Mastering Averages of a Series for Competitive Exams

Averages are a fundamental concept in quantitative aptitude, frequently tested in competitive exams like the CAT. Understanding how to calculate and manipulate averages of series is crucial for solving a wide range of problems efficiently. This module will break down the core concepts and provide strategies for tackling average-related questions.

What is an Average?

The average (or arithmetic mean) of a set of numbers is the sum of all the numbers divided by the count of the numbers. It represents a central or typical value for a set of data.

What is the formula for calculating the average of a series of numbers?

Average = (Sum of all numbers) / (Count of numbers)

Key Properties of Averages

The sum of deviations from the average is always zero.

If you subtract the average from each number in a series and then sum up these differences, the result will be zero. This property is useful in certain algebraic manipulations.

Let the series be $x_1, x_2, ..., x_n$ and their average be $\bar{x}$ . The sum of deviations from the average is given by $\sum_{i=1}^{n} (x_i - \bar{x})$ . Expanding this, we get $\sum_{i=1}^{n} x_i - \sum_{i=1}^{n} \bar{x}$ . Since $\sum_{i=1}^{n} x_i = n \times \bar{x}$ and $\sum_{i=1}^{n} \bar{x} = n \times \bar{x}$ , the expression becomes $n \times \bar{x} - n \times \bar{x} = 0$ . This confirms that the sum of deviations from the mean is zero.

Adding a constant to each number increases the average by that constant.

If every value in a series is increased by a fixed number, the average of the new series will be the original average plus that fixed number.

Consider a series $x_1, x_2, ..., x_n$ with average $\bar{x}$ . If we add a constant $k$ to each term, the new series becomes $(x_1+k), (x_2+k), ..., (x_n+k)$ . The sum of the new series is $\sum_{i=1}^{n} (x_i+k) = \sum_{i=1}^{n} x_i + \sum_{i=1}^{n} k = (n \times \bar{x}) + (n \times k)$ . The new average is $\frac{(n \times \bar{x}) + (n \times k)}{n} = \bar{x} + k$ .

Multiplying each number by a constant multiplies the average by that constant.

If every value in a series is multiplied by a fixed number, the average of the new series will be the original average multiplied by that fixed number.

Let the original series be $x_1, x_2, ..., x_n$ with average $\bar{x}$ . If we multiply each term by a constant $k$ , the new series is $kx_1, kx_2, ..., kx_n$ . The sum of the new series is $\sum_{i=1}^{n} kx_i = k \sum_{i=1}^{n} x_i = k \times (n \times \bar{x})$ . The new average is $\frac{k \times (n \times \bar{x})}{n} = k \times \bar{x}$ .

Calculating Average of a Series with Missing Values

A common problem type involves a series where one or more values are unknown or have changed. The key is to use the relationship: Sum = Average × Count.

When a value is replaced, the change in the sum is equal to the change in the average multiplied by the number of terms.

Example: The average of 10 numbers is 50. If one number is replaced by 70, the average becomes 52. Find the original number.

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Initial Sum = 10 numbers × 50 (average) = 500. New Sum = 10 numbers × 52 (average) = 520. Change in Sum = New Sum - Initial Sum = 520 - 500 = 20. Change in Average = 52 - 50 = 2. Since the number of terms is constant (10), the change in sum is also (Change in Average × Number of Terms) = 2 × 10 = 20. This matches. Let the original number be 'x' and the new number be 'y' (which is 70). Initial Sum = Sum of other 9 numbers + x New Sum = Sum of other 9 numbers + y New Sum - Initial Sum = y - x 20 = 70 - x x = 70 - 20 = 50. The original number was 50.

Average of a Series with Added/Removed Elements

When elements are added to or removed from a series, the count of numbers changes, affecting the average. Again, the core principle is Sum = Average × Count.

Consider a scenario where the average weight of 5 boys is 45 kg. If a new boy joins the group, the average weight increases by 2 kg. What is the weight of the new boy?

Initial state: 5 boys, average weight = 45 kg. Initial total weight = 5 * 45 = 225 kg.

New state: 6 boys (5 original + 1 new), average weight = 45 + 2 = 47 kg. New total weight = 6 * 47 = 282 kg.

Weight of the new boy = New total weight - Initial total weight = 282 - 225 = 57 kg.

This visual represents the change in total weight due to the addition of a new member, impacting the overall average.

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Text-based content

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If the average of 'n' items is 'A', and 'k' items with an average of 'B' are added, what is the average of the new combined series?

New Average = (nA + kB) / (n+k)

Weighted Average

A weighted average is used when different items in a series contribute differently to the overall average. Each item is multiplied by its 'weight' (importance or frequency) before summing, and then divided by the sum of the weights.

Concept	Simple Average	Weighted Average
Calculation	Sum of values / Count of values	Sum of (value * weight) / Sum of weights
Application	All values have equal importance	Values have different levels of importance or frequency
Example	Average marks in a single test	Average marks in a course where assignments, midterms, and finals have different weightages

Practice Strategies

To excel in average-related problems:

Understand the core formula: Always remember Sum = Average × Count.
Identify changes: Carefully note if numbers are added, removed, or replaced, and how this affects the count and the sum.
Use properties: Leverage the properties of averages to simplify calculations.
Practice diverse problems: Solve questions involving different scenarios, including weighted averages and series with missing data.

Learning Resources

Understanding Average (Arithmetic Mean)(video)

Khan Academy provides a clear and concise video explanation of what an average is and how to calculate it.

Averages - Concepts and Questions(documentation)

IndiaBIX offers a comprehensive overview of average concepts, formulas, and solved examples relevant to competitive exams.

Average Formula and Examples(blog)

This blog post from Toppr explains the average formula and provides practical examples with step-by-step solutions.

CAT Quantitative Aptitude: Averages(documentation)

Cracku provides topic-wise explanations and practice questions specifically tailored for the CAT exam, including a detailed section on averages.

Weighted Average Explained(documentation)

Investopedia explains the concept of weighted average, its calculation, and its applications, which is useful for understanding variations of the average concept.

Average Problems for Competitive Exams(documentation)

CareerBliss offers a collection of important average problems with solutions, focusing on common patterns found in competitive exams.

Average of a Series - Concepts and Tricks(blog)

Examveda provides useful tricks and concepts for solving average problems efficiently, often seen in banking and other aptitude tests.

How to Calculate Average: Step-by-Step(tutorial)

A simple, step-by-step guide on how to calculate an average, useful for reinforcing the basic definition.

Average of Numbers in Arithmetic Progression(documentation)

GeeksforGeeks explains a specific case: calculating the average of numbers in an arithmetic progression, a common variation.

Average - Practice Questions(documentation)

Oliveboard offers practice questions on averages with detailed explanations, helping learners test their understanding and application skills.