Discussion
The minute-hand and second-hand of a clock cross each other _______ times between 09:15:00 AM and 09:45:00 AM on a day.
Answer: A
The hands cross 30 times.
The second hand turns at 6° per second and the minute hand at 0.1° per second, so the second hand gains 5.9° per second. At 09:15:00 the minute hand points at 3 (90°) and the second hand at 12 (0°), so the second hand starts 90° behind. It overtakes the minute hand each time it has gained 90° + 360°k:
\[ \begin{aligned} t_{k} &= \frac{90 + 360k}{5.9} \le 1800 \\ 90 + 360k &\le 10620 \Rightarrow k \le 29.25 \end{aligned} \]
So k = 0, 1, ..., 29: 30 crossings, the first about 15 s after 09:15:00 and the last about 15 s before 09:45:00. Over 30 minutes the second hand gains 29.5 laps, and 29 is that figure rounded down, missing the crossing that the 90° starting gap adds. 31 counts one lap too many, and 15 is half the true count.
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