The hour hand of a clock is 6 inches long and the minute hand is 8 inches long. What is the ratio of the distance in inches traveled by the tip of the hour hand to the distance in inches traveled by the tip of the minute hand from noon to 3 p.m.? Express your answer as a common fraction.
In three hours, the hour hand travels $\frac{1}{4}$ of a revolution while the minute hand travels 3 revolutions.  So the ratio of the number of revolutions traveled by the hour hand to the number of revolutions traveled by the minute hand is $\frac{1}{12}$.  However, the ratio of distances traveled is even smaller, because for each revolution the hour hand travels $\frac{2\pi(6\text{ in.})}{2\pi(8\text{ in.})}=\frac{3}{4}$ as far as the minute hand.  Therefore, the ratio of the total distance traveled by the hour hand to the total distance traveled by the minute hand is $\dfrac{1}{12}\cdot \dfrac{3}{4}=\frac{1}{16}$.