Average: Basic Understanding and Properties Today I'm going to start a new topic of quantitative aptitude i.e. Average. It is a very simple topic and

The term "average" simply refers to the mean value of all provided observations, or, alternatively, the arithmetic mean of observations.

Average = (Sum of all observations)/ (Number of observations)


Example1: Calculate the average of the subsequent observations:

3, 4, 8, 12, 2, 5, 1

Solution: Average = (Sum of all observations)/ (Number of observations)

Average = (3+ 4+ 8+ 12 +2+ 5+ 1)/7 = 35/7 = 5

So, Average = 5

i) Between the maximum and least observation is the average.

ii) The average will also be multiplied by the same value if the value of each observation is multiplied by some value N. i.e.N.
 

For example: Assume the prior collection of findings. If all observations are multiplied by 2 then the new observations will be as follows.:
6, 8, 16, 24, 4, 10, 2
New Average $=(70)/7=10=2(5)=2\times$ Old Average
 

iii) The average will change by the same amount whether the value of each observation is increased or decreased by some amount.
 

For example: Continuing with the same example. If 2 is added to all observations, then new observations will be as follows:
5, 6, 10, 14, 4, 7, 3
New Average = (49)/7 = 7 = (5 + 2) = 2 + Old Average
 

iv) Similarly, if each observation is divided by some number, then average will also be divided by same number.
For example: If 2 is divided from all observations, then new observations will be as follows:
1.5, 2, 4, 6, 1, 2.5, 0.5
 

New Average = (17.5)/7 = 2.5 = 5/2 =  Old Average/ 2
I can therefore state that any general procedure done to observations 
 

Example2: Find an average of first 20 natural numbers.
 

Solution: Average =(Sum of first  20  natural numbers)/ (20)
Now, we know that Sum of first n natural numbers = ((n)(n+1))/2
Therefore, Sum of first 20 natural numbers $=(20\times 21)/2$
Average $=(20\times 21)(2\times 20)=10.5$
 

Example3: The second number in a set of three numbers is twice the first and three times the third. Find the greatest number if the average of these numbers is 44.
 

Solution: Let x be the third number
According to question, second number = 3x = 2(first number)
Therefore, first number = (3x)/2 second number = 3x and third number = x
Now, average = 44 = (x + 3x + (3x)/2)/3 
$\Rightarrow (11x)/2=44\times 3\Rightarrow x=24$
So, largest number i.e. (3x) = 72      
 

Example4: Average of four consecutive even numbers is 27. Find the numbers.
 

Solution: Let x, x+2, x+4 and x+6 be the four consecutive even numbers.
According to question, ((x) + (x+2) + (x+4) + (x+6))/4 = 27
(4x + 120)/4 = 27 x = 24 Therefore, numbers are 24, 26, 28, 30

Special Case

finding the average speed

Suppose a man covers a certain distance at x km/hr and covers an equal distance at y km/hr. The average speed during the whole distance covered will be (2xy)/ (x+y)
 

Example5: A bike covers certain distance from A to B at 50 km/hr speed and returns back to A at 56 km/hr. Find the average speed of the bike during the whole journey.

Solution: Average speed $=((2xy)(x+y))=(2\times (50)\times (56))/(50+56)\Rightarrow 52.83$ km/hr