**GETTING STARTED**

☛ An average is a central value for a set of numbers.

☛ It is calculated by adding all the numbers and dividing it by the total count of numbers.

**Eg. Calcualate average of 85, 95 and 129.**

(85+95+129)/3 = 309/3 = 103

**Important Properties of Averages**

☛ It can be observed that **average value always lies between the smallest and the largest numbers** of a given set.

In the above example, the average is 103 and it lies between 85 and 129.

☛ If we have 5 numbers a, b, c,d,e whose average is n, it is to be noted that** (n-a)+(n-b)+(n-c)+(n-d)+(n-e) = 0**

**Let us understand with the help of an example:**

Consider 5 nos. – 72, 98, 84,51,35

Average of 5 numbers = (72+98+84+51+35)/5 = 340/5 = 68

Now, ( 68-72)+ (68-98)+(68-84)+(68-51)+(68-35)

= (-4)+(-30)+(-16)+(17)+(33)

= 0

This property will help in solving many problems in short time.

☛ For n numbers in Arithmetic Progression with any difference,

**Average of n numbers is the middle term of the series.**

**Eg. Consider a series 3,6,9,12,15**.

Average of series = (3+6+9+12+15)/5 =9

It is same as the third term which also happens to be the middle term

For n numbers in Arithmetic Progression with any difference,

Average of n numbers =( [ First Term] + [Last Term])/2

**Eg. Consider an AP, 7,14,21,28,35.**

Common Difference = 7

Average of AP = Middle Term = 21

( First term + Last Term)/2 = (7+35)/2= 21

**One of the most frequently asked type of question is:**** **

**☛ Average of a group changing due to addition/ removal of a person.**

**Eg.** **Average of a group of 5 boys is 20 years. On adding one more boy in the group, average age increases by 1 year. What is the average age of new person?**

** **

**Conventional Method:**** **

Average age of 5 boys = 20 yrs

Total age of 5 boys = 20 x 5 = 100 yrs

New average age of 6 boys = 21 yrs

Total new age of 6 boys = 126 yrs

**Average of new boy = New Total age – Old Total Age = 126-100 = 26 years**

A simpler method to solve the above example can be :

The average age of the group increases by 1.

**So add increase of 1 year per person for 5 members to the new average ie. 21.**

We get **5+21 = 26 years** without much calculation and effort.

**☛ Age of new member = New Average + [Increase in Average x Initial no. Of members]**

Similarly, if average age of the group is decreased by 1 with 1 person adding to the group,

Again add new average to increase (-1 per person) for 5 persons.