Understanding Average Speed Calculation

This problem asks us to find the average speed for a complete journey where the speeds for going and returning are different. The key is that the distance covered in both directions is the same.

Key Information and Formula

• Speed going (Bhopal to Indore): Let's call this $v_1$. $v_1 = 9$ km/hr.
• Speed returning (Indore to Bhopal): Let's call this $v_2$. $v_2 = 18$ km/hr.
• Route: The journey back follows the same route, so the distance is the same each way.
• Objective: Calculate the average speed for the entire journey.

The fundamental formula for average speed is:

$ \text{Average Speed} = \frac{\text{Total Distance Covered}}{\text{Total Time Taken}} $

It's important to remember that average speed is not simply the average of the two speeds ($ (9 + 18) / 2 $), because the time spent traveling at each speed is different.

Calculating Total Distance and Total Time

To solve this, let's represent the distance between Bhopal and Indore as '$d$' kilometers.

• Total Distance: The man travels from Bhopal to Indore (distance $d$) and then returns from Indore to Bhopal (distance $d$). So, the total distance for the entire journey is $d + d = 2d$ km.
• Time for the first leg (Going): We use the formula Time = Distance / Speed. The time taken to go from Bhopal to Indore ($t_1$) is: $ t_1 = \frac{d}{v_1} = \frac{d}{9} \text{ hours} $
• Time for the second leg (Returning): Similarly, the time taken to return from Indore to Bhopal ($t_2$) is: $ t_2 = \frac{d}{v_2} = \frac{d}{18} \text{ hours} $
• Total Time: The total time for the entire journey is the sum of the times for both legs: $ \text{Total Time} = t_1 + t_2 = \frac{d}{9} + \frac{d}{18} \text{ hours} $

Step-by-Step Calculation of Average Speed

1. Calculate Total Time: First, we need to add the fractions representing the time taken for each part of the journey. The least common denominator for 9 and 18 is 18. $ \text{Total Time} = \frac{2 \times d}{2 \times 9} + \frac{d}{18} = \frac{2d}{18} + \frac{d}{18} $ $ \text{Total Time} = \frac{2d + d}{18} = \frac{3d}{18} $ Simplifying the fraction gives: $ \text{Total Time} = \frac{d}{6} \text{ hours} $
2. Calculate Average Speed: Now we plug the Total Distance ($2d$) and Total Time ($\frac{d}{6}$) into the average speed formula: $ \text{Average Speed} = \frac{2d}{\frac{d}{6}} $ To divide by a fraction, we multiply by its inverse (reciprocal): $ \text{Average Speed} = 2d \times \frac{6}{d} $ Notice that the distance '$d$' cancels out in the equation: $ \text{Average Speed} = 2 \times 6 $ $ \text{Average Speed} = 12 $

Therefore, the average speed for the entire journey from Bhopal to Indore and back is 12 km/hr.