3550.24 – Put Your Hands Together

The hands will cross sometime after 1:00 PM where the dashed line in diagram below indicates. To be more precise, let $x$ be the angle past 1:00 PM, measured in degrees, when this happens.

Then, $30^\circ$ is the distance between hours and $\dfrac{x}{30^\circ}$ is the fraction of the hour (between 1:00 and 2:00) that the hour hand traverses to meet the minute hand. Meanwhile, the minute hand traverses the fraction $\dfrac{x + 30^\circ}{360^\circ}$ of the whole clock-circle. These fractions must be equal for the hands to meet, therefore

$$\dfrac{x}{30^\circ} = \dfrac{x+30^\circ}{360^\circ}$$

Solving this equation gives $x = \dfrac{30^\circ}{11}$. In other words, the hands meet one eleventh of an hour past 1:00 PM. In minutes this is $\dfrac{60^\circ}{11} = 5.4545\ldots$ or 5 minutes and about 27 seconds past 1:00 PM.

How many times in 12 hours do the hands meet (besides the moment at the beginning when both are vertical)? Does this suggest another approach to the problem?