**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