Discussion

Find at what time between 8 and 9 o'clock will the hands of a clock be in the same straight line but not together ?

Answer: B

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

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

\[ \left| 240 - 5.5M \right| = 180 \]

\[ 5.5M = 60 \quad\Longrightarrow\quad M = \frac{120}{11} \]

That is \( 10\tfrac{10}{11} \) minutes past 8. The other root, \( 5.5M = 420 \), gives \( M = 76.4 \), beyond the hour.

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