Average: Notes, Concept & Shortcut (Quantitative Aptitude) Part 1

Average of the series in AP.
Average of AP or the consecutive number is (First term + Last term)/2

Example-101, 104, 107, 110, 113
Average is= 101+113=214/2=107

Now since the example has 5 terms (ODD terms) the average is the middle number

And here notice that if we take 104 and 110 then also average is 107
So, we can say an average of consecutive series having odd terms, the average is the middle term
And when the average of the number having even terms then the average is calculated by the help of an example

Example- 101, 104, 107, 110, 113, 116
Average is= 101+116= 108.5 which lies somewhere between the middle terms (107, 110)

Example based on this concept-

Example- The average of 5 consecutive integers starting with m is n. What is the average of 6 consecutive integers starting with (m+2).

Solution- The numbers are m, m+1, m+2, m+3, m+4
Average is given=n
We know the average of a consecutive odd number is middle term= m+2
So m+2=n
Now the average of 6 consecutive numbers stating with m+2
m+2, m+3, m+4, m+5, m+6, m+7
Here the number of terms is even so the average lies between the middle term or average is (last term+ first term)/2.
(m+4+m+5)/2= (2m+9)/2=m+4.5 or 2n+5/2 is answer

Example2- Find the average of 5 consecutive even numbers.
Statement1: The difference of fifth number and the first number is 8.
Statement2: The sum of first two numbers is 6 more than the fifth number.

Solution- let the even number is x, x+2, x+4, x+6, x+8
Stmt1- (x+8)-x=8
we can't find solution by stmt 1
Stmt2- (x+x+2)=6+(x+8)
2x+2=6+x+8
x=12
average of 5 number is middle number i.e.= 16
so only 2 is required

Example3- Smallest of a five-digit consecutive odd number is 3 more than second largest number of five consecutive even number. Find the average of consecutive even number is how much less than the average of an odd number.

Solution-let the odd number is- x, x+2, x+4, x+6, x+8
Average of odd number= x+4
ATQ
smallest of five-digit consecutive odd number is 3 more than the second-largest even number
so the even number is
x-9, x-7, x-5, x-3, x-1
Now the average is
Average of even number= x-5

Average of an odd number is 9 more than the average of even number

Example4- The sum of five consecutive even number of Set A is 220.
What is the different Set B of five consecutive positive integers whose 2nd lowest number is 37 less than double of the lowest of Set A.

Solution-First find the average= 220/5=44
we know the average of 5 consecutive even number is the middle term
middle term= 44
so the five numbers of SetA are 40, 42, 44, 46, 48
for set B
given that 2nd lowest number of SetB= 2*40-37=43
i.e Set B is 42, 43, 44, 45, 46

Example5- The average of the group of 8 friends is 52 kg. What will be the average weight when a person of weight 76 kg joins the group?

Solution- If you know the concept of deviance you can solve it in seconds
here average is 52 kg means all 8 person is 52 kg
when a new man joins its weight is (76-52) kg more than the group=24 kg more
this extra 24 kg is divided into 9 persons now
Average = 52+ 24/9

Do it for yourself-

Example1- Shubham was conducting an experiment in which the average of 11 observations came to 90, while the average of first observation was 87, and that of last five was 84 What was the 6th observation?

Example2- The average of marks obtained by 120 candidates was 35. If the average of the passed candidates was 39 and that the failed candidates were 15, then the number of those candidates who have passed the exam?