Here's a solution I came up with, and as usual it is likely to be wrong. So, do let me know what is the right answer.
I think the problem as stated is highly confusing. It can be restated in a different way without taking away the essence. The problem can be restated as follows: Let's say Prashant has reached the summit after climbing for three hours, and starts the descent on the same day (he doesn't stay overnight). At the very same time when he starts his descent there is another person, Vikram, who starts climbing up. Let's say that the speeds of Prashant and Vikram are the same. Hence, Vikram will need three hours to reach the top and two hours to climb down. The path followed by both of them is the same.
So, here we have Vikram climbing up and Prashant descending. At some point of time they will surely meet. The spot where they meet is, then, the spot we are interested in. Of course, since we don't have a description of the spot we will try to find the time required to reach that particular spot.
Going back to the original statement, let's assume that Prashant starts his descent at 9am while Vikram starts the ascent at the same time, 9am. Let the distance to be travelled in each direction be "x". There is one more thing we need to assume; let's assume they meet after "t" hours after starting. This is the time that we wish to determine.
Vikram needs three hours to climb to the top, so in "t" hours he would have covered tx/3 of the distance. Prashant needs two hours to climb down, so he will have covered tx/2 of the distance in "t" hours.
But, the total distance is "x". Therefore, tx/3 and tx/2 will add up to "x" (for instance, if at the point of meeting, Prashant has covered 2 kilometres from the top and Vikram has covered 1 kilometre from the bottom, then the total distance is, of course, 1 + 2 = 3 kilometres).
Thus, we have tx/2 + tx/3 = x.
So, 5tx/6 = x.
Thus, 5tx = 6x, or t = 6/5. Thus, t = 1.2 hours, or one hour and twelve minutes. Thus, if both Prashant and Vikram start their descent and ascent respectively at 9am, they will meet at 10.12 am. The spot where they meet at 10.12 am is exactly the spot where Prashant would have arrived on both the days. Do you think I did it right?