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

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

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***

**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

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