Table of Contents

## Formula of Average

Average! The standard we never want to fit into but to become excellent we have to study Average, as average is a favorite topic of the examiner and competitive exams are incomplete without questions asked on average , so here are all the things you need to know about Average, Students! focus on the topic and prepare it well for the Exams, it will immensely help you in scoring well.

## Average

Averages can be defined as the central value in a set of data. Average can be calculated simply by dividing the sum of all values in a set by the total number of values. In other words, an average value represents the middle value of a data set. The data set can be of anything like age, money, runs, etc.

**Average = [ Sum of Data (or observations) in a set / Number of data (or observations) in that set]**

## Weighted Average

When two groups of Parts or objects are combined together, then we can talk of the average of the entire group. However, if we know only the average of the two groups individually, we cannot find out the average of the combined group of objects.

## Formula of Average and Tricks

- Average = Sum of quantities/ Number of quantities
- Sum of quantities = Average * Number of quantities
- The average of first n natural numbers is (n +1) / 2
- The average of the squares of first n natural numbers is (n +1)(2n+1 ) / 6
- The average of cubes of first n natural numbers is n(n+1)
^{2}/ 4 - The average of first n odd numbers is given by (last odd number +1) / 2
- The average of first n even numbers is given by (last even number + 2) / 2
- The average of squares of first n consecutive even numbers is 2(n+1)(2n+1) / 3
- The average of squares of consecutive even numbers till n is (n+1)(n+2) / 3
- The average of squares of squares of consecutive odd numbers till n is n(n+2) / 3
- If the average of n consecutive numbers is m, then the difference between the smallest and the largest number is 2(m-1)
- If the number of quantities in two groups be n
_{1}and n_{2}and their average is x and y respectively, the combined average is (n_{1}x+n_{2}y) / (n_{1}+ n_{2}) - The average of n quantities is equal to x. When a quantity is removed, the average becomes y. The value of the removed quantity is n(x-y) + y
- The average of n quantities is equal to x. When a quantity is added, the average becomes y. The value of the new quantity is n(y-x) + y

## Average Formulas: Average Sample Questions

**Q1. The average weight of 24 students of section A of a class is 58 kg whereas the average weight of 26 students of section B of the same class is 60.5 kg. Find the average weight of all the 50 students of the class.**

**Sol.** Here n₁ = 24, n₂ = 26, x = 58 and y = 60.5.

∴ Average weight of all the 50 students

=(n₁ x+n₂ y)/(n₁+n₂ )

=(24×58+24×60.5)/(24+26)

=(1392+1573)/50=2965/50

= 59.3 kg

**Q2. The average weight of 25 persons is increased by 2 kg when one of them whose weight is 60 kg is replaced by a new person. What is the weight of the new person?**

**Sol.** The weight of the new person

= p + n(y – x)

= 60 + 25(2) = 110 kg

**Q3.**** What is the average of odd numbers from 1 to 40?**

**Sol.** The required average

=(last odd number+1)/2

=(39+1)/2

= 20

**Q4.**** Find the average of squares of first 19 consecutive even numbers.**

**Sol.** The required average

=(2 (n+1)(2n+1))/3=(2(19+1)(2×19+1))/3

=(2×20×39)/3=1560/3=520

**Q5. Find the average of squares of consecutive odd numbers from 1 to 31. **

**Sol.** The required average

=(n (n+2))/3=(31×(31+2))/3=(31×33)/3=341