jmc

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 $\frac{1}{12}\cdot \frac{3}{4}=\boxed{\frac{1}{16}}$.