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

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

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