¶ Mixtures and Alligations
¶ Aptitude GK
¶ Introduction to Mixtures and Alligations: A Human-Centric Journey Into Logical Thinking, Ratios, and Everyday Problem-Solving
There are parts of mathematics that feel abstract, distant, or heavily theoretical. And then there are areas like mixtures and alligations—topics that quietly shape the way we understand proportions, balance, and the clever mathematics hidden inside everyday decisions. Whether we notice it or not, mixtures play a role in countless moments of life: blending ingredients in a kitchen, adjusting the strength of a solution, balancing percentages in business, determining the average speed of a journey, or forming the right combination of elements to achieve a goal. Behind all these situations lies one central idea: how do we combine things correctly, and how do we reason about the result?
This course, spread across a hundred detailed articles, will explore mixtures and alligations in the wider domain of Aptitude and General Knowledge—not as a dry mathematical topic, but as a framework for sharp reasoning, elegant logic, and intuitive problem-solving. Before diving into formulas, shortcuts, ratio diagrams, weighted averages, replacement problems, and the classic alligation rule, it’s important to first understand why this topic is so valuable, why it appears so often in aptitude examinations, and why it strengthens the mind far beyond mathematics.
At its heart, mixtures and alligations is a subject built on balance. When two or more components blend together, they create a new entity with characteristics shaped by each part’s contribution. This idea of balance is fundamental not only to mathematics but to critical thinking. When you learn to work with mixtures, you’re also learning how to think proportionally—how to judge the influence of different elements, how to identify the right ratio, how to estimate outcomes, and how to apply logic to real-world variations.
What makes this topic fascinating is that even though the underlying mathematics is simple—often involving ratios, averages, and proportionality—the reasoning behind it can become beautifully intricate. You may start with basic concepts like combining milk and water or mixing two varieties of tea, but soon you’ll find yourself solving problems of concentration, price mixtures, alloy combinations, percentage adjustments, cost-profit optimization, and even indirect applications such as weighted averages in data and decision-making.
This subject strengthens logical aptitude in a way few others do. When you solve mixture problems, you’re not just performing calculations—you’re analyzing relationships. You’re understanding how components influence each other. You’re identifying hidden patterns behind numbers. And you’re doing it all through reasoning that feels natural once you unlock the underlying intuition.
One of the reasons mixtures and alligations appears so frequently in competitive examinations is that it tests multiple skills at once. It checks your numerical fluency, but it also checks your ability to see patterns. It evaluates your understanding of ratios and percentages, but it also evaluates your ability to think quickly and make the right inferences. It combines conceptual clarity with creative reasoning, which is exactly why mastering this topic can significantly improve your overall aptitude performance.
This course is designed to help you build that mastery in a gradual, intuitive way. You’ll begin by understanding what a mixture actually is—not just mathematically, but logically. You’ll explore how the strength, price, or composition of a mixture is determined by weighted contributions. You’ll learn why averages behave the way they do when two components are blended. And you’ll develop the ability to solve problems through reasoning instead of memorizing formulas.
One of the most powerful tools in this area is the alligation method—a simple yet elegant rule that makes many mixture problems almost effortless. The moment you understand how alligation diagrams work, mixture problems begin to feel less like arithmetic and more like a puzzle waiting to be solved. Alligation teaches you to see balance visually: high values on one side, low values on the other, and the desired average in the middle. The ratio emerges not through calculation but through logical comparison. Many students find this moment of understanding to be a turning point in their aptitude preparation.
But mixtures and alligations go far beyond the classical rule. As you progress through the course, you’ll work through a wide range of scenarios: mixtures with replacement, repeated dilution, mixing in stages, finding unknown quantities, adjusting concentrations, comparing prices, and much more. You’ll encounter problems where the ratio changes gradually, where components evaporate or are removed, where cost adjustments alter the overall price, or where a mixture must be achieved with minimal waste or maximum efficiency.
Through these topics, you’ll begin to recognize how mathematical reasoning reflects real-world logic. For instance, when a shopkeeper blends two varieties of rice to achieve a target price, he is essentially using the alligation rule. When a chemist mixes solutions of different strengths, she is applying the same weighted average principles. When a business calculates average profit across divisions with different revenue contributions, it is again a mixture problem—though disguised under financial terminology.
