Discussion

At what time between 7 and 8 o'clock will the hands of a clock be in the same straight line but,not together?

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

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

"Same straight line but not together" means \( \theta = 180 \), with \( H = 7 \):

\[ \left| 210 - 5.5M \right| = 180 \]

\[ 5.5M = 30 \quad\Longrightarrow\quad M = \frac{60}{11} = 5\tfrac{5}{11} \]

The other root, \( 5.5M = 390 \), gives \( M = 70.9 \), beyond the hour.

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