Maclaurin 2019 Q1

Suppose that the first train covers a distance \(u\) in one minute and the second train a distance \(v\) in one minute, and that the trains pass \(T\) minutes after 09:30. Call the crossing point X.

The first train takes \(T\) minutes to go from K to X, and the second train takes 100 minutes to travel from X to K. Hence, equating distances, \(Tu = 100v\).

The first train takes 40 minutes to travel from X to L, and the second train takes \(T - 30\) minutes to travel from L to X, since it started 30 minutes later than the first train. Hence, again equating distances, \(40u = (T - 30)v\).

Each of these two equations allows us to find \(\dfrac{u}{v}\). Equating the resulting expressions, we obtain a quadratic equation for \(T\).

We have \(\dfrac{u}{v} = \dfrac{T - 30}{40} = \dfrac{100}{T}\).

Rearranging, we get \(T^2 - 30T - 4000 = 0\), and factorising, we obtain \((T - 80)(T + 50) = 0\) so that \(T = 80\) (ignoring the negative answer, which is not practicable here).

Hence the trains pass at 10:50.