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,... in that order several times. The scooter travels at a constant speed of 25 kMPH and the bicycle at 15 kMPH. After some time, they meet at a vertex of the square PQRS for the first time. This vertex is

The perimeter of the square is 4 times 45 km, which is 180 km, so the scooter takes 180 divided by 25, or 7.2 hours, to complete one full lap; the bicycle takes 180 divided by 15, or 12 hours. They will meet at a vertex at a time that is a common multiple of their individual vertex-to-vertex times: 1.8 hours for the scooter and 3 hours for the bicycle. The least common multiple of 18 and 30 is 90, making the least common multiple of 1.8 and 3 equal to 9 hours. In 9 hours, the scooter covers 225 km, which is exactly five sides of the square (225 divided by 45 equals 5), bringing it to vertex R; the bicycle covers 135 km, which is exactly three sides, also placing it at vertex R.

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