Two towns P and Q are 275 km apart. A motorcycle rider starts from P towards Q at 10:00 a.m. at the speed of 25 km/h. Another rider starts from Q towards P at 12 noon on the same day at the speed of 20 km/h. At what time will they cross each other?

To find the time at which the two riders cross each other, we can break down their journeys step-by-step.

Step 1: Calculate the distance traveled by the first rider before the second rider starts.
The first rider starts from town P towards Q at 10:00 a.m. at a speed of 25 km/h.
The second rider starts from town Q towards P at 12 noon on the same day.
The time elapsed between 10:00 a.m. and 12 noon is:
$12-10=2\text{hours}$

During these 2 hours, the distance traveled by the first rider is:
$\text{Distance}=\text{Speed}\times \text{Time}$
$\text{Distance}=25\text{km/h}\times 2\text{hours}=50\text{km}$

Step 2: Determine the remaining distance between the two riders at 12 noon.
The total distance between towns P and Q is 275 km.
At 12 noon, the remaining distance between them is:
$\text{Remaining Distance}=275\text{km}-50\text{km}=225\text{km}$

Step 3: Calculate the relative speed of the two riders.
Since the two riders are traveling towards each other, their relative speed is the sum of their individual speeds:
$\text{Relative Speed}=25\text{km/h}+20\text{km/h}=45\text{km/h}$

Step 4: Find the time taken to cover the remaining distance.
The time taken for them to meet after 12 noon is:
$\text{Time}=\frac{\text{Remaining Distance}}{\text{Relative Speed}}$
$\text{Time}=\frac{225\text{km}}{45\text{km/h}}=5\text{hours}$

Step 5: Determine the crossing time.
Adding 5 hours to the starting time of 12 noon:
$12\text{noon}+5\text{hours}=5:00\text{p.m.}$

Therefore, the riders will cross each other at 5:00 p.m.