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 just involves simple mathematical calculations. Average concept has various applications. I will discuss its applications in next session. Firstly I will try to make you understand the basics of this topic.

Average is just a mean value of all the given observation or i can say it is an arithmetic mean of observations.

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


Example1: Find an average of  following 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`

***But, remember that this formula does not directly applies on average speed. Discussed in special cases***

Properties of Average

i) Average lies between maximum and minimum observation.

ii) If value of each observation is multiplied by some value 'N', then average will also be multiplied by the same value i.e.N.

For example: Assume the previous set of observations. If `2` is multiplied with all observations, then 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) If value of each observation is increased or decreased by some number, then average will also be increased or decreased by the same number.

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`

Therefore, I can say any general operation applied on observations will have same effect on average.

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: Out of three numbers, second number is twice the first and is also thrice the third. If average of these numbers if `44`, then find the largest number.

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` 

⇒`(11x)/2 = 44 times 3`

⇒`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

To find 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)`

*I will soon update a video lesson of this concept that how this formula has been derived.*

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)` 

⇒ `52.83` km/hr