Discussion

At what time between 5 and 6 o'clock are the hands of a clock 3 minutes apart?

Answer: A

The hour hand moves \( 0.5^{\circ} \) a minute and the minute hand \( 6^{\circ} \), so at \( H \) hours \( M \) minutes the angle between them is

\[ \theta = \left| 30H - 5.5M \right| \]

Three minute-divisions is \( 3 \times 6 = 18^{\circ} \), with \( H = 5 \):

\[ \left| 150 - 5.5M \right| = 18 \]

\[ 5.5M = 132 \quad\Longrightarrow\quad M = 24 \]

The other root gives \( M = 30\tfrac{6}{11} \), on the far side of the coincidence at \( 27\tfrac{3}{11} \) minutes past 5.

The 24-minute answer is the exact one, which is what makes it the intended reading.

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