Question

At what exact time between 3:00 PM and 4:00 PM do the hour and minute hands of a standard clock coincide?

• A. 3:18:27 PM
• B. 3:08:14 PM
• C. 3:09:07 PM
• D. 3:12:06 PM

Answer 1

The Correct Option is C

Relative speed of the minute hand with respect to the hour hand:
The minute hand moves 12 times faster than the hour hand.
So, the relative speed of the minute hand with respect to the hour hand is 11 times the speed of the hour hand.
Angle between the hour and minute hands at 3:00 PM:
The hour hand covers 3 \(\times\) 30 = 90 degrees from 12:00.
The minute hand starts at 0 degrees.
So, the angle between them is 90 degrees.
Time taken for the minute hand to cover 90 degrees relative to the hour hand:
Let the time taken be x minutes.
In x minutes, the minute hand gains 11x degrees relative to the hour hand.
We need this to be equal to 90 degrees.
So, 11x = 90
x = \(\frac{90}{11}\) minutes
Therefore, the minute hand will coincide with the hour hand at 90/11 minutes past 3:00 PM.
Converting \(\frac{90}{11}\) minutes to minutes and seconds, we get approximately:
8 minutes and 7.27 seconds
So, the hands will coincide at approximately 3:08:07 PM.