Discussion
At what time between 7 o'clock and 8 o'clock, will the hands of a clock be 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
\[ \theta = \left| 30H - 5.5M \right| \]
The hands are together when \( \theta = 0 \), with \( H = 7 \):
\[ 30(7) = 5.5M \quad\Longrightarrow\quad 210 = 5.5M \]
\[ M = \frac{210}{5.5} = \frac{420}{11} = 38\tfrac{2}{11} \]
So they meet at \( 38\tfrac{2}{11} \) minutes past 7. The hands coincide once in each hour except between 11 and 12, which is why there are 22 coincidences a day and not 24.
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