Discussion
Immediately after 12 o'clock at what time duration both the minute hand and the hour hand of a watch exactly overlap ?
Answer: C
They overlap after 1 hour five minutes and before one hour six minutes.
The minute hand gains \(5.5^\circ\) per minute on the hour hand. After 12 o'clock it must gain a full \(360^\circ\) to overlap again:
\[ \begin{aligned} t &= \frac{360}{5.5} = \frac{720}{11}\ \text{min} \\ &= 65\tfrac{5}{11}\ \text{min} \\ &\approx 1\ \text{h}\ 5.45\ \text{min} \end{aligned} \]
That is between 1:05 and 1:06. At exactly one hour the hour hand is at 1 while the minute hand is at 12, and at 1:05 the hour hand has moved on past the 1, so the exact one hour and one hour five minutes options are both too early.
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