## Average Notes

**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