Understanding mixtures is also a powerful way to strengthen your sense of proportional reasoning. You begin to see how small changes in one component can have a large impact on the whole. You understand why sometimes adding a small amount of something strong can significantly increase concentration, while in other cases even large additions barely make a difference. This sensitivity to proportion improves decision-making beyond mathematics. It helps you evaluate information more carefully, compare situations more objectively, and approach problems with sharper analytical instincts.
Another reason mixtures and alligations is valuable in aptitude training is that it naturally builds mental agility. The problems often require quick comparisons, ratio formation, estimation, and reverse reasoning. Sometimes the solution lies not in the obvious direction but in stepping back and analyzing what the problem is really asking. You learn when to use the average method, when to use the alligation rule, and when to break a problem into simpler parts. You learn how to avoid traps, how to interpret data correctly, and how to reach answers with minimal computation.
This kind of thinking is what aptitude tests reward—not speed alone, but clarity.
And the beauty of this subject is that it grows with you. When you first encounter it, the problems may feel unfamiliar. But as you progress, patterns emerge. You begin to understand the language of the problems. You can see the solution often before touching pen to paper. You can predict the outcome just by looking at values. This instinctive sense of balance—developed through practice—is one of the greatest strengths you’ll gain from this course.
But beyond aptitude and exams, mixtures and alligations carry a quiet elegance that makes the subject enjoyable. It shows you how mathematics can mirror life. It reminds you that outcomes are rarely shaped by one factor alone—they are blends of many influences. It teaches you that understanding the contribution of each part helps you understand the whole. In that sense, this topic becomes not just a mathematical tool but a way of thinking.
Throughout this course, you’ll explore real-world analogies that anchor the concepts firmly in your mind. You’ll see how mixtures appear in cooking, chemistry, finance, manufacturing, agriculture, and even in average calculations used in everyday decisions. You’ll understand why proportions matter, how averages behave, and how logic guides the blending of components. You’ll also uncover the mathematical clarity behind everyday intuitions that we often use without realizing.
By the time you complete all 100 articles, mixtures and alligations will no longer feel like a specialized topic—it will feel like a natural extension of your reasoning abilities. You’ll understand the deeper logic behind every problem. You’ll be able to approach even the most complex mixture scenarios with confidence. And you will carry the kind of mathematical clarity that strengthens not only your aptitude performance but also your general problem-solving skills.
For now, consider this introduction the beginning of a thoughtful journey—one that blends logic, intuition, and real-world understanding. Mixtures and alligations might begin as a mathematical topic, but as you dive deeper, you’ll discover that it is actually a gateway into sharper reasoning and clearer thinking.
Welcome to the course.
Welcome to the art and logic behind every mixture.
¶ Beginner Level (1-30)
1. Introduction to Mixtures and Alligations: Basic Concepts
2. Understanding Mixtures: Components and Their Properties
3. What is Alligation? The Rule of Mixture
4. Simple Mixtures: Two Ingredients with Different Quantities
5. Calculating the Mean Price of a Mixture
6. Basics of Alligation: Weighted Averages
7. Mixtures with Two Ingredients: Equal Quantities
8. Mixtures with Two Ingredients: Unequal Quantities
9. Finding the Ratio of Ingredients in a Mixture
10. Replacing a Part of a Mixture: Simple Replacement Problems
11. Mixtures with Water: Dilution Problems
12. Mixtures with Alcohol: Concentration Problems
13. Mixtures with Sugar and Water: Sweetness Problems
14. Mixtures with Milk and Water: Basic Problems
15. Mixtures with Two Liquids: Basic Alligation Problems
16. Mixtures with Solids and Liquids: Basic Problems
17. Word Problems on Mixtures: Beginner Level
18. Understanding Percentage Concentration in Mixtures
19. Mixtures with Profit and Loss: Basic Problems
20. Mixtures with Cost Price and Selling Price: Basic Problems
21. Mixtures with Ratios: Simple Ratio Problems
