1980 AMC 12 Problem 10

Question

The number of teeth in three meshed circular gears

A,

B,

C

are

x,

y,

z,

respectively. (The teeth on all gears are the same size and regularly spaced as in the figure.) The angular speeds, in revolutions per minute, of

A,

B,

C

are in the proportion

x:y:z

z:y:x

y:z:x

yz:xz:xy

xz:yx:zy

Meshed teeth have the same tangential speed at their points of contact

Angular speed is inversely proportional to circumference and hence to the number of teeth

The gears’ circumferences are proportional to

x,

y,

and

z,

while their tangential speeds have equal magnitude. Their angular speeds are therefore proportional to

\frac1x:\frac1y:\frac1z.

Multiplying all three terms by

xyz

gives

yz:xz:xy.

Therefore, the correct answer is D.

Problem 10 in other years