Question

on a particular clock, the ratio of the length of the hour hand to that of the minute hand is 3 to 4. if h and m are the distances that the tips of the hour and minute hands travel in 3 hours, respectively, what is the ratio of h to m

Answer 1

You're looking at the arc length of each hand, and you're going to set up a ratio based on the two arc lengths.
The formula for the circumference of a circle is C=pi*2*radius

The hour hand has a (relative) circumference of
\[C_h=2*3\pi=6\pi\]However, the hour hand only travels 1/4 of the circumference. This means the arc length is only a quarter of this number, or
\[3\pi/2\]

The minute hand has a circumference of \[C_m=2*4\pi=8\pi\]However, it goes three times around the circle in 3 hours. This means we triple 8pi to get the arc length.
\[24\pi\]

The ratio between the two hands is \[(3\pi/2)/(24\pi)=3/48=1/16\]

Let me know if you need any clarification.