A man is crossing a narrow bridge. When he is 3/8 of the way across the bridge, he hears a train traveling 60 miles per hour coming behind him. He figures that he has exactly enough time either to run back or to run forward and miss the train. How fast can the man run?
To find the speed at which the man can run, we need to analyze the situation.
Let's assume the length of the bridge is 'L' miles. The man is 3/8 of the way across the bridge, which means he has covered (3/8) * L miles.
Now, we need to determine the time it would take for the train to reach the bridge. The train is traveling at a speed of 60 miles per hour, so we can set up the equation:
Time = Distance / Speed
Since the man is at (3/8) * L miles away from the start of the bridge, the distance the train needs to cover is the remaining portion of the bridge, which is (5/8) * L miles. Plugging in the values:
Time = (5/8) * L miles / 60 miles per hour
Now, we know from the problem that the man has enough time either to run back or to run forward and miss the train. This means the train takes the same amount of time to reach the man, regardless of the direction in which he runs.
Therefore, we can set up another equation to represent the time taken by the man:
Time = Distance / Speed
If the man decides to run back, he needs to cover a distance of (3/8) * L miles. Let's say the man's running speed is 'R' miles per hour:
Time = (3/8) * L miles / R miles per hour
If the man decides to run forward, he needs to cover a distance of (5/8) * L miles:
Time = (5/8) * L miles / R miles per hour
Since both times are equal, we can equate them:
(3/8) * L miles / R miles per hour = (5/8) * L miles / 60 miles per hour
Now, we can solve this equation to find the speed at which the man can run. Multiplying both sides of the equation by R, we have:
(3/8) * L = (5/8) * L * (R / 60)
Simplifying further:
3 * R = 5 * 60
3R = 300
R = 100
Therefore, the man can run at a speed of 100 miles per hour.