Average

Average

The average of a number of items of the same type is their sum divided by the number of those items.

Properties of Average;
1. If you observe the result of average closely it should be less than the greatest observation and greater than the smallest observation.
2. Suppose if the given observations are equal then the average is also the same as the observation.
3. If the zero is one of the observations of given data we have to include that zero in the calculation.
4. If all the numbers get increased by x then their average should be increased by x. This point is also applicable for subtraction, multiplication and divide properties.

Basic Formulas
• Average = Sum of observation /Number of observation

Example sum
Ramya obtained 56, 65, 72, 86 and 92 marks ( out of 100 ) in English, maths, physics, chemistry, and biology. What are his average marks?
Average mark = Sum of mark / Number ofsubject

=(56+65+72+86+91)/5
=370/5
Average mark = 74

Basic Formulas
• Average of consecutive first n natural numbers = (n +1 ) / 2
• Average of first n natural even numbers= (n +1 )
• Average of natural even numbers up to n = (n/ 2) +1

·  Average of first n natural odd numbers = n
• Average of natural odd numbers upto n = (n +1
• Average of n multiples of p = (p (n +1))/2

• Average of consecutive numbers

 = (First number+ Last number) / 2
• Average of 1 to n even numbers 

= ( Last even number + 2 ) / 2
• Average of 1 to n odd numbers 

= ( Last odd number + 1 ) / 2

Example sum
Find the average of first 20 natural numbers?
Sum of first n natural number = (n * (n +1))/2
Sum of first 40 natural numbers = ( 20 * (20+1))/ 2
=420/2
= 210
Required average = 210/ 20
= 10.5

Basic Formulas
• If a person covered a certain distance at a speed of x kmph and y kmph respectively, then his average speed will be [ ( 2xy ) / (x + y)] kmph

Example sum
A motorist travels to a place 150 km away at an average speed of 50kmph and returns at 30 kmph. His average speed for the whole journey in kmph?
Average speed = [ 2xy / ( x + y)] kmph
=[(2*50*30)/(50+30)] kmph
= ( 3000 / 80 )kmph
= 37.5 kmph

Examples For Shortcut Method

1. The average age of 10 students is 16. When the teacher joins the class then the average increases by 1. Then what is the teacher's age?
Solution : 

[Number of students * Average Age] -[Number of Students (Including Teacher) *Average Age (Including Teacher)] 

=Answer

=10*16=160

=11*17=187

Teacher’s age=187-160 =27

Shortcut Method :

Number of Students + Average Age + 1 = Answer
10+16+1=27
Teacher's age = 27 years

2. The average of 20 students is 12 years if the teacher's age is included, average increases by one. The age of the teacher is:
Solution :
[Numberof students * Average Age]-[Number of Students (Including Teacher) *Average Age (Including Teacher)]

 = Answer.
20 * 12 = 240
21*13=273
Teacher's age = 273 - 240
= 33

Shortcut Method :
Number of Students + Average Age + 1 =Answer

20+12+1 = 33

Teacher's age = 33 years
Note: This Trick works only for the problems in the above format.