To find the time when Tom and Ben meet each other, we need to determine how long it takes for them to cover the distance between their houses.

Let's assume that they meet after t hours.

In that time, Tom would have traveled a distance of 6t km, and Ben would have traveled a distance of 4t km.

Since they are cycling in opposite directions, the sum of the distances they travel should be equal to the total distance between their houses, which is 36 km.

So, we can write the equation: 6t + 4t = 36

Combining like terms, we get: 10t = 36

Dividing both sides by 10, we get: t = 3.6

Since t represents the time in hours, we can convert it to minutes by multiplying by 60: t = 3.6 * 60 = 216 minutes.

Therefore, Tom and Ben meet each other after 216 minutes, which is equivalent to 3 hours and 36 minutes.

Adding this time to the initial time of 1:00 PM, we get: 1:00 PM + 3 hours and 36 minutes = 4:36 PM.

So, Tom and Ben meet each other at 4:36 PM.

Therefore, the correct answer is B. 4 P.M.