3550.51 – A Wheel Within A Wheel

Tags:Problem Set 13Algebra/TrigonometryGeometry-Euclidean

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


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 5/275/27 so that 2727 copies of the little segment will exactly measure 55 copies of the big line segment. The answer is 2727.