5. The clock shows 00:00, at which the hour and minute hands of the clock coincide. Counting this coincidence as number 0, determine how much time (in minutes) will pass before they coincide for the 21st time. Round your answer to the nearest hundredth.
Solution. The minute hand passes 1 circle in an hour, while the hour hand passes $1 / 12$ of a circle, so their closing speed is $11 / 12$ of a circle per hour, one closing takes $1 /(11 / 12)=12 / 11$ hours or $720 / 11$ minutes. The 21st closing will occur after $21 \cdot \frac{720}{11}=\frac{15120}{11}$ minutes.

Answer: $1374.55$.