Discussion

At what time between 4 and 5 o'clock will the hands of a watch point in opposite directions?

Answer: D

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

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

Opposite directions means \( \theta = 180 \), with \( H = 4 \):

\[ \left| 120 - 5.5M \right| = 180 \]

\[ 5.5M = 300 \quad\Longrightarrow\quad M = \frac{600}{11} = 54\tfrac{6}{11} \]

The other root, \( 5.5M = -60 \), is negative and discarded.

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