Two trains start at the same time, one from London to Liverpool, the other from Liverpool to London. If they arrive at their destinations one hour and four hours respectively after passing one another, how much faster is one train running than the other?

One train was running twice as fast as the other.

Let:
speed of the fast train = F
speed of the slow train = S
time it takes for the trains to meet (pass each other) = T

Since both trains travel the same total distance and distance = time x speed:
F(T+1) = S(T+4)

We’re trying to figure out F/S which is equal to (T+4) / (T+1) from the equation above. So we need to figure out the value of T.

After they meet, the fast train travels one more hour at speed F and covers the same distance the slow train covered in T hours:
F1 = ST or F = ST

After they meet, the slow train travels for 4 more hours and covers the same distance the fast train covered in T hours:
S4 = FT

Substituting ST from the first equation in for F in the 2nd equation:
4S = STT
4 = TT
2 = T

Substitute 2 in for T in the (T+4) / (T+1) equation to get 6/3 or 2. The fast train is going twice as fast as the slow train.

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