13_Averages

Category: Quantitative Aptitude
Generated on: 2025-07-15 09:19:30
Source: Aptitude Mastery Guide Generator

---

A Comprehensive Guide to Averages for Competitive Exams

Welcome to your ultimate guide on Averages! This document is designed to equip you with the knowledge and skills necessary to tackle any average-related question in competitive exams and placement tests. We’ll cover the fundamentals, essential formulas, powerful shortcuts, solved examples, and practice problems to solidify your understanding. Let’s begin!

1. Foundational Concepts

The term “average” (also known as the “mean”) represents a central value within a set of numbers. It’s a way to summarize a collection of data into a single, representative figure.

Definition: The average of a set of n numbers (x₁, x₂, …, xₙ) is calculated by summing all the numbers and then dividing by n.

Why does this work? Imagine distributing a total amount equally among several individuals. The average represents the amount each individual would receive. This concept is crucial for understanding weighted averages and other variations.

Types of Averages:

• Arithmetic Mean (AM): The most common type of average, calculated as described above. This guide primarily focuses on the arithmetic mean.
• Geometric Mean (GM): Useful for finding the average rate of change over time. (Less common in basic aptitude tests but worth knowing).
• Harmonic Mean (HM): Useful when dealing with rates and ratios, especially when the denominator is constant. (Less common in basic aptitude tests but worth knowing).

In most competitive exams, “average” implicitly refers to the Arithmetic Mean.

2. Key Tricks & Shortcuts

This section is the heart of this guide. Mastering these shortcuts will significantly improve your speed and accuracy.

• Trick 1: The Deviation Method (Finding the Average Quickly)

  • How it works: Assume an average. Calculate the deviation of each number from this assumed average. Sum the deviations. Divide the sum of deviations by the total number of values. Add this result to your assumed average to get the actual average.
  • When to use it: When dealing with large numbers or a long list of numbers. It simplifies calculations by working with smaller deviations.
  • Example: Find the average of 52, 56, 58, 60, and 64.
    1. Assume the average is 58.
    2. Deviations: -6, -2, 0, +2, +6
    3. Sum of deviations: -6 - 2 + 0 + 2 + 6 = 0
    4. Average = 58 + (0/5) = 58
  • Why it works: You’re essentially redistributing the ‘excess’ or ‘deficit’ from your assumed average equally among all the numbers.

• Trick 2: Weighted Average Shortcut

  • How it works: If you have two groups with averages A₁ and A₂ and sizes N₁ and N₂ respectively, the combined average is (N₁A₁ + N₂A₂) / (N₁ + N₂). This generalizes to more than two groups.
  • When to use it: When dealing with groups of numbers with different sizes and averages.
  • Example: The average score of 30 students in a class is 70. The average score of another 20 students is 80. What is the average score of all 50 students?
    • Average = (30 * 70 + 20 * 80) / (30 + 20) = (2100 + 1600) / 50 = 3700 / 50 = 74

• Trick 3: Average Speed (Constant Distance)

  • How it works: If a person travels the same distance at two different speeds, x and y, the average speed is 2xy / (x + y). This is the Harmonic Mean.
  • When to use it: Only when the distances are equal.
  • Example: A man travels from A to B at 40 kmph and returns from B to A at 60 kmph. What is his average speed for the entire journey?
    • Average speed = (2 * 40 * 60) / (40 + 60) = 4800 / 100 = 48 kmph

• Trick 4: Replacement/Inclusion Shortcut

  • How it works: If an element is replaced or added to a group, the change in the total sum divided by the total number of elements gives the change in the average.
  • When to use it: When a value is added, removed, or replaced within a set, and you need to find the change in average.
  • Example: The average weight of 8 people is increased by 2.5 kg when one of them who weighs 56 kg is replaced by a new man. What is the weight of the new man?
    • Increase in total weight = 8 * 2.5 = 20 kg
    • Weight of new man = 56 + 20 = 76 kg

• Trick 5: Consecutive Numbers

  • How it works: The average of a set of consecutive numbers (or consecutive even/odd numbers) is simply the average of the first and last number. It’s also the middle number if there’s an odd number of elements.
  • When to use it: When dealing with consecutive integers, consecutive even integers, or consecutive odd integers.
  • Example: Find the average of the consecutive numbers 1, 2, 3, 4, 5.
    • Average = (1 + 5) / 2 = 3

• Trick 6: Weighted average when only the ratio of counts is given:

  • How it works: If we know that the ratio of the two groups is a:b, then we can directly use the averages A1 and A2 to compute the new average as : (aA1 + bA2)/(a+b)
  • When to use it: When we are not given the number of groups, just their ratio.
  • Example: In a school, the ratio of boys and girls is 3:2. If the average weight of the boys is 50kg and the average weight of the girls is 40kg, then what is the average weight of all students in the school?
    • Average weight = (350 + 240)/(3+2) = (150+80)/5 = 230/5 = 46kg

