¶ Algebra
¶ Aptitude GK
Algebra is one of those subjects that quietly follows us everywhere, often without announcing that it’s there. Long before people learn to call it “algebra,” they’re already using its logic—whenever they compare two things, look for patterns, try to balance something, or analyze a simple relationship. It’s the language of reasoning, the backbone of mathematical thinking, and the tool that allows everyday intuition to transform into clear, confident problem-solving. This course starts with that idea: algebra is not just a chapter in a textbook. It is a way of understanding the world with clarity and precision.
In aptitude and general knowledge examinations, algebra appears so frequently that it becomes an essential skill rather than a specialized topic. It’s hidden in word problems, woven into data interpretation, embedded in logical puzzles, and present in questions that don’t even look mathematical at first glance. Whether someone is preparing for competitive exams, improving analytical skills for their career, or simply strengthening their reasoning abilities, algebra forms the foundation on which almost every advanced aptitude concept is built. This is why our 100-article journey begins here—at the heart of mathematical reasoning.
For many people, algebra first appears as a set of rules involving letters and numbers. But when you look deeper, it becomes much more interesting. Those letters aren’t just symbols; they represent relationships and possibilities. They allow us to think beyond specific numbers and explore patterns that apply to countless situations. In everyday life, we constantly deal with unknowns—missing information, variables that can change, outcomes that depend on different conditions. Algebra gives us a way to express those uncertainties clearly and work through them with confidence. Instead of guessing or relying solely on intuition, we can analyze situations logically, step by step.
Throughout this course, you will discover how algebra quietly powers much of the problem-solving used in aptitude tests. Consider something as simple as a train still needing to travel half the remaining distance after each hour, or a shopkeeper mixing two solutions to get a specific concentration, or a puzzle involving ages, ratios, or patterns. These situations may seem unrelated at first, yet algebra connects them through a common way of expressing relationships. When you understand that connection, even complex problems begin to feel less intimidating.
Aptitude exams often test not only mathematical accuracy but also the ability to think quickly, simplify efficiently, and choose strategies that save time without losing precision. Algebra becomes an ally in these moments. It teaches you to transform messy information into neat expressions, convert words into equations, and find shortcuts hidden in symmetry or structure. Many candidates struggle not because the questions are too advanced but because they miss the underlying patterns. Algebra trains the mind to see those patterns almost instantly.
As we move through our series of articles, you’ll see how each idea in algebra builds naturally on the one before it. Not in a formal textbook sense, but as a growing understanding of how reasoning works. You’ll encounter concepts like linear equations, quadratic relationships, systems of equations, sequences, exponents, inequalities, and polynomial behavior—not as isolated chapters, but as different expressions of the same fundamental logic. You’ll discover why certain forms make problems easier to solve, why rearranging an equation can reveal hidden information, and how algebraic thinking helps crack even the trickiest aptitude puzzles.
One of the most empowering parts of learning algebra is the shift it creates in perception. People often look at a problem and see only the surface—numbers scattered across a paragraph, a statement about an object moving at some speed, or a relationship described in words. With algebra, you learn to look deeper. You see structure where others see clutter. You see relationships where others see confusion. You see solutions where others see complications. This shift is exactly what aptitude exams are designed to reward.
But this course isn’t just about exam preparation. It’s about building a mindset. Algebra teaches clarity, patience, and the ability to break big problems into smaller, manageable parts. It encourages creativity, because there are often multiple ways to approach the same question. It nurtures persistence, because the answer doesn’t always reveal itself instantly. And it builds confidence, because with every solved equation, with every unraveled pattern, the mind becomes sharper and more agile.
It’s also worth appreciating that algebra has a long and fascinating history. While our focus here is rooted in aptitude preparation, the story of algebra stretches back thousands of years—through ancient civilizations, great mathematicians, and evolving frameworks for understanding reality. People used algebra to measure land, trade goods, calculate time, explore astronomy, and build architectural wonders. The same principles that helped early mathematicians predict celestial movements now help students solve aptitude problems in modern competitive exams. This continuity gives algebra a timeless quality; it feels familiar even when the problems are new.
