Calculating Average Speed for a Round Trip

The problem involves calculating the average speed when a journey is made over the same distance at two different speeds.

Understanding Average Speed

Average speed is defined as the total distance covered divided by the total time taken. It is not simply the average of the two speeds, especially when the time spent at each speed is different.

Formula for Average Speed

Let the distance covered in one direction be '$d$'. The speed during the outward journey is $v_1 = 15$ km/hr, and the speed during the return journey is $v_2 = 10$ km/hr.

• Time taken for the outward journey ($t_1$): $t_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{d}{15}$ hours.
• Time taken for the return journey ($t_2$): $t_2 = \frac{\text{Distance}}{\text{Speed}} = \frac{d}{10}$ hours.
• Total distance covered = $d + d = 2d$.
• Total time taken = $t_1 + t_2 = \frac{d}{15} + \frac{d}{10}$.

Average Speed Calculation Steps

1. Calculate the total time: $ \text{Total Time} = \frac{d}{15} + \frac{d}{10} $ Find a common denominator (30): $ \text{Total Time} = \frac{2d}{30} + \frac{3d}{30} = \frac{5d}{30} = \frac{d}{6} \text{ hours} $
2. Calculate the average speed: $ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} $ $ \text{Average Speed} = \frac{2d}{\frac{d}{6}} $ $ \text{Average Speed} = 2d \times \frac{6}{d} $ $ \text{Average Speed} = 12 \text{ km/hr} $

Alternative Method: Harmonic Mean

For a journey covering the same distance at two different speeds ($v_1$ and $v_2$), the average speed is the harmonic mean:

Substitute the given speeds:

Both methods yield the same result.