Puzzle – The Hands of a Clock First Overlap

Question: 

An hour hand and a minute hand are standard on a clock. At 12 midnight the hands are exactly aligned. When will they next precisely align or overlap? How frequently will they cross paths each day?

Solution:

a) The hour hand revolves more slowly than the minute hand.

In one minute, the minute hand made an angle of 360/60 = 6°. 

In one minute, an hour hand made an angle of 360/720 = 0.5°. 

The difference between the angle made by an hour hand and a minute hand is 6° – 0.5° = 5.5°, 

The difference increases by 5.5° each minute.

The difference will be 11° after 2 minutes and 16.5° after 3 minutes.

The angle between the hour hand and minute hand at 12 midnight is zero degrees.

The difference between both hands must be 360° for them to overlap once more.

As both hands rotate continuously, we can use a fundamental ratio to get the difference of 360°.

1: 5.5° =? : 360°

i.e., 360/5.5 = 65.4545 minutes.

Therefore, they will overlap each other after 65.4545 minutes. It will happen when you convert it in the time given.

12am + 65.4545 minutes = 01: 05: 27 am (approximately) {0.4545 × 60=27.27}

b) Let’s start at 12:00 a.m. Hands were therefore overlapping there. And then, it overlaps once more between 1-2 AM. So, it goes on until 10 or 11 AM. i.e., It overlaps once per hour.
However, it overlaps at 12 PM for 11 AM to 12 PM. They overlap 11 times in 12 hours.
1) 12:00 AM
2) 1AM – 2 AM
3) 2AM – 3AM
4) 3AM – 4AM
5) 4AM – 5AM
6) 5AM – 6AM
7) 6AM – 7AM
8) 7AM – 8AM
9) 8AM – 9AM
10) 9AM – 10ÀM
11) 10AM – 11AM

They won’t overlap after 11 AM until after 12 PM.
The cycle then repeats from noon until 11:59 at night. Since our first unit began at 12 a.m. (midnight), the work should have been finished by 12 AM. the following day.
11 + 11 = 22 in total.

That means an hour hand and a minute hand will overlap or meet 22 times in a day.