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Average

Q: What is the average of the first 50 natural numbers?

25.5. The sum of the first N natural numbers is N(N+1)/2. The average is (N(N+1)/2) / N = (N+1)/2. For N=50, average = (50+1)/2 = 51/2 = 25.5.

Q: The average of 5 numbers is 20. If one number is excluded, the average becomes 23. What is the excluded number?

11. Initial total sum = 5 * 20 = 100. New total sum (4 numbers) = 4 * 23 = 92. Excluded number = 100 - 92 = 8.

Q: The average of 11 players of a cricket team is 27 years. If the age of the coach is also included, the average age increases by 1 year. What is the age of the coach?

41 years. Total age of 11 players = 11 * 27 = 297 years. After including the coach, total persons = 12. New average = 27 + 1 = 28 years. New total age = 12 * 28 = 336 years. Coach's age = 336 - 297 = 39 years.

Q: A person travels from P to Q at a speed of 30 km/hr and returns from Q to P at a speed of 20 km/hr. What is his average speed for the whole journey?

24 km/hr. For equal distances, Average Speed = (2xy) / (x + y). Here x=30, y=20. Average Speed = (2 * 30 * 20) / (30 + 20) = 1200 / 50 = 24 km/hr.

Q: The average of seven consecutive even numbers is 30. Find the largest number.

34. For consecutive even numbers, the average is the middle number. So, the 4th number is 30. The numbers are 24, 26, 28, 30, 32, 34, 36. The largest number is 36.

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Introduction

Have you ever wondered how your favorite cricketer's performance is measured over several matches? Or how schools calculate the overall performance of students? The answer lies in 'Average'. Average is a fundamental concept in mathematics that helps us find a typical or central value for a set of numbers. It simplifies complex data into a single, understandable figure, making it incredibly useful in competitive exams and real-life scenarios like managing finances or analyzing sports statistics. Let's dive in and master this essential topic!

Avg

Total Sum / Number of Items

Full Explanation

What is Average?

In simple terms, the Average (also known as the arithmetic mean) is a single value that represents a group of numbers. Imagine you have different amounts of money: Rs 10, Rs 20, and Rs 30. If you wanted to distribute this money equally among three people, how much would each person get? You'd add them up (10 + 20 + 30 = 60) and divide by the number of people (3), giving you Rs 20. So, Rs 20 is the average amount.

The concept of average is crucial for competitive exams like SSC, Banking, Railway, and UPSC because it allows you to quickly gauge the typical value in a dataset, which is frequently tested in various problem types.

Basic Formula of Average

The most fundamental formula for calculating average is:

Average = (Sum of all observations) / (Number of observations)

Let's take an example: Find the average of 5, 10, 15, 20, 25.
Sum of observations = 5 + 10 + 15 + 20 + 25 = 75
Number of observations = 5
Average = 75 / 5 = 15

Important Properties of Average

If each number in a set is increased by a certain value 'x', the average of the set also increases by 'x'.
If each number in a set is decreased by 'x', the average also decreases by 'x'.
If each number in a set is multiplied by 'x', the average also gets multiplied by 'x'.
If each number in a set is divided by 'x' (where x ≠ 0), the average also gets divided by 'x'.
The average of a group always lies between the smallest and largest values in that group.

Types of Average Problems

Competitive exams often feature several types of average problems:

Problems based on adding/removing items: When a new member joins or leaves a group, how does the average change? If N items have an average of A, and a new item 'x' is added, the new average becomes (N*A + x) / (N+1).
Weighted Average: This is used when different observations have different 'weights' or importance. For example, calculating the average marks of a class where different subjects have different maximum marks. If you have groups with averages A1, A2, ... Ak and number of items N1, N2, ... Nk, the weighted average is (N1*A1 + N2*A2 + ... + Nk*Ak) / (N1 + N2 + ... + Nk).
Average Speed: This is a common application where speed is not constant. If a person travels a distance D1 at speed S1 and another distance D2 at speed S2, the average speed is (Total Distance) / (Total Time). If equal distances are covered at different speeds, say 'x' km/hr and 'y' km/hr, the average speed is (2xy / (x+y)) km/hr.
Average of Consecutive Numbers: For a series of consecutive numbers (arithmetic progression), the average is simply the middle number. If there's an even number of terms, it's the average of the two middle numbers. For example, the average of 1, 2, 3, 4, 5 is 3 (the middle number). The average of 2, 4, 6, 8 is (4+6)/2 = 5.

Understanding these variations and their associated formulas will equip you to tackle a wide range of average questions efficiently in exams.

Formulas & Rules

Basic Average Formula

Average = (Sum of observations) / (Number of observations)

Weighted Average Formula

Weighted Avg = (Σ(Weight × Value)) / (ΣWeight)

Average of Consecutive Numbers

Average = (First Term + Last Term) / 2

Average Speed (Equal Distance)

Average Speed = (2xy) / (x + y)

Average after Adding a New Item

New Avg = (Old Total + New Item) / (Old Count + 1)

Comparison

Scenario	Description	Formula/Approach
Simple Average	Average of a basic set of numbers.	Sum of observations / Number of observations
Weighted Average	Observations have different importance or frequency.	(Σ(Weight × Value)) / (ΣWeight)
Average Speed	Covering different distances at different speeds.	Total Distance / Total Time
Consecutive Numbers	Numbers in an arithmetic progression.	(First Term + Last Term) / 2
Item Added/Removed	Change in average when a value is added or removed from a group.	(New Total) / (New Count)

Solved Examples

Example 1

Q: The marks obtained by 5 students in a test are 60, 75, 80, 65, and 90. Find their average marks.

