Mastering Proportions and Continued Proportions for Competitive Exams
Welcome to this module on Proportions and Continued Proportions, a fundamental topic in quantitative aptitude for competitive exams like the CAT. Understanding these concepts is crucial for solving a wide range of problems efficiently.
Understanding Ratios and Proportions
A ratio compares two quantities. A proportion states that two ratios are equal. For example, if the ratio of boys to girls in a class is 2:3, and another class has a ratio of 4:6, these are proportional because 2/3 = 4/6. This equality is the essence of a proportion.
A proportion is an equality of two ratios.
If a:b = c:d, then a, b, c, and d are in proportion. This can be written as a/b = c/d.
In a proportion a:b = c:d, 'a' and 'd' are called the 'extremes' and 'b' and 'c' are called the 'means'. A key property is that the product of the extremes equals the product of the means: a × d = b × c. This property is fundamental for solving proportion-based problems.
Extremes are 3 and 15. Means are 5 and 9.
Types of Proportions
Proportions can be categorized based on how quantities change in relation to each other.
| Type | Description | Example |
|---|---|---|
| Direct Proportion | As one quantity increases, the other quantity increases proportionally, or as one decreases, the other decreases proportionally. | If you buy more apples, the total cost increases. |
| Inverse Proportion | As one quantity increases, the other quantity decreases proportionally, and vice versa. | If more workers are hired for a job, the time taken to complete the job decreases. |
Remember: Direct proportion means the ratio remains constant (a/b = c/d), while inverse proportion means the product remains constant (ab = cd).
Continued Proportions
A continued proportion is a special case where three or more quantities are in proportion, with the second term of the first ratio being the first term of the second ratio, and so on. This is often seen with three terms.
Three quantities a, b, and c are in continued proportion if a:b = b:c.
This means b is the mean proportional between a and c. The relationship can be written as a/b = b/c, which simplifies to b² = ac.
In a continued proportion a:b = b:c, 'b' is called the mean proportional or geometric mean between 'a' and 'c'. This concept is vital for problems involving geometric progressions and certain types of ratio analysis.
x² = 4 * 9 => x² = 36 => x = 6 (assuming positive values).
Visualizing the relationship in a continued proportion a:b = b:c helps solidify understanding. Imagine a line segment divided into three parts. If the ratio of the first part to the second part is the same as the ratio of the second part to the third part, then the second part is the mean proportional. This is analogous to how terms grow or shrink consistently in a geometric sequence.
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Applying Proportions in Problem Solving
Many competitive exam questions can be solved by setting up proportions. Look for scenarios where quantities are related by a constant ratio or a constant product.
Tip: Always ensure the units and the order of terms are consistent when setting up your proportions to avoid errors.
Practice Scenarios
Common applications include problems involving mixtures, work and time, speed and distance, and currency conversions. For instance, if 5 kg of a mixture costs ₹250, how much would 12 kg cost? This is a direct proportion: 5/250 = 12/x.
This is an inverse proportion. 8 men * 15 days = 12 men * x days. So, 120 = 12x, which means x = 10 days.
Learning Resources
Provides a clear explanation of ratios and proportions with examples, suitable for beginners.
A foundational video explaining the concept of proportions and how to identify them.
Details the concept of continued proportions and the mean proportional with illustrative examples.
Offers a wide range of practice problems on ratios and proportions with solutions.
A comprehensive guide specifically tailored for CAT aspirants, covering key concepts and strategies.
Explains the geometric mean, which is directly related to continued proportions.
Provides a step-by-step approach to solving word problems involving proportions.
A quick reference for essential formulas related to ratios and proportions.
An interactive tutorial with clear explanations and examples for understanding ratios and proportions.
A forum discussion with insights and tips on advanced ratio and proportion problems for the CAT exam.