Arithmetic Cheat Sheet - CDS 2027 - Paper 3: Elementary Mathematics

1. Exam Context & PYQ Weightage Analysis

In the UPSC Combined Defence Services (CDS) examination, Paper 3 tests candidates on Elementary Mathematics. Arithmetic forms the bedrock of this paper, typically accounting for 20–25% of the total 100 questions. Mastery over these fundamental rules is essential not only for direct questions but also for quick computation in Data Interpretation, Mensuration, and Commercial Mathematics.

UPSC Pattern Insights: Conceptual questions on rational versus irrational numbers and properties of real numbers frequently appear as multi-statement conceptual assertions (e.g., "Consider the following statements regarding rational numbers..."). Quick application of exponent laws and accurate evaluation using BODMAS ensure precision and save time under exam conditions.

2. Number Systems & Core Properties

Understanding the hierarchy of numbers is critical for solving logical set-theoretic math problems in CDS Paper 3.

Classifications of Real Numbers

• Natural Numbers (\(\mathbb{N}\)): Counting numbers starting from 1: \(\{1, 2, 3, \dots\}\).
• Whole Numbers (\(\mathbb{W}\)): Natural numbers including zero: \(\{0, 1, 2, 3, \dots\}\).
• Integers (\(\mathbb{Z}\)): Complete set of positive and negative whole numbers: \(\{\dots, -2, -1, 0, 1, 2, \dots\}\).
• Rational Numbers (\(\mathbb{Q}\)): Numbers representable in the form \(\frac{a}{b}\) where \(a, b \in \mathbb{Z}\) and \(b eq 0\). Decimal expansions are either terminating or repeating.
• Irrational Numbers (\(\mathbb{I}\)): Numbers that cannot be expressed as \(\frac{a}{b}\); their decimal expansions are non-terminating and non-repeating (e.g., \(\sqrt{2}\), \(\pi\)).

Fundamental Algebraic Properties

For any real numbers \(a\), \(b\), and \(c\):

• Commutative Property: \(a + b = b + a\) and \(a \cdot b = b \cdot a\)
• Associative Property: \((a + b) + c = a + (b + c)\) and \((a \cdot b) \cdot c = a \cdot (b \cdot c)\)
• Distributive Property: \(a(b + c) = ab + ac\)

3. PEMDAS / BODMAS Order of Operations

To avoid ambiguity when evaluating mathematical expressions, follow the standard order of operations strict sequence:

1. P / B (Parentheses / Brackets): Evaluate operations within grouping symbols first, moving from innermost to outermost brackets.
2. E / O (Exponents / Orders): Compute all powers, square roots, and indices.
3. MD (Multiplication & Division): Perform multiplication and division sequentially from left to right as they appear.
4. AS (Addition & Subtraction): Perform addition and subtraction sequentially from left to right as they appear.

4. Fractions, Decimals & Percentage Formulas

Fraction Arithmetic

Standard operating rules for operations on algebraic and numeric fractions:

• 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: \[ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} \]

Percentage Conversions & Calculations

Percentages quantify proportion per hundred and form the basis for profit/loss, simple interest, and compound interest problems:

5. Exponents & Radical Rules

Exponential identities simplify complex numerical and algebraic calculations in Paper 3:

• Product Rule: \(x^a \cdot x^b = x^{a+b}\)
• Quotient Rule: \(\frac{x^a}{x^b} = x^{a-b}\)
• Power of a Power: \((x^a)^b = x^{ab}\)
• Negative Exponent: \(x^{-a} = \frac{1}{x^a}\)
• Zero Exponent: \(x^0 = 1\) (where \(x eq 0\))
• Power of a Product: \((xy)^a = x^a y^a\)
• Radical Notation: \(\sqrt[n]{x} = x^{1/n}\)
• Product under Radical: \(\sqrt[n]{xy} = \sqrt[n]{x} \cdot \sqrt[n]{xy}\)

6. Frequently Asked Questions (FAQ)