A man covers half part of a certain distance at 45 km/hr. He covers 40 percentage of remaining distance at the speed of 8 km/hr and covers rest of the distance at the speed of 24 km/hr. What is the average speed for the entire journey?

a) 164/7 km/hr
b) 144/7 km/hr
c) 156/7 km/hr
d) 128/7 km/hr

Ans : b) 144/7 km/hr

Solution :

Let’s assume the total distance is ( 2D ).

Step 1: First part of the journey

The man covers half of the total distance, i.e., ( D ) at a speed of 45 km/h.

• Distance covered = ( D )
• Speed = 45 km/h
• Time taken for this part: $ t_1 = \frac{D}{45} $

Step 2: Second part of the journey

Now, 40% of the remaining distance is covered at 8 km/h. The remaining distance after the first part is ( D ), and 40% of this distance is ( 0.4D ).

• Distance covered = ( 0.4D )
• Speed = 8 km/h
• Time taken for this part: $ t_2 = \frac{0.4D}{8} = \frac{D}{20} $

Step 3: Third part of the journey

The rest of the remaining distance is covered at 24 km/h. The remaining distance after the second part is ( 0.6D ).

• Distance covered = ( 0.6D )
• Speed = 24 km/h
• Time taken for this part: $ t_3 = \frac{0.6D}{24} = \frac{D}{40} $

Step 4: Total time taken

The total time taken is the sum of the time taken for each part:
$ t_{\text{total}} = t_1 + t_2 + t_3 = \frac{D}{45} + \frac{D}{20} + \frac{D}{40} $
To simplify, take the LCM of 45, 20, and 40, which is 360:
$ t_{\text{total}} = \frac{8D}{360} + \frac{18D}{360} + \frac{9D}{360} = \frac{35D}{360} $
So, the total time taken is: $ t_{\text{total}} = \frac{35D}{360} = \frac{7D}{72} $

Step 5: Total distance covered

The total distance is ( 2D ).

Step 6: Calculate the average speed

The average speed is given by:
$ \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{2D}{\frac{7D}{72}} = \frac{2\times 72}{7} = \frac{144}{7} $

Thus, the average speed for the entire journey is 144/7 km/h.