Solution:

Step 1: Sum all the marks obtained by the students.
Sum = 60 + 75 + 80 + 65 + 90 = 370
Step 2: Count the number of students.
Number of students = 5
Step 3: Apply the average formula (Sum / Number).
Average = 370 / 5 = 74

Answer: The average marks of the 5 students are 74.

Example 2

Q: The average weight of 10 boys is 45 kg. If a new boy joins the group, the average weight becomes 46 kg. What is the weight of the new boy?

Solution:

Step 1: Calculate the total weight of the initial 10 boys.
Total weight = Average × Number of boys = 45 kg × 10 = 450 kg
Step 2: After the new boy joins, the number of boys becomes 11 and the new average is 46 kg.
New total weight = New Average × New Number of boys = 46 kg × 11 = 506 kg
Step 3: The weight of the new boy is the difference between the new total weight and the old total weight.
Weight of new boy = 506 kg - 450 kg = 56 kg

Answer: The weight of the new boy is 56 kg.

Example 3

Q: A car travels from city A to city B at a speed of 60 km/hr and returns from city B to city A at a speed of 40 km/hr. What is the average speed for the entire journey?

Solution:

Step 1: Recognize that the distance is equal for both parts of the journey (A to B and B to A).
Step 2: Use the formula for average speed when equal distances are covered at different speeds: Average Speed = (2xy) / (x + y).
Here, x = 60 km/hr and y = 40 km/hr.
Step 3: Substitute the values into the formula.
Average Speed = (2 × 60 × 40) / (60 + 40)
Average Speed = (4800) / (100)
Average Speed = 48 km/hr

Answer: The average speed for the entire journey is 48 km/hr.

Example 4

Q: The average age of 24 students and their teacher is 15 years. If the teacher's age is excluded, the average age of the students becomes 14 years. What is the teacher's age?

Solution:

Step 1: Calculate the total age of 24 students and the teacher.
Total age (24 students + teacher) = 25 × 15 = 375 years
Step 2: Calculate the total age of 24 students when the teacher's age is excluded.
Total age (24 students) = 24 × 14 = 336 years
Step 3: The teacher's age is the difference between the two totals.
Teacher's age = 375 - 336 = 39 years

Answer: The teacher's age is 39 years.

🌍 Real-World Scenarios

🏏Cricket's Consistency Score

Rohit scored 50, 80, 20, 100, and 50 runs in five T20 matches. What is his average score per match, which tells us how consistent he is?

📚Exam Report Card

In your final exams, you scored 70 in Maths, 85 in Science, 60 in English, and 75 in Social Studies. What's your average score across these subjects?

💰Pocket Money Planner

You receive pocket money: ₹100 in Week 1, ₹150 in Week 2, and ₹50 in Week 3. If you want to save the same amount each week from now on, what's your average weekly pocket money?

🚗Road Trip Rally

Your family car covered 120 km in 2 hours on the highway and then 40 km in 1 hour in city traffic. What was the car's average speed for the entire trip?

Tricky Questions

The average of 7 consecutive numbers is 20. The largest of these numbers is:

The average age of a husband, wife, and their child 3 years ago was 27 years. The average age of the wife and child 5 years ago was 20 years. What is the present age of the husband?

A student was asked to find the average of 10 numbers. He correctly calculated the average of 9 numbers as 25 but made a mistake in one number, reading 45 instead of 54. What is the correct average?

The average of 11 results is 50. If the average of the first 6 results is 49 and that of the last 6 results is 52, find the 6th result.

Practice MCQ Quiz

1What is the average of the first 50 natural numbers?

2The average of 5 numbers is 20. If one number is excluded, the average becomes 23. What is the excluded number?

3The average of 11 players of a cricket team is 27 years. If the age of the coach is also included, the average age increases by 1 year. What is the age of the coach?

4A person travels from P to Q at a speed of 30 km/hr and returns from Q to P at a speed of 20 km/hr. What is his average speed for the whole journey?

5The average of seven consecutive even numbers is 30. Find the largest number.

6The average monthly income of A and B is ₹5050. The average monthly income of B and C is ₹6250. The average monthly income of A and C is ₹5200. What is the monthly income of A?

7The average weight of 8 persons increases by 2.5 kg when a new person comes in place of one of them weighing 65 kg. What is the weight of the new person?

8The average of 30 results is 20 and the average of other 20 results is 30. What is the average of all the results?

9If the average of M numbers is N^2 and that of N numbers is M^2, then the average of (M+N) numbers is:

10In a certain school, the average age of all the students is 10 years, and the average age of the 20 teachers is 35 years. If the average age of all students and teachers combined is 12 years, how many students are there in the school?

More in Maths

Number System HCF & LCM (Highest Common Factor & Least Common Multiple)Simplification: Rules, Formulas, and Examples Percentage: Basics to Advanced Profit Loss & Discount: Concepts and Tricks Decimal & Fraction Ratio & Proportion: Master the Art of Comparison

Table of Contents

Introduction In Short Full Explanation Formulas Comparison Solved Examples Tricky Questions Practice Quiz

Quick Formulas

Basic Average Formula

Average = (Sum of observations) / (Number of observations)

Weighted Average Formula

Weighted Avg = (Σ(Weight × Value)) / (ΣWeight)

Average of Consecutive Numbers

Average = (First Term + Last Term) / 2

+2 more formulas below

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Scenario

Description

Formula/Approach

Simple Average

Average of a basic set of numbers.

Sum of observations / Number of observations

Weighted Average

Observations have different importance or frequency.

(Σ(Weight × Value)) / (ΣWeight)

Average Speed

Covering different distances at different speeds.

Total Distance / Total Time

Consecutive Numbers

Numbers in an arithmetic progression.

(First Term + Last Term) / 2

Item Added/Removed

Change in average when a value is added or removed from a group.

(New Total) / (New Count)