Two men travel between two cities by different routes, one route covering 240 miles and the other covering 255 miles. The man taking the shorter route travels 10 miles per hour faster than the other and completes the trip in $2 \frac{1}{2}$ hours less time than the other. At what rates did the men travel? (Answer: 30 miles per hour and 40 miles per hour)

Step 1 ：Let's denote the speed of the man taking the longer route as \(x\) (miles per hour) and the speed of the man taking the shorter route as \(x+10\) (miles per hour).

Step 2 ：The time it takes for the man taking the longer route to complete his trip is \(255/x\) hours and the time it takes for the man taking the shorter route to complete his trip is \(240/(x+10)\) hours.

Step 3 ：According to the problem, the man taking the longer route takes 2.5 hours more than the man taking the shorter route. So we have the equation \(255/x = 240/(x+10) + 2.5\).

Step 4 ：We can solve this equation to find the value of \(x\).

Step 5 ：The solution to the equation is \(x = -34\) and \(x = 30\). However, speed cannot be negative, so we discard the negative solution.

Step 6 ：Therefore, the speed of the man taking the longer route is 30 miles per hour. The speed of the man taking the shorter route is 10 miles per hour faster, which is 40 miles per hour.

Step 7 ：Final Answer: The speeds at which the men traveled are \(\boxed{30}\) miles per hour and \(\boxed{40}\) miles per hour.