Arithmetic Master Cheat Sheet - IIM CAT 2026

Welcome to the ultimate chapter-wise breakdown and open educational resource for IIM CAT 2026 Arithmetic. Designed in the LibreTexts structural format, this guide provides rigorous conceptual explanations, essential formulas, quick calculation tricks, and previous year question (PYQ) insights tailored to the evolving exam patterns set by the Indian Institutes of Management (IIMs).

1. CAT Exam Pattern & Weightage Analysis

Arithmetic forms the backbone of the Quantitative Aptitude (QA) section in the CAT exam. Historically, out of 22 questions in QA, Arithmetic accounts for 8 to 10 questions (approximately 35% to 45% weightage). Mastering these core chapters guarantees a high percentile, as these topics range from direct formula-based application to multi-concept word problems.

2. Chapter 1: Number Systems & Properties

Understanding the fundamental classifications of numbers and their structural properties is essential for solving higher-level quantitative problems efficiently.

Types of Numbers

• Natural Numbers (\(\mathbb{N}\)): Counting numbers starting from 1: \(1, 2, 3, \dots\)
• Whole Numbers (\(\mathbb{W}\)): Non-negative integers including zero: \(0, 1, 2, 3, \dots\)
• Integers (\(\mathbb{Z}\)): Complete set of whole numbers and their negative counterparts: \(\dots, -2, -1, 0, 1, 2, \dots\)
• Rational Numbers (\(\mathbb{Q}\)): Numbers expressible in the form \(\frac{p}{q}\), where \(p, q \in \mathbb{Z}\) and \(q eq 0\).
• Irrational Numbers: Real numbers that cannot be written as simple fractions; they exhibit non-terminating, non-repeating decimals (e.g., \(\pi\), \(\sqrt{2}\)).

Essential Divisibility Rules

Quick divisibility checks save critical time during CAT calculations:

• Divisibility by 2: The last digit of the number must be even.
• Divisibility by 3: The sum of all digits must be divisible by 3.
• Divisibility by 4: The last two digits formed as a two-digit number must be divisible by 4.
• Divisibility by 5: The last digit must be 0 or 5.
• Divisibility by 9: The sum of all digits must be divisible by 9.
• Divisibility by 11: The absolute difference between the sum of digits at odd places and the sum of digits at even places must be either 0 or divisible by 11.

3. Chapter 2: Fractions, Decimals & Percentages

Manipulating fractional expressions and executing rapid percent conversions form the foundation of fast mental math.

Algebraic Operations on Fractions

Addition, subtraction, multiplication, and division are governed by basic algebraic identity rules:

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

Fraction-to-Percentage Conversions

• Fraction to Percentage: Multiply the fraction by \(100\%\).
• Percentage to Fraction: Divide the percentage value by \(100\) and simplify to its lowest terms.
• Standard Equivalents:
  • \(\frac{1}{2} = 50\%\)
  • \(\frac{1}{4} = 25\%\)
  • \(\frac{1}{5} = 20\%\)
  • \(\frac{1}{8} = 12.5\%\)

4. Chapter 3: Ratio, Proportion & Variation

Ratios compare quantities of similar units, whereas proportions establish equality between two ratios.

Ratio & Proportion Basics

If two ratios are equivalent, written as \(a:b = c:d\), then by cross-multiplication:

\[ ad = bc \]

Variations

• Direct Variation: Quantity \(y\) varies directly with \(x\) (written as \(y \propto x\)). The governing equation is \(y = kx\), where \(k\) is a non-zero constant.
• Inverse Variation: Quantity \(y\) varies inversely with \(x\) (written as \(y \propto \frac{1}{x}\)). The governing equation is \(xy = k\).

Partnership Sharing Profit

When multiple investors pool capital for varying durations, profit distribution is proportional to the product of investment capital and duration:

\[ \text{Profit Ratio} = (\text{Capital}_1 \times \text{Time}_1) : (\text{Capital}_2 \times \text{Time}_2) \]

5. Chapter 4: Percentages, Profit & Loss

Commercial arithmetic questions in CAT frequently test percentage changes, profit margins, loss percentages, and discount structures.

Percentage Change Formula

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

Profit, Loss, and Discount Relations

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

Note: \(CP\) = Cost Price, \(SP\) = Selling Price, \(MP\) = Marked Price.

6. Chapter 5: Averages, Mixtures & Alligations

Averages represent central tendency, while alligation provides a shortcut for solving weighted average problems involving mixtures.

Arithmetic Mean & Weighted Average

\[ \text{Average} = \frac{\text{Sum of Observations}}{\text{Number of Observations}} \] \[ \text{Avg}_{wg} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2} \]

Rule of Alligation

When mixing two ingredients of quantities \(Q_1\) and \(Q_2\) with unit costs \(C_1\) and \(C_2\) respectively to obtain a mixture of mean price \(M\):

\[ \frac{Q_1}{Q_2} = \frac{C_2 - M}{M - C_1} \]

7. Chapter 6: Simple & Compound Interest

Financial arithmetic centers around simple and compound interest computations over fixed periods.

Simple Interest Formula

Simple interest grows linearly based only on the initial principal \(P\), annual rate \(R\), and time period \(T\):

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

8. People Also Ask (Frequently Asked Questions)