The velocity ratio (ratio of angular speeds) of the hour hand and minute hand of a clock is:

Let us compute the angular velocities ($\omega$) of both hands of a clock:
1. Minute hand: Completes one full rotation ($360^{\circ }$) in $60\text{minutes}$.

$${\omega }_{\text{minute}}=\frac{360^{\circ }}{60\text{min}}=6^{\circ }/\text{min}$$

2. Hour hand: Completes one full rotation ($360^{\circ }$) in $12\text{hours}=12\times 60=720\text{minutes}$.

$${\omega }_{\text{hour}}=\frac{360^{\circ }}{720\text{min}}=0.5^{\circ }/\text{min}$$

Now, taking the ratio of the velocity of the hour hand to the minute hand:

$$\text{Ratio}=\frac{{\omega }_{\text{hour}}}{{\omega }_{\text{minute}}}=\frac{0.5}{6}=\frac{1}{12}$$

Hence, the velocity ratio is $1:12$.