Question: The time gap between the two instants, one before and one after $12:00$ noon, when the angle between the hour hand and the minute hand is $66^°$ , is
$1.~~~~ 12 ~\text{min}.$
$2.~~~~ 16~\text{min}.$
$3.~~~~ 18 ~\text{min}.$
$4.~~~~ 24 ~\text{min}.$
My thought: We know that at $12:00$ noon, both the hour and the minute hand are at position $0^°$.
Also after $x$ hours of time, the hour hand travels $x / 12$ rotations around the clock. So after $x$ minutes, it travels $x / (60 \cdot 12) = x / 720$.
After $x$ minutes of time, the minute hand travels $x / 60$ rotations around the clock.
Now how to proceed the further ? Please help.
Best Answer
The hour hand travels $360^\circ$ in $12$ hours, or $720$ minutes. That is $\frac 12^\circ$ per minute. The minute hand travels $360^\circ$ in $60$ minutes or $6^\circ$ per minute, so the minute hand gains $\frac {11}2$ degrees/minute. To gain $66^\circ$ therefore takes $12$ minutes. Similarly, going backwards, the minutes hand loses $66^\circ$ in $12$ minutes, so the times are $24$ minutes apart.