Are your pupils creative enough to solve this one? How fast are the trains?
I’ve yet to find a pupil that can solve the problem below but what is lovely about this problem is that most pupils can access it and have some ideas about how to solve it. It hooks them but provides a real challenge to solve it. There are many methods you can use to solve the problem and it is lovely to see some pupils using algebra, some using speed = distance / time, some using trial and error, some using speed-time graphs, some using inequalities etc.
The solution is that one train is twice as fast as the other. Here’s the algebraic proof that I came up with:
Two trains A and B.
Speed of train A = A
Speed of train B = B
Speed = distance/ time therefore distance = speed X time
T = time from start of train journey until the passing point.
Distance from London to passing point
We don’t know the exact distance from London to the passing point but we do know that this is the same for both trains. Using distance = speed X time for each train and the fact that the distances are the same we can say:
1 X A = B X T (i.e. train A is travelling at speed A and has 1 hour to go, train B is travelling at speed B and has been travelling for T hours)
A = BT
Distance from Liverpool to passing point
Using the same approach that the distances are the same for both trains (so speed X time for each train is equal):
A X T = 4 X B (i.e. train A is travelling at speed A for T hours and train B is travelling at speed B for 4 hours)
AT = 4B
A = BT
AT = 4B
Rearrange the first formula in terms of T (T=A/B) and substitute into the second formula to give:
A^2 / B = 4B
A^2 / B^2 = 4
A/B = 2
Train A is travelling at twice the speed of B