3. Essential Formulas & Rules

Formula/Rule	Description
Average (Arithmetic Mean)	Sum of observations / Number of observations
Sum of observations	Average * Number of observations
Weighted Average	(N₁A₁ + N₂A₂ + … + NₙAₙ) / (N₁ + N₂ + … + Nₙ) where A₁, A₂, …, Aₙ are averages of groups with sizes N₁, N₂, …, Nₙ
Average Speed (Constant Distance)	2xy / (x + y) where x and y are the speeds for the same distance.
Sum of first n natural numbers	n(n+1)/2
Sum of squares of first n natural numbers	n(n+1)(2n+1)/6
Sum of cubes of first n natural numbers	[n(n+1)/2]^2
Average of first n natural numbers	(n+1)/2
Average of first n even numbers	n+1
Average of first n odd numbers	n

4. Detailed Solved Examples

Example 1: Basic Average (Deviation Method)

Find the average of the following numbers: 120, 130, 145, 150, 165.

Solution:

1. Assume the average is 140.
2. Deviations: -20, -10, +5, +10, +25
3. Sum of deviations: -20 - 10 + 5 + 10 + 25 = 10
4. Average = 140 + (10/5) = 140 + 2 = 142

Method Used: Deviation Method (Trick 1)

Example 2: Weighted Average

A class has 40 students. The average weight of the boys is 60 kg, and the average weight of the girls is 50 kg. If the overall average weight of the class is 56 kg, find the number of boys in the class.

Solution:

Let the number of boys be ‘b’ and the number of girls be ‘g’. We know that b + g = 40, so g = 40 - b. Using the weighted average formula: (60b + 50g) / 40 = 56 60b + 50(40 - b) = 56 * 40 60b + 2000 - 50b = 2240 10b = 240 b = 24

Therefore, there are 24 boys in the class.

Method Used: Weighted Average (Trick 2) and Basic Algebra

Example 3: Average Speed (Constant Distance)

A car travels from city A to city B at a speed of 60 km/h and returns from city B to city A at a speed of 40 km/h. Find the average speed of the car for the entire journey.

Solution:

Using the average speed formula for constant distance: Average speed = (2 * 60 * 40) / (60 + 40) = 4800 / 100 = 48 km/h

Method Used: Average Speed (Constant Distance) (Trick 3)

Example 4: Replacement

The average weight of 10 people is 60 kg. When one person is replaced by a new person, the average weight increases by 2 kg. Find the weight of the new person if the weight of the replaced person was 55 kg.

Solution:

Increase in total weight = 10 * 2 = 20 kg Weight of new person = 55 + 20 = 75 kg

Method Used: Replacement/Inclusion Shortcut (Trick 4)

5. Practice Problems (Graded Difficulty)

[Easy] The average of five numbers is 27. If one number is excluded, the average becomes 25. Find the excluded number.

[Easy] What is the average of the first 10 natural numbers?

[Medium] The average age of 8 men is increased by 2 years when two of them, whose ages are 21 and 23 years, are replaced by two new men. What is the average age of the two new men?

[Medium] A batsman has an average of 50 runs in 12 innings. How many runs must he score in his next innings to increase his average to 52 runs?

[Hard] The average weight of a group of students is 65 kg. If 10 students with an average weight of 70 kg join the group, and 5 students with an average weight of 60 kg leave the group, the average weight of the group becomes 66 kg. Find the original number of students in the group.

[Hard] A train travels from station A to station B at a speed of 80 km/h. It stops at station B for 30 minutes and then travels from station B to station C at a speed of 100 km/h. The distance from A to B is equal to the distance from B to C. If the average speed of the train for the entire journey is 88 km/h, find the distance from A to B.

[Medium] In a class, the average score of boys is 70 and the average score of girls is 80. If the ratio of boys to girls is 2:3, find the overall average score of the class.

6. Advanced/Case-Based Question

A company has three departments: Production, Marketing, and Sales. The number of employees in each department and their average salaries are given below:

• Production: 50 employees, Average salary = $60,000
• Marketing: 30 employees, Average salary = $80,000
• Sales: 20 employees, Average salary = $100,000

The company decides to implement the following changes:

1. 10 employees are transferred from Production to Sales.
2. 5 new employees are hired in Marketing with an average salary of $70,000.
3. All employees receive a 5% raise.

Calculate the following:

a) The overall average salary before the changes. b) The overall average salary after the changes. c) The percentage increase in the overall average salary after the changes (rounded to two decimal places).

This guide should provide a robust foundation for excelling in average-related questions. Good luck with your preparation! Remember to practice regularly and apply the tricks and shortcuts learned.