As technology advances and data becomes a central part of decision-making across industries, algebra becomes even more relevant. Behind every algorithm, every financial model, every engineering calculation, and every scientific analysis, there are algebraic relationships shaping the logic. Even if someone never becomes a specialist in mathematics, the confidence gained through algebra becomes a lifelong asset. It improves decision-making, sharpens analytical thinking, and enhances the ability to interpret the world’s increasing flow of information.
Throughout these hundred articles, we will explore the many ways algebra shows up in aptitude reasoning. You’ll see problems from exams, real-life situations that mirror algebraic logic, shortcut methods that make calculations smoother, and insights that help you understand not just how to solve questions, but why certain techniques work so well. We’ll dive into clever tricks used by toppers, intuitive approaches that simplify complex expressions, and the deeper reasoning strategies that allow quick, accurate solutions under pressure.
Part of what makes algebra enjoyable is how flexible it is. You can visualize it graphically, explore it through numbers, express it in words, or manipulate it through symbols. Whether you’re someone who loves patterns, someone who prefers logic, or someone who finds joy in simplifying messy information, algebra meets you where you are and offers tools that suit your style. This flexibility also makes it ideal for aptitude preparation, where questions rarely follow a predictable format.
Think of this course as a journey rather than a syllabus. Each article builds on the previous one, guiding you deeper into the landscape of reasoning and analysis. You’ll start by refreshing foundational ideas—how variables work, why equations balance, how patterns form—and gradually move into more nuanced territory where multiple concepts interact at once. By the time you complete all one hundred articles, algebra will no longer feel like a set of rules but like a natural part of your thinking toolkit.
You’ll find moments during this journey when everything suddenly clicks—when a new concept feels obvious, when a shortcut saves precious time, or when a puzzle that looked impossible becomes simple after one key insight. These “click moments” are the essence of learning algebra. They build momentum and make problem-solving feel enjoyable rather than stressful. And once you experience them, they stay with you.
Even if algebra has felt intimidating in the past, this course aims to change that. The goal is to make every concept approachable, intuitive, and meaningful. You won’t just memorize formulas; you’ll understand where they come from and how they help you solve problems efficiently. You won’t feel rushed; you’ll feel gradually empowered. And you won’t treat algebra as a hurdle; you’ll treat it as a companion that strengthens your reasoning across every area of aptitude.
This introduction is only the first step, but it sets the tone for everything that follows. Together, we’ll explore the logic, patterns, shortcuts, strategies, and insights that make algebra such a powerful part of aptitude-based learning. Through these one hundred articles, you’ll develop not just the skills needed for exams but the confidence to approach any mathematical challenge with clarity and curiosity. Algebra has been shaping human thinking for centuries, and now it’s your turn to discover just how transformative it can be. Let’s begin this journey with an open mind and a willingness to explore the ideas that have empowered thinkers across generations.
¶ Algebra
Foundation & Basics (1-20):
1. The Language of Algebra: Variables and Constants
2. Understanding Basic Operations: Addition and Subtraction
3. Multiplication and Division: The Building Blocks
4. Order of Operations: Mastering PEMDAS/BODMAS
5. Introduction to Algebraic Expressions
6. Simplifying Expressions: Combining Like Terms
7. Evaluating Expressions: Plugging in Values
8. Introduction to Equations: Balancing Both Sides
9. Solving One-Step Equations: The Inverse Operation
10. Solving Two-Step Equations: A Systematic Approach
11. Introduction to Inequalities: Greater Than and Less Than
12. Solving Basic Inequalities: Flipping the Sign
13. Number Systems: Integers, Rationals, and Reals
14. Absolute Value: Understanding Distance from Zero
15. Introduction to Exponents: Powering Up
16. Basic Exponent Rules: Multiplication and Division
17. Introduction to Roots: Square Roots and Beyond
