Average in Maths: Definition, Symbol, Formula & Examples

All of us are somewhat familiar with the concept of Average. Questions related to the Average Quantitative Aptitude section is one of the easiest sections but sometimes the questions are framed in a tricky way so it is very necessary to be well aware of all the key concepts related to the Averages section. We are going to cover the key concepts of the Averages section along with the various types of questions, important formulas along with various tips and tricks. We have also added a few solved examples that candidates will find beneficial in their exam preparation. Read the article thoroughly to clear all the doubts regarding the same.

What are Averages?

Average is the mean value which is equal to the ratio of the sum of the number of a given set of values to the total number of values present in the set. We apply an average in various areas of real life. So, Average is defined as the sum of the observations divided by the number of observations.

Average = Sum of observations / Number of observations

Averages Symbol

Averages can basically be defined as the mean value which can be expressed as x bar (x̄), it is also known as the Average Symbol. The average symbol can also be denoted by μ.

Average of Negative Numbers

If there is a negative number present in a given list of numbers then candidates can calculate the average by using the same formula mentioned above

Average = Sum of the observations / Number of observations

How to Calculate Average of a List?

The formula which we use to calculate the Average of a list of numbers or values is very easy to use. Students can follow the below-mentioned steps to successfully calculate the average of a given list of numbers of values.

Step 1: At first, add all the numbers given in the list.

Step 2: Divide the calculated sum by the number of terms given in the list.

Step 3: Calculate and conclude the result by using the Average formula. The average of number can be expressed as:

Average = Sum of the observations / Number of observations

Difference Between Mean and Average

The main difference between mean and average are given below:

Types of Questions based on Averages 

Let us see different types of questions that may come in the Average section one by one:

1. Mathematical Based on Averages: Questions of this type are mathematically based, which may or may not be true in the real world.

2. Real-Life Based on Averages: Questions of this type are real-life based, which is always based on real-world situations.

Tips and Tricks to Solve Questions Based on Averages

Students can find different tips and tricks for solving the questions related to Averages from below:

Tip 1: Average = Sum of the observations / Number of observations

Tip 2: The average of any consecutive series is the middle term (median).

⇒ For example 8, 10, 12 in this series middle term is 10 which is also the average of the series.

Required percentage = [(12 – 8)/8] × 100 = 50%

Averages Formula

Students can find the important average formulas to solve questions based on Averages:

• • Average of first n natural numbers = (n+1) / 2
  • Average of squares of first n natural numbers = (n+1) (2n+1) / 6
  • Average of cubes of first n natural numbers = n (n+1)2 / 4
  • Average of first n even numbers = n + 1
  • Average of squares of first n even numbers = 2 (n+1) (2n+1) / 3
  • Average of cube of first n even numbers = 2n (n+1)2
  • Average of first n odd numbers = n
  • Average of squares of first n odd numbers = (2n+1) (2n-1) / 3
  • Average of cube of first n odd numbers = n(2n2 – 1)

Solved Examples of Averages

Example 1: The average age of a cricket team of eleven players is 27 years. If two more players are included in the team the average becomes 26 years, then the

average age (in years) of the two included players is∶

Solution: Average age of eleven players is = 27 years

⇒ Sum of age of 11 players = 27 × 11 = 297 years

⇒ If two players are included in the team, then average age of 13 players = 26 years

⇒ Sum of age of 13 players = 13 × 26 = 338 years

⇒ Sum of age of two players (included) = 338 – 297 = 41 years

Hence, Average age of two players (included) = 41/2 = 20.5 years

Example 2: The average age of a cricket team of eleven players is 27 years. If two more players are included in the team the average becomes 26 years, then the average age (in years) of the two included players is∶

Solution: Let the average expenditure of all be Rs. x According to the problem,

ÞAverage expenditure= Total expenditure

Number of people

⇒ 25x = 24 × 30 + x + 48

⇒ x = 32

Hence, Total money spent by all of them = 25 × 32 = Rs 800

Example 3: The average of 3 consecutive even numbers is 10, then the third number is by what percent more than the first number?

Solution: Let the first number be x.

So, the second and the third number will be (x + 2) and (x + 4) respectively. According to the question

⇒ x + x + 2 + x + 4 = 10 × 3

⇒ x = 8

The first number is 8.

⇒ The second number be = 8 + 2 = 10

⇒ The third number be = 8 + 4 = 12

Hence, Required percentage = [(12 – 8)/8] × 100 = 50%

Example 4: The averages of the first 15 odd numbers are by what percent less/more than the average of the first 15 even numbers?

Solution: As we know,

⇒ Average of the first n odd numbers = n

⇒ Average of the first n even numbers = n + 1

⇒ Average of the first 15 odd numbers = 15

⇒ Average of the first 15 even numbers = 15 + 1 = 16

Hence, Required percentage = [(16 – 15)/16] × 100 = 6.25%

Example 5: Find the averages of cubes of the first 10 odd numbers.

Solution: The average of cubes of first n odd numbers = n (2n2 – 1)

Hence, The average of cubes of first 10 odd numbers = 10(2 × (10)2 – 1) = 10 × 199 = 1990

Example 6: A cricketer had a certain average of runs for his 43 innings. In his 44th innings, he is bowled out for no score on his part. This brings down his average by three runs. Find his new average of runs.

Solution: Concept:

Total runs = Number of wickets × Average

Averages of bowler = Total runs / Number of wickets

Let the average of runs be x

Total runs = Number of innings × Average

⇒ 43 × x = 44 × (x – 3)

⇒ x = 132

New average = x – 3 = 132 – 3 = 129

Example 7: The average age of a family of 6 members is 25 years. If the age of the youngest member of the family is 8 years, then find the average age of the members of the family just before the birth of the youngest member.

Solution: Total age of a family of 6 members = 6 × 25 = 150 years

⇒ Total age of a family of 6 members before 8 years = 150 – (8 × 6) = 102 years

Hence, The average age of the family just before the birth of the youngest = 102/(6 – 1) = 20.4 years

Example 8: In a hostel, 52 students are living. If 18 students are joined in this hostel then average expenditure will be 3 rupees less whenever total expenditure 510 rupees will be increased. Find the total expenditure initially.

Solution: Let the averages expenditure of initially students be x,

⇒ 52x + 510 = (52 + 18) × (x – 3)

⇒ 18x = 720

⇒ x = 40

Hence, Total expenditure of 52 students = 52 × 40 = 2080

Learn about Algebraic Equations

We hope you found this article regarding average was informative and helpful, and please do not hesitate to contact us for any doubts or queries regarding the same. You can also download the Testbook App, which is absolutely free and start preparing for any government competitive examination by taking the mock tests before the examination to boost your preparation.

FAQs on Averages

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