quantitative aptitude problems in CAT 2026.

• Natural Numbers (\(\mathbb{N}\)): The counting numbers starting from 1. Set representation: \(\{1, 2, 3, \dots\}\).
• Whole Numbers: Includes all natural numbers along with zero. Set representation: \(\{0, 1, 2, 3, \dots\}\).
• Integers (\(\mathbb{Z}\)): Consists of negative numbers, zero, and positive numbers. Set representation: \(\{\dots, -2, -1, 0, 1, 2, \dots\}\).
• Rational Numbers (\(\mathbb{Q}\)): Numbers that can be expressed in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q eq 0\).
• Irrational Numbers: Numbers that cannot be written as a simple fraction \(\frac{p}{q}\). Their decimal expansions are non-terminating and non-repeating (e.g., \(\pi\), \(e\), \(\sqrt{2}\)).

2. Divisibility Rules

Quick divisibility checks speed up calculation during the QA section of CAT 2026.

3. HCF and LCM Principles

Fundamental relation for any two positive integers \(a\) and \(b\):

HCF and LCM of Fractions

When computing HCF or LCM for fractional quantities \(\frac{a}{b}\) and \(\frac{c}{d}\):

4. Fractions and Decimals Operations

Standard algebraic rules governing fundamental fraction manipulations:

• Addition and 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 building block for arithmetic modules like Profit & Loss, Simple Interest, and Compound Interest.

• Percentage to Fraction Conversion: \[x\% = \frac{x}{100}\]
• Percentage Change Formula: \[\% \text{ Change} = \frac{\text{Final} - \text{Initial}}{\text{Initial}} \times 100\]
• Successive Percentage Change: If a value undergoes two successive changes of \(a\%\) and \(b\%\), the net percentage change is given by: \[\text{Net Change} = \Bigl(a + b + \frac{ab}{100}\Bigr)\%\]

6. Ratio, Proportion & Variation

Ratios measure relative size, whereas proportion represents the equality of two ratios.

• Cross Product Rule: If \(a:b = c:d\), then \(ad = bc\).
• Componendo and Dividendo Rule: \[\text{If } \frac{a}{b} = \frac{c}{d}, \text{ then } \frac{a+b}{a-b} = \frac{c+d}{c-d}\]
• Direct Proportion: \(y\) varies directly with \(x\) (\(y = kx\)), yielding: \[\frac{y_1}{x_1} = \frac{y_2}{x_2}\]
• Inverse Proportion: \(y\) varies inversely with \(x\) (\(xy = k\)), yielding: \[x_1 y_1 = x_2 y_2\]

7. Profit, Loss, and Discount Mechanics

Core formulas for financial arithmetic questions in CAT 2026:

• Profit (\(\text{P}\)): \(\text{SP} - \text{CP}\)
• Loss (\(\text{L}\)): \(\text{CP} - \text{SP}\)
• Profit Percentage (\(\text{P}\%\)): \[\text{P}\% = \Bigl(\frac{\text{SP} - \text{CP}}{\text{CP}}\Bigr) \times 100\]
• Loss Percentage (\(\text{L}\%\)): \[\text{L}\% = \Bigl(\frac{\text{CP} - \text{SP}}{\text{CP}}\Bigr) \times 100\]

8. IIM CAT 2026 PYQ Analysis & NTA Pattern

Chapter Weightage: Arithmetic consistently accounts for 40% to 50% of the Quantitative Aptitude section in IIM CAT exams. Key emphasis areas include Percentage Applications, Profit & Loss, Ratios, and HCF/LCM application problems.

NTA Pattern Trends: CAT exam papers test conceptual depth over memory. Expect multi-concept questions combining Ratio-Proportion with Percentages, or Divisibility Rules integrated into Number System constraints.

9. Frequently Asked Questions (FAQ)