Mastering the Rule of Alligation for Competitive Exams

The Rule of Alligation is a powerful technique used in quantitative aptitude sections of competitive exams, particularly for problems involving mixtures and their concentrations. It provides a shortcut to solve problems that would otherwise require lengthy algebraic manipulations. This module will guide you through understanding and applying this rule effectively.

What is the Rule of Alligation?

The Rule of Alligation is a method used to find the ratio in which two ingredients of different prices/qualities are mixed to produce a mixture of a desired mean price/quality. It's particularly useful when dealing with two quantities that are to be mixed to achieve a specific average value.

The Alligation Cross Method

The most common way to visualize and apply the Rule of Alligation is through the 'Alligation Cross'. This visual tool helps in quickly calculating the required ratio.

The Alligation Cross involves placing the prices (or qualities) of the two ingredients on the left side and the desired mean price (or quality) in the center. On the right side, you calculate the differences diagonally. The difference between the higher price and the mean price gives the quantity of the cheaper ingredient, and the difference between the mean price and the lower price gives the quantity of the dearer ingredient. These differences represent the ratio in which the ingredients should be mixed.

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Let's illustrate with an example:

Problem: In what ratio should rice costing ₹30/kg be mixed with rice costing ₹25/kg so that the mixture, when sold at ₹28/kg, yields a profit of 10%?

Solution Steps:

Calculate the Cost Price (CP) of the mixture: The selling price (SP) is ₹28/kg, and the profit is 10%. Therefore, CP = SP / (1 + Profit%) = 28 / (1 + 0.10) = 28 / 1.10 = ₹25.45/kg (approximately).
Apply the Alligation Cross:
- Cheaper rice CP: ₹25/kg
- Dearer rice CP: ₹30/kg
- Mean CP: ₹25.45/kg
code
```
25 ---- 
     \  / 
      25.45 
     /  \n    30 ---- 
```
Calculate the differences diagonally:
- Difference between Dearer CP and Mean CP: |30 - 25.45| = 4.55
- Difference between Mean CP and Cheaper CP: |25.45 - 25| = 0.45
Determine the ratio: The ratio of cheaper rice to dearer rice is 4.55 : 0.45, which simplifies to 455 : 45, or 91 : 9.

Therefore, the rice should be mixed in the ratio 91:9.

Key Considerations and Variations

The Rule of Alligation can be applied to various scenarios beyond just prices, including:

Concentrations: Mixing solutions of different concentrations (e.g., acid and water).
Alcohol and Water: Determining the ratio to mix spirits of different strengths.
Averages: Finding the average score of a class when two groups with different average scores are combined.

Always ensure that the units for all quantities (prices, concentrations, averages) are consistent before applying the alligation cross. If dealing with profit or loss, always convert the selling price to the cost price first.

What is the fundamental principle behind the Rule of Alligation?

The weighted average of two components lies between their individual values, and the differences from the mean are inversely proportional to the quantities.

Practice Problems

To solidify your understanding, practice solving a variety of problems involving mixtures, ratios, and averages. Focus on identifying the two components, the desired mean value, and applying the alligation cross correctly. Pay close attention to profit/loss percentages and unit conversions.

Learning Resources

Alligation Rule for CAT Exam | Quantitative Aptitude(blog)

This blog post provides a clear explanation of the alligation rule with examples relevant to competitive exams like CAT.

Alligation - Mixtures and Alligations(documentation)

IndiaBIX offers a comprehensive section on alligation and mixtures, including theory, solved examples, and practice questions.

Alligation Rule Explained with Examples | CAT Quant(video)

A video tutorial that visually explains the alligation rule and demonstrates its application with practical examples for CAT preparation.

Mixtures and Alligations - Quant for CAT(blog)

This resource covers mixtures and alligations, including the alligation rule, with explanations and practice problems tailored for CAT aspirants.

Rule of Alligation - Concept and Examples(blog)

Examveda provides a detailed explanation of the rule of alligation with step-by-step examples and a focus on competitive exam patterns.

Alligation and Mixtures - Quantitative Aptitude(blog)

Toppr's Bytes section offers a good overview of alligation and mixtures, explaining the core concepts and providing solved examples.

Quantitative Aptitude - Alligation(blog)

Oliveboard's blog post delves into the alligation rule, explaining its use in solving mixture problems with clear examples.

Alligation and Mixtures - Practice Questions(blog)

This resource provides practice questions specifically on alligation and mixtures, allowing learners to test their understanding and application of the rule.

Alligation Rule - How to Solve Mixture Problems(video)

Another helpful video tutorial that breaks down the alligation rule and demonstrates how to solve mixture problems efficiently.

Alligation and Mixtures - Concepts and Tricks(forum)

While a forum, this discussion often contains valuable insights, tips, and alternative approaches to solving alligation and mixture problems from fellow aspirants.