钟表指针再次重合需要多久

设经过 $h$ 小时。

• 分针转过 $360h$ 度。
• 时针转过 $30h$ 度。

下一次重合时，分针比时针多转过一整圈：

$$360h=30h+360\Rightarrow 330h=360\Rightarrow h=\frac{12}{11}.$$

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Original Explanation

Let $h$ be the number of hours. The minutes hand will move $360h$ degrees. In addition, we know that the hours hand moves $30h$ degrees. However, as we need to account for rotation, the first time they meet, the minutes hand will have travelled an additional $360$ degrees beyond the hours hand, as it completes a rotation. Therefore, we get the equation $360h = 30h + 360$, which yields $h = \frac{12}{11}$.