Henry Ernest Dudeney/Modern Puzzles/39 - The Two Trains/Solution

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Modern Puzzles by Henry Ernest Dudeney: $39$

The Two Trains
Two railway trains, one $400$ feet long and the other $200$ feet long, ran on parallel rails.
It was found that when they went in opposite directions they passed each other in $5$ seconds,
but when they ran in the same direction the faster train would pass the other in $15$ seconds.
Now, a curious passenger worked out from these facts the rate per hour at which each train ran.
Can the reader discover the correct answer?


Solution

The faster train went at $54 \tfrac 6 {11}$ miles per hour.

The slower train went at $27 \tfrac 3 {11}$ miles per hour.


Proof

Let $v_1$ feet per second denote the speed of the faster train.

Let $v_2$ feet per second denote the speed of the slower train.


We have:

\(\text {(1)}: \quad\) \(\ds 5 \paren {v_1 + v_2}\) \(=\) \(\ds 600\) when they went in opposite directions they passed each other in $5$ seconds
\(\text {(2)}: \quad\) \(\ds 15 \paren {v_1 - v_2}\) \(=\) \(\ds 600\) but when they ran in the same direction the faster train would pass the other in $15$ seconds
\(\text {(3)}: \quad\) \(\ds \leadsto \ \ \) \(\ds 15 \paren {v_1 + v_2}\) \(=\) \(\ds 1800\) $3 \times (1)$
\(\ds \leadsto \ \ \) \(\ds 30 v_1\) \(=\) \(\ds 2400\) $(3) + (2)$
\(\ds \leadsto \ \ \) \(\ds v_1\) \(=\) \(\ds 80\)
\(\ds \leadsto \ \ \) \(\ds v_2\) \(=\) \(\ds 40\)

The result follows on converting $80$ feet per second and $40$ feet per second into miles per hour.

$\blacksquare$


Sources

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