Problem 3. The hands of the clock overlap at 12 o'clock. After how many hours will the hands of the clock overlap again?
Solution. The hand that shows the minutes rotates 12 times faster than the hand that shows the hours. If we denote by $x$ the angle between two overlaps expressed in $\frac{1}{60}$-th of a full circle (the angle the minute hand rotates in 1 minute), then $x=\frac{60+x}{12} ; x=\frac{60}{11}=5+\frac{5}{11}$. Therefore, the next overlap will occur in $1 \frac{1}{11}$ hours.