Well have you tried to evaluate the amount of time it takes for the two trains to cross?
Two trains 75 km apartapproach each other on parallel tracks at a rate of 15km/h. A bird flies back and forth between the tracks at 20 km/h until the trains pass each other. what is the total distance travelled by the bird?
Hint: the bird is in the air for the exact amount of time it takes the trains to pass each other.
and distance=rate*time.
Well the time the bird will travel is the same with the time the 2 trains will meet. That makes it easy to solve....
A little more interesting(to see how the student will manage with unit conversions and to see just another one thing) one, is a small variation of it:
2 trains 75 km apart, approach each other in the same track at a rate of 15 km/h. A fly of body length 1 cm, flies back and forth between the trains at 20 km/h in the following way: When the fly's face(it travels with its body length parallel to the ground/track/train direction and its face is in the leading part of its body) meets a train, it instantly changes direction(so in time=0 its "face" goes from the trains front position to the fly's tail) and travels towards the other train.
What is the total distance traveled by the fly(let's say by its middle point of body-or we just have to say that we don't care about its head movement when it instantly changes direction and we don't consider that as a movement fro the fly) just before it will get squashed(when the 2 trains have a distance of 1 cm)? The initial position of the fly's face is in the middle of the distance.
The two trains are 75 km apart and approaching each other at 15 km/hr. Surely you don't need someone to set up a formula for you to decide how long it will take the two trains to meet! if you were driving at 15 km/hr, how long would it take you to go 75 km?
I just have a comment to make about the presentation of this problem. If I were assigning this problem to my students, I certainly would not give this hint...having the student come to this realization on their own is much more instructive and rewarding in my opinion than the resulting mundane distance/rate/time problem given the hint.
I think the time of crossing is quite simple since the two trains are approaching each other implies that the distance is covered at the rate of 30 km per hour. Thus the time taken for the trains to start crossing would be 75/30 h. The complicated issue appears to be distance covered by the fly. Would think about it?
It's interesting the approaches that the posters have taken. I would just say that since the trains are travelling at the same speed, they will cover the same distance at any point in time, so they have to meet halfway...
There is an old story (apocryphal no doubt) that a college professor gave this problem to John Von Neumann (or put the name of your favorite mathematician here) who thought for a moment and then gave the correct answer. The professor chuckled and said "a lot of people try to do that problem as an infinite series. Von Neumann looked puzzled and said "But I summed it as an infinite series!"
That is not how I interpreted the problem. We were told that "Two trains 75 km apart approach each other on parallel tracks at a rate of 15 km/h" and I interpreted this as meaning that the two trains are closing upon each other at 15 km/h- that is their two speeds, which are not necessarily the same, sum to 15 km/h.
(That makes the "time of crossing" even simpler- no fractions!)