# Average: Introduction

Average is the mean value of N numbers. We can understand this as, Let we have N numbers like $$K_1, K_2, K_3,.........K_n$$, then

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

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

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

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

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

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

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

### Average Speed:

Case (1):

If any movable thing covers a certain distance $$d_1$$ with the speed of $$x_1 \ m/sec$$ and the same movable thing covers equal distance with a speed $$x_2 \ m/sec$$, then

$$Average \ Speed = \frac{2 x_1 x_2}{(x_1 + x_2)}$$

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, $$x_1 = 5 \ m/minute$$ and $$x_2 = 10 \ m/minute$$ by using average speed formula from case (1),$$Average \ Speed = \frac{2 x_1 x_2}{(x_1 + x_2)}$$ $$Average \ Speed = \frac{2 \times 5 \times 10}{(5 + 10)} \\ = \frac{100}{15} = 6.667 \ m/minute$$

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, $$x_1 = 100 \ m\sec$$ and $$x_2 = 150 \ m\sec$$ by using average speed formula from case (1),$$Average \ Speed = \frac{2 x_1 x_2}{(x_1 + x_2)}$$ $$Average \ Speed = \frac{2 \times 100 \times 150}{(100 + 150)} \\ = \frac{30000}{250} = 120 \ m/sec$$

Case (2):

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

Example (1): 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),$$Average \ Speed = \frac{Total \ distance \ (D)}{Total \ time \ (T)}$$ $$Average \ Speed = \frac{(10 + 20 + 20) \ km}{(20 + 30 + 20) \ minutes}$$ $$= \frac{50 \ km}{70 \ minutes} = 0.714 \ km/minute$$

Example (2): 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),$$Average \ Speed = \frac{Total \ distance \ (D)}{Total \ time \ (T)}$$ $$Average \ Speed = \frac{(20 + 25 + 25) \ km}{(50 + 60 + 70) \ minutes}$$ $$= \frac{70 \ km}{180 \ minutes} = 0.389 \ km/minute$$