3550.24 – Put Your Hands Together

The hands will cross sometime after 1 where the dashed line in diagram below indicates. To be more precise, let $x$ be the distance past 1 when this happens. We measure distances around the circular face of the clock in degrees. Then $30°$ is the distance between hours and $x/30°$ is the fraction of the hour (between 1:00 and 2:00) that the hour hand traverses to meet the minute hand. The minute hand, in the same time, meanwhile traverses the fraction $(x + 30)/360$ of the whole clock-circle. These fractions must be equal for the hands to meet, therefore

$\frac{x}{30} = \frac{x+30}{360}.$

Solving this equation gives $x = 30/11$. In other words, the hands meet one eleventh of an hour past 1. In minutes this is $60/11 = 5.454545 . . .$ or 5 minutes and about 27 seconds past 1.

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?