☛ 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)
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?
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.