Arithmetic Cheat Sheet - CDS 2027 Study Material & Formulas

1. Number Systems

Understanding classification of numbers is fundamental for the CDS (Combined Defence Services) exam. Concepts include sets of numbers and their foundational properties.

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

2. Divisibility Rules

Divisibility shortcuts allow candidates to quickly solve prime factorization, remainder, and factor problems under strict time constraints during CDS 2027.

• Divisibility by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Divisibility by 4: A number is divisible by 4 if the last two digits form a multiple of 4.
• Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
• Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
• Divisibility by 11: A number is divisible by 11 if the alternating sum of its digits (difference between sum of odd position digits and even position digits) is divisible by 11.

3. Fractions & Decimals

Arithmetic operations on fractions form the bedrock of simplification questions in the UPSC CDS Mathematics paper.

Fundamental Operations

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

4. LCM & HCF (GCD)

Lowest Common Multiple (LCM) and Highest Common Factor (HCF) are extensively used in time and work, speed-distance-time, and remainder problems.

Key Formulas & Relations

• For any two positive integers \(a\) and \(b\): \[a \times b = \text{HCF}(a,b) \times \text{LCM}(a,b)\]
• HCF of Fractions: \[\text{HCF}\Bigl(\frac{a}{b}, \frac{c}{d}\Bigr) = \frac{\text{HCF}(a,c)}{\text{LCM}(b,d)}\]
• LCM of Fractions: \[\text{LCM}\Bigl(\frac{a}{b}, \frac{c}{d}\Bigr) = \frac{\text{LCM}(a,c)}{\text{HCF}(b,d)}\]

5. Percentages

Percentages serve as a baseline concept across commercial mathematics, including profit & loss, simple interest, and compound interest.

Core Formulas

• Basic Percentage Formula: \[\text{Percentage} = \Bigl(\frac{\text{Part}}{\text{Whole}}\Bigr) \times 100\]
• Percentage Increase / Decrease: \[\text{\% Change} = \Bigl(\frac{\text{New} - \text{Old}}{\text{Old}}\Bigr) \times 100\]

6. Ratio & Proportion

Ratios compare two quantities by division, while proportions equate two ratios. Direct application occurs in partnership and mixture problems.

• Cross Product Rule: If \(a:b = c:d\) (i.e., \(\frac{a}{b} = \frac{c}{d}\)), then: \[ad = bc\]
• Direct Proportion: \(y\) varies directly as \(x\) when \(y = kx\) for a constant \(k\).
• Inverse Proportion: \(y\) varies inversely as \(x\) when \(y = \frac{k}{x}\) for a constant \(k\).

7. Simple & Compound Interest

Commercial arithmetic in CDS frequently tests calculations involving simple interest and compound interest schemes.

Simple Interest (SI)

Where \(P\) is Principal, \(R\) is Rate of Interest per annum, and \(T\) is Time in years.

Compound Interest (CI)

Where \(A\) is Amount and \(t\) is time period in years.

8. Exam Pattern, PYQ Analysis & Chapter Weightage

The UPSC CDS Mathematics paper consists of 100 objective questions worth 100 marks. Arithmetic accounts for a significant portion of this paper.

UPSC CDS Arithmetic Weightage

• Number Systems & Divisibility: 15-20 Questions
• LCM & HCF: 3-5 Questions
• Percentages, Ratio & Proportion: 8-10 Questions
• Simple & Compound Interest: 3-5 Questions
• Fractions & Decimals: 2-4 Questions

NTA / UPSC Trend Note: Recent exams focus on multi-concept questions combining divisibility rules with remainder theorems, as well as multi-year compound interest calculations with varying rates.

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