Wheel Rolling Inside Wheel

A wheel of radius 10 inches rolls inside a wheel of radius 54 inches. Point P1 on the little rolling wheel starts at point P on the fixed wheel. After how many revolutions of the little wheel do points P and P1 coincide once more?

Show/Hide Solution

Points P and P1 will coincide when the little wheel gets back to P after a whole number of complete rotations. So we need to see how many times the little wheel will traverse the inside of the big wheel in an exact whole number of rotations.

The circumference of the big wheel is 108pi, while the circumference of the little wheel is 20 pi, so the little wheel must circle the big wheel 5 times before the distance traversed (540pi) is an exact multiple (27) of 20pi.

The little wheel will make 27 revolutions before P and P1 coincide again.

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