## Averages Part 3

Problem 1: The average of 5 numbers is 52. If one number is excluded, the average becomes 50. Find the excluded number?

Solution:

Sum = 52*5 = 260

Sum = 50*4 = 200

260-200 = 60

Shortcut

52+4*2=60

Problem 2: The average age of 30 students is 12 years. When the teacher’s age is included in it, the average increased by 1. What is the teacher’s age?

Solution:

12+31*1=43

Problem 3: Average marks 0f 22 students is 75. The average is reduced by 2 if we exclude the highest and lowest marks. Find the sum of the highest and lowest marks?

Solution:

75*2+2*20 = 150+40 =190

## Averages Part 2

Problem 1: Find the average of 1,2,3,….91?

Solution:

$\frac{n+1}{2}=\frac{91+1}{2}=46$

Problem 2: Find the average of first 39 even numbers?

Solution:

2,4,6,8,…….

n+1 = 39+1 =40

Problem 3: Find the average 4+16+36+……..+400?

Solution:

4+16+36+…….+400

$2^{2}+4^{2}+6^{2}+.......+20^{2}$

$2^{2}(1+2^{2}+3^{2}+......+10^{2})$

$\frac{(n+1)(2n+1)}{6}$

$\frac{4\times&space;11\times&space;21}{6}=154$

Problem 4: The average of 6 consecutive even number is 63. Find the largest number?

Solution:

63=a+6-1

a = 63-5 = 58

## Averages Part 1

Important Formulae:

• Average = $\frac{Sum&space;Of&space;Observations&space;}{No.of&space;Observations}$
• The average of first ‘n’ natural numbers = $\frac{n+1}{2}$
• The average of first ‘n’ even numbers = n+1
• The average of first ‘n’ odd numbers = n
• The average of square of first ‘n’ natural numbers = $\frac{(n+1)(2n+1)}{6}$
• The average of consecutive
• a , a+1 , a+2 , a+3 ,………….,a+(n-1) = $a+\frac{(n-1)}{2}$
• a is smallest number   a+(n-1) is largest number
• Consecutive even or odd numbers
• a,a+2,a+4,a+6,………….a+(n-1) = a+2(n-1)