3550.51 – A Wheel Within A Wheel

A wheel of radius 10" rolls inside a wheel of radius 54". Point $P'$ on the little wheel starts at point $P$ on the big wheel. After how many revolutions of the little wheel do the points $P'$ and $P$ coincide again?

Solution

The problem may be re-conceived as follows. Unroll each wheel into a straight line segment. Then the problem asks how many copies of the little line segment are needed to exactly measure a whole number of copies of the big line segment. The ratio of the radii (in lowest terms) is $5/27$ so that $27$ copies of the little segment will exactly measure $5$ copies of the big line segment. The answer is $27$ .