# Average Important Formulas, Definitions, & Examples:

#### Overview:

 Topic Included: Formulas, Definitions & Exmaples. Main Topic: Quantitative Aptitude. Quantitative Aptitude Sub-topic: Average Aptitude Notes & Questions. Questions for practice: 10 Questions & Answers with Solutions.

#### What is Average:

In mathematics the average is the arithmetic mean value of N numbers. We can understand this as, suppose we have N numbers $$K_1, K_2, K_3,.........K_n$$, then the average

$$= \frac{K_1 + K_2 + K_3 +.........K_n}{N}$$

#### Formula of Average:

$$Average = \frac{Sum \ of \ observations}{Number \ of \ observations}$$ $$= \frac{K_1 + K_2 + K_3 +.........K_n}{N}$$

#### Average Examples:

Example (1): Find the average value of the first five natural numbers?

Solution: The first five natural numbers are 1, 2, 3, 4, 5, then

$$Average = \frac{1 + 2 + 3 + 4 + 5}{5} \\ = \frac{15}{5} = 3$$

Example (2): Find the average value of the first four even numbers?

Solution: The first four even numbers are 2, 4, 6, 8, then

$$Average = \frac{2 + 4 + 6 + 8}{4} \\ = \frac{20}{4} = 5$$

#### Average Speed:

Case (1):

If a man covers a certain distance $$d_1$$ with the speed of $$x \ m/sec$$ and the same man also covers equal distance with a speed $$y \ m/sec$$, then

$$Average \ Speed = \frac{2 \ x \ y}{(x + y)} \ m/sec$$

Case (2):

If we have, total distance travelled by a man $$D = d_1 + d_2 + d_3 +......d_n$$ and total time taken by the man by travelling the total distance $$T = t_1 + t_2 + t_3 +......t_n$$, then $$Average \ Speed = \frac{Total \ distance \ (D)}{Total \ time \ (T)}$$

#### Average Speed Examples:

Example (1): A person named Ram, covers $$500$$ $$m$$ distance at the speed of $$5$$ $$m/minute$$ and covers another equal distance at the speed of $$10$$ $$m/minute$$, find the average speed of Ram?

Solution: Given values, $$x = 5 \ m/minute$$ and $$y = 10 \ m/minute$$ by using average speed formula from case (1),$$Average \ Speed = \frac{2 x y}{(x + y)}$$ $$Average \ Speed = \frac{2 \times 5 \times 10}{(5 + 10)} \\ = \frac{100}{15} = 6.667 \ m/minute$$

Example (2): A man travelles $$50 \ km$$ distance, one half distance travelles at the speed of $$100 \ m/sec$$ and another half distance at the speed of $$150 \ m/sec$$ find the average speed of the man?

Solution: Given values, $$x = 100 \ m/sec$$ and $$y = 150 \ m/sec$$ by using average speed formula from case (1),$$Average \ Speed = \frac{2 x y}{(x + y)}$$ $$Average \ Speed = \frac{2 \times 100 \times 150}{(100 + 150)} \\ = \frac{30000}{250} = 120 \ m/sec$$

Example (3): Mr. George travelled $$10 \ km$$ distance in $$20 \ minutes$$, another $$20 \ km$$ distance in $$30 \ minutes$$ and another $$20 \ km$$ distance in $$20 \ minutes$$, then find the average speed of Mr. George?

Solution: Given values, distances travelled by Mr. George, $$d_1 = 10 \ km$$, $$d_2 = 20 \ km$$, and $$d_3 = 20 \ km$$, time taken by Mr. George to cover the distance, $$t_1 = 20 \ minutes$$, $$t_2 = 30 \ minutes$$, and $$t_3 = 20 \ minutes$$, by using average speed formula from case (2),$$Average \ Speed = \frac{Total \ distance \ (D)}{Total \ time \ (T)}$$ $$= \frac{(10 + 20 + 20) \ km}{(20 + 30 + 20) \ minutes}$$ $$= \frac{50 \ km}{70 \ minutes}$$ $$= 0.714 \ km/minute$$

Example (4): A woman travelled $$20 \ km$$, $$25 \ km$$, and another $$25 \ km$$ and the time taken by the woman $$50 \ minutes$$, $$60 \ minutes$$, and $$70 \ minutes$$, successively, find the average speed of the woman?

Solution: Given values, distances travelled by woman, $$d_1 = 20 \ km$$, $$d_2 = 25 \ km$$, and $$d_3 = 25 \ km$$, time taken by the woman to cover the distance, $$t_1 = 50 \ minutes$$, $$t_2 = 60 \ minutes$$, and $$t_3 = 70 \ minutes$$, by using average speed formula from case (2),$$Average \ Speed = \frac{Total \ distance \ (D)}{Total \ time \ (T)}$$ $$= \frac{(20 + 25 + 25) \ km}{(50 + 60 + 70) \ minutes}$$ $$= \frac{70 \ km}{180 \ minutes}$$ $$= 0.389 \ km/minute$$

#### Some Important results of Average:

The average of the first ten natural numbers is $$= \frac{1 + 2 + 3 + 4 +....+ 10}{10}$$ $$= \frac{55}{10} = 5.5$$

The average of the first ten whole numbers is $$= \frac{0 + 1 + 2 + 3 +....+ 9}{10}$$ $$= \frac{45}{10} = 4.5$$

The average of the first ten even numbers is $$= \frac{2 + 4 + 6 + 8 +....+ 20}{10}$$ $$= \frac{110}{10} = 11$$

The average of the first ten odd numbers is $$= \frac{1 + 3 + 5 + 7 +....+ 19}{10}$$ $$= \frac{100}{10} = 10$$

The average of the first ten prime numbers is $$= \frac{2 + 3 + 5 + 7 +....+ 29}{10}$$ $$= \frac{129}{10} = 12.9$$

The average of the first n natural numbers is $$= \frac{n + 1}{2}$$

The average of the squares of the first n natural numbers is $$= \frac{(n + 1) (2n + 1)}{6}$$

The average of the cubes of the first n natural numbers is $$= \frac{n \ (n + 1)^2}{4}$$

The average of first n odd natural numbers is $$= n$$

The average of first n even natural numbers is $$= n + 1$$