22. Mixtures with Averages: Basic Weighted Average Problems
23. Mixtures with Fractions: Basic Fractional Problems
24. Mixtures with Decimals: Basic Decimal Problems
25. Mixtures with Proportions: Basic Proportionality Problems
26. Mixtures with Time and Work: Basic Problems
27. Mixtures with Speed and Distance: Basic Problems
28. Mixtures with Simple Interest: Basic Problems
29. Mixtures with Compound Interest: Basic Problems
30. Mixtures with Probability: Basic Problems
¶ Intermediate Level (31-60)
31. Advanced Alligation: Three or More Ingredients
32. Mixtures with Three Ingredients: Ratio Calculations
33. Mixtures with Three Ingredients: Mean Price Calculations
34. Mixtures with Replacement: Intermediate Problems
35. Mixtures with Repeated Replacements: Intermediate Problems
36. Mixtures with Alcohol and Water: Intermediate Problems
37. Mixtures with Milk and Water: Intermediate Problems
38. Mixtures with Sugar and Water: Intermediate Problems
39. Mixtures with Profit and Loss: Intermediate Problems
40. Mixtures with Cost Price and Selling Price: Intermediate Problems
41. Mixtures with Ratios: Intermediate Ratio Problems
42. Mixtures with Averages: Intermediate Weighted Average Problems
43. Mixtures with Fractions: Intermediate Fractional Problems
44. Mixtures with Decimals: Intermediate Decimal Problems
45. Mixtures with Proportions: Intermediate Proportionality Problems
46. Mixtures with Time and Work: Intermediate Problems
47. Mixtures with Speed and Distance: Intermediate Problems
48. Mixtures with Simple Interest: Intermediate Problems
49. Mixtures with Compound Interest: Intermediate Problems
50. Mixtures with Probability: Intermediate Problems
51. Mixtures with Algebraic Equations: Intermediate Problems
52. Mixtures with Linear Equations: Intermediate Problems
53. Mixtures with Quadratic Equations: Intermediate Problems
54. Mixtures with Simultaneous Equations: Intermediate Problems
55. Mixtures with Inequalities: Intermediate Problems
56. Mixtures with Graphs: Intermediate Problems
57. Mixtures with Tables: Intermediate Problems
58. Mixtures with Charts: Intermediate Problems
59. Mixtures with Diagrams: Intermediate Problems
60. Mixtures with Real-Life Applications: Intermediate Problems
¶ Advanced Level (61-90)
61. Advanced Alligation: Multiple Replacements
62. Mixtures with Multiple Replacements: Advanced Problems
63. Mixtures with Alcohol and Water: Advanced Problems
64. Mixtures with Milk and Water: Advanced Problems
65. Mixtures with Sugar and Water: Advanced Problems
66. Mixtures with Profit and Loss: Advanced Problems
67. Mixtures with Cost Price and Selling Price: Advanced Problems
68. Mixtures with Ratios: Advanced Ratio Problems
69. Mixtures with Averages: Advanced Weighted Average Problems
70. Mixtures with Fractions: Advanced Fractional Problems
71. Mixtures with Decimals: Advanced Decimal Problems
72. Mixtures with Proportions: Advanced Proportionality Problems
73. Mixtures with Time and Work: Advanced Problems
74. Mixtures with Speed and Distance: Advanced Problems
75. Mixtures with Simple Interest: Advanced Problems
76. Mixtures with Compound Interest: Advanced Problems
77. Mixtures with Probability: Advanced Problems
78. Mixtures with Algebraic Equations: Advanced Problems
79. Mixtures with Linear Equations: Advanced Problems
80. Mixtures with Quadratic Equations: Advanced Problems
81. Mixtures with Simultaneous Equations: Advanced Problems
82. Mixtures with Inequalities: Advanced Problems
83. Mixtures with Graphs: Advanced Problems
84. Mixtures with Tables: Advanced Problems
85. Mixtures with Charts: Advanced Problems
86. Mixtures with Diagrams: Advanced Problems
87. Mixtures with Real-Life Applications: Advanced Problems
88. Mixtures with Competitive Exams: Advanced Problems
89. Mixtures with Time-Saving Techniques: Advanced Problems
90. Mixtures with Puzzles and Riddles: Advanced Problems
¶ Expert Level (91-100)
91. Mixtures with Research Problems: Open-Ended Challenges
92. Mixtures with Future Applications: Exploring New Frontiers
93. Mixtures with Advanced Mathematics: Connections to Algebra
94. Mixtures with Advanced Mathematics: Connections to Calculus
95. Mixtures with Advanced Mathematics: Connections to Probability
96. Mixtures with Advanced Mathematics: Connections to Statistics
97. Mixtures with Advanced Mathematics: Connections to Geometry
98. Mixtures with Advanced Mathematics: Connections to Trigonometry
99. Mixtures with Advanced Mathematics: Connections to Number Theory
100. Mixtures with Advanced Mathematics: Connections to Linear Algebra