At what time between 3 oclock and 4 oclock both the hour hand and minute hand coincide with each other?

Answer:- 1

Explanation:-

Solution:

55 min. are gained in 60 min.

15min.aregainedin(6055×15)min=16411min∴Thehandsarecoincidentat16411min.past315min.aregainedin(6055×15)min=16411min∴Thehandsarecoincidentat16411min.past3

Let $x$ be the number of minutes after 3:00 that the hour and minute hands overlap.

Since the hour hand moves $360^\circ$ twice over a $24$-hour period, so it moves $\frac{360^\circ}{12\times60}=0.5^\circ$ per minute. The hour hand would be at $90^\circ$ at time 3:00, therefore, it would be at $90^\circ+0.5^\circ\times x$ at the time that the hands overlap.

For the minute hand: it moves $360^\circ$ every hour, so it moves $\frac{360^\circ}{60}=6^\circ$ every minute. And since it starts at an angle of $0^\circ$ at 3:00, the minute hand should be at the angle of $6^\circ\times x$ when they overlap.

Now, set them equal to each other, you get:

\begin{align*} 90+0.5x&=6x\\ 180+x&=12x\\ 180&=11x\\ x&=\frac{180}{11} \end{align*}

$\therefore\text{The hour and minute hands overlap each other }180/11\text{ minutes after 3:00}.$

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