A man started from home at 14:30 hours and drove to village, arriving there when the village clock indicated 15:15 hours. After staying for 25 minutes, he drove back by a different route of length 1.25 times the first route at a rate twice as fast reaching home at 16:00 hours. As compared to the clock at home, the village clock is

The correct option is 5 minutes fast.

Let us break down the journey step-by-step and analyze the times recorded by the home clock and the village clock to find the discrepancy.

Let:
- $d_1$ be the distance of the first route (from home to the village).
- $v_1$ be the speed during the onward journey.
- $t_1$ be the actual time taken to travel from home to the village.

The actual time taken for the onward journey is:
$t_1=\frac{d_1}{v_1}$

For the return journey, we are given that:
- The distance of the return route $d_2=1.25\times d_1$
- The return speed $v_2=2\times v_1$

Thus, the actual time taken for the return journey, $t_2$, is:
$t_2=\frac{d_2}{v_2}=\frac{1.25\times d_1}{2\times v_1}=0.625\times t_1$

Now, let us look at the total elapsed time according to the home clock.
The man started from home at 14:30 and returned at 16:00.
Total time elapsed at home = 16:00 - 14:30 = 1 hour 30 minutes = 90 minutes.

During this trip, the man stayed in the village for 25 minutes.
Therefore, the total actual driving time (onward + return) is:
Total driving time = 90 minutes - 25 minutes = 65 minutes.

We can write this relationship as:
$t_1+t_2=65$

Substitute $t_2=0.625\times t_1$ into the equation:
$t_1+0.625\times t_1=65$
$1.625\times t_1=65$
$t_1=\frac{65}{1.625}=40\text{minutes}$

Since the onward journey took exactly 40 minutes, and the man started from home at 14:30 (home clock time), he must have arrived at the village when the home clock was at:
14:30 + 40 minutes = 15:10.

However, when he arrived at the village, the village clock indicated 15:15.
Comparing the village clock reading (15:15) to the home clock reading (15:10), we find that:
Village clock - Home clock = 15:15 - 15:10 = 5 minutes.

Therefore, as compared to the clock at home, the village clock is 5 minutes fast.