18. Introduction to Fractions in Algebra
19. Decimal Operations in Algebraic Contexts
20. Word Problems: Translating Words to Algebra
Linear Equations & Inequalities (21-40):
21. Solving Multi-Step Equations: Distributive Property
22. Equations with Variables on Both Sides
23. Solving Linear Inequalities: Compound Inequalities
24. Solving Inequalities with Absolute Values
25. Applications of Linear Equations: Real-World Scenarios
26. Ratios and Proportions: Solving for Unknowns
27. Direct and Inverse Variation: Relationships Between Variables
28. Linear Equations and Slope: Visualizing Change
29. Finding the Equation of a Line: Slope-Intercept Form
30. Point-Slope Form: Another Way to Write Lines
31. Parallel and Perpendicular Lines: Geometric Relationships
32. Systems of Linear Equations: Finding Intersection Points
33. Solving Systems by Substitution: A Step-by-Step Guide
34. Solving Systems by Elimination: Adding and Subtracting
35. Applications of Systems of Equations: Mixture Problems
36. Linear Inequalities in Two Variables: Graphing Solutions
37. Systems of Linear Inequalities: Finding Common Regions
38. Rate, Time, and Distance Problems: Applications of Linear Equations
39. Work Rate Problems: Combining Efforts
40. Percentage Problems: Algebraic Solutions
Polynomials & Quadratics (41-60):
41. Introduction to Polynomials: Terms and Degrees
42. Adding and Subtracting Polynomials: Combining Like Terms
43. Multiplying Polynomials: FOIL and Distribution
44. Special Products: Squaring and Cubing Binomials
45. Factoring Polynomials: Finding Common Factors
46. Factoring by Grouping: Rearranging Terms
47. Factoring Trinomials: The AC Method
48. Difference of Squares and Sum/Difference of Cubes
49. Introduction to Quadratic Equations: The Standard Form
50. Solving Quadratic Equations by Factoring
51. Solving Quadratic Equations by Completing the Square
52. The Quadratic Formula: A Universal Solution
53. Discriminant of a Quadratic: Nature of Roots
54. Applications of Quadratic Equations: Projectile Motion
55. Graphing Quadratic Functions: Parabolas
56. Vertex Form of a Quadratic: Finding Maximum/Minimum
57. Quadratic Inequalities: Solving and Graphing
58. Polynomial Division: Long Division and Synthetic Division
59. Remainder and Factor Theorems: Shortcuts for Polynomials
60. Rational Root Theorem: Finding Possible Solutions
Advanced Concepts & Applications (61-80):
61. Exponential Equations: Solving for the Unknown Exponent
62. Logarithmic Equations: The Inverse of Exponentials
63. Properties of Logarithms: Simplifying Expressions
64. Solving Exponential and Logarithmic Inequalities
65. Functions: Domain, Range, and Composition
66. Inverse Functions: Reversing the Mapping
67. Rational Expressions: Simplifying and Operating
68. Solving Rational Equations: Clearing Denominators
69. Rational Inequalities: Finding Intervals of Solutions
70. Radical Equations: Isolating the Radical
71. Radical Inequalities: Combining Algebraic and Inequality Techniques
72. Absolute Value Functions and Graphs
73. Piecewise Functions: Combining Different Rules
74. Sequences and Series: Arithmetic and Geometric
75. Summation Notation: Representing Series
76. Permutations and Combinations: Counting Techniques
77. Binomial Theorem: Expanding Binomials to Powers
78. Matrices and Determinants: Introduction to Linear Algebra
79. Solving Systems of Equations Using Matrices
80. Complex Numbers: Introduction to Imaginary Units
Aptitude & Problem Solving (81-100):
81. Advanced Word Problems: Multi-Variable Scenarios
82. Rate and Time Problems: Advanced Concepts
83. Work and Efficiency Problems: Combined Efforts
84. Age Problems: Setting Up Equations
85. Geometry and Algebra: Combining Concepts
86. Data Interpretation and Algebra: Extracting Information
87. Logical Reasoning and Algebraic Puzzles
88. Optimization Problems: Finding Maximum/Minimum Values
89. Pattern Recognition and Algebraic Sequences
90. Algebraic Proofs: Justifying Solutions
91. Advanced Inequality Techniques: AM-GM Inequality
92. Modular Arithmetic: Remainders and Congruences
93. Number Theory and Algebra: Prime Factorization
94. Problem Solving Strategies: Guess and Check, Work Backwards
95. Algebraic Manipulation: Simplifying Complex Expressions
96. Advanced Factoring Techniques: Sum and Difference of Higher Powers
97. Algebraic Modeling: Creating Equations from Real-World Data
98. Advanced Graphing Techniques: Transformations of Functions
99. Test-Taking Strategies for Algebra Aptitude
100. Comprehensive Practice: Integrating All Concepts.