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Study Notes On Average With Examples

Average Notes

Average can be defined as the mean of a certain set of numbers. Questions based on average are seen in various competitive exams especially government job examinations. Candidates must be aware of the method required to solve the questions on Average. We are providing you with the study notes on Average along with examples.

Average Formula & Tricks

Average= (Sum of quantities)/(Number of quantities) / 
  • Average of two or more groups taken together 

(a) If the number of quantities in two groups be n₁ and n₂ and their average is x and y, respectively, the combined average (average of all of them put together) is /

(n₁ x+n₂ y)/(n₁+n₂ )


(b) If the average of n₁ quantities is x and the average of n₂ quantities out of them is y, the average of remaining group (rest of the quantities) is –

(n₁ x-n₂ y)/(n₁ – n₂ )


Q. 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

  • The average of n quantities is equal to x. If one of the given quantities whose value is p, is replaced by a new quantity having value q, the average becomes y,  then
q = p + n(y – x)


Q. 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

  • 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 y. When a quantity is added, the average will become y. The value of the new quantity is
n(y – x) + y.


Q. The average age of 24 students and the class teacher is 16 years. If the class teacher’s age is excluded, the average age reduces by 1 year. What is the age of the class teacher?

Sol. The age of class teacher
= n (x – y) + y
= 25 (16 – 15) + 15
= 40 years

  • The average of first n natural numbers is
(n + 1)/2.
  • The average of square of natural numbers till n is
((n + 1)  (2n + 1))/6
  • The average of cubes of natural numbers till n is
(n (n + 1)²)/4.
  • The average of odd numbers from 1 to n is
(last odd number+1)/2.
  • The average of even numbers from 1 to n is
(last even number + 2)/2.

 

Click Here to Attempt More Questions From Quantitative Aptitude For SSC CGL 


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

Sol. The required average
=(last odd number+1)/2
=(39+1)/2
= 20

Q. What is the average of even numbers from 1 to 81?

Sol. The required average
=(last even number+2)/2
=(80+2)/2
= 41

  • If n is odd: The average of n consecutive numbers, consecutive even numbers or consecutive odd numbers is always the middle number.
  • If n is even: The average of n consecutive numbers, consecutive even numbers or consecutive odd numbers is always the average of the middle two numbers.
  • The average of first n consecutive even numbers is   (n + 1)
  • The average of first n consecutive odd numbers is n.
  •  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 consecutive odd numbers till n is
(n (n + 2))/3.
  • If the average of n consecutive number is m, then the difference between the smallest and the largest number if
2 (n – 1).


Q. 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

Q. 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

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