Discussion
At what time between 3 and 4 o'clock will the hands of a watch point in opposite directions ?
Answer: A
The hands point in opposite directions at \(49\frac{1}{11}\) minutes past 3.
At 3 o'clock the hour hand is 15 minute-divisions ahead of the minute hand. In m minutes the minute hand moves m divisions and the hour hand \(\frac{m}{12}\). For the hands to be opposite, the minute hand must be 30 divisions ahead of the hour hand:
\[ \begin{aligned} m &= 15 + \frac{m}{12} + 30 \\ \frac{11m}{12} &= 45 \\ m &= \frac{540}{11} = 49\frac{1}{11} \end{aligned} \]
The minute hand gains \(5\frac{1}{2}\) degrees a minute and must gain 90 + 180 = 270 degrees, which gives the same \(\frac{270}{5.5} = 49\frac{1}{11}\) minutes. 50 minutes ignores the hour hand's movement, and the other fractions come from arithmetic slips.
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