relative speeds and finding the final position of the moving bodies

Question

A scooter and a bicycle travel along the perimeter of a square $PQRS$ of side length $45$ km. They both start at the vertex $P$ and go through $Q,R,S,P,Q,R,S,\dots$ in that order several times. The scooter travels at a constant speed of $25$ km/h and the bicycle at $15$ km/h. After some time, they meet at a vertex of the square $PQRS$ for the first time. This vertex is?

Answer 1

If the number of sides covered by the scooter is $n$, then the number of sides covered by the bicycle must be $n-4$

The ratio of these distances is the ratio of their respective speeds.

i.e.

$ \dfrac{n}{n-4} = \dfrac{25}{15} $

From which, $15 n = 25 n - 100$, so $n = 10$

That is, the scooter has travelled $10$ sides, and that is $2$ complete cycles plus $2$ sides. The ordered labelling is $PQRS$, so they meet at $R$.