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}=\boxed{\frac{1}{16}} $.