Clock Teasers

Question 1:

Find the angle between the hands of a clock.

Question 2:

The hour and minute hands are at equal distance from the 6 hour, what time will it be exactly?

Question 3:

Find out how many times the minute hand and hour hand exactly match over a 12-hour cycle.How often the second hand and minute hand match each other exactly

  

Solution 1:

Minutes Angle = (360 * m) / 60 = 6m where m is the minutes.
Hour Angle = ((360 * h) / 12) + (360 * m / 12 * 60)
Hour Angle - Minutes Angle = 30h - 11m/2

Solution 2:

Say answer is "8 hour X minute". According as proposition, the angle between the minute hand and "mark 4" of the watch is equal to the angle between the hour hand and "mark 8" of the watch. 

We know in 60 minutes the minute hand makes 360 degrees (360/60=6 degrees per minute) and the hour hand makes 360/12=30 degrees (30/60=1/2 degrees per minute). 

Therefore, (20-X) minutes corresponds to 6(20-X) degrees (this is the angle between the minute hand and "mark 4"). 

And in X minutes the hour hand makes X/2 degrees with "mark 8". 

Thus, X/2=6(20-X) gives X=18 minutes 27 and 9/13 second. 

So, the answer is 8 hour, 18 minutes, 27 9/13 second.

Solution 3:

Between 12:00 and 1:00, the minute hand is always ahead of the hour hand. Then somewhere slightly past 5 minutes after 1:00, the hour and minute hands are in the exact same position. If you have a clock or watch on which you can manipulate the time, try this for yourself.

In this way, the minute hand will pass over the hour hand ten more times, once each hour between 1 and 2, 3 and 4, and so on, until the hour between 10 and 11. Between 11 and 12, the minute hand never catches up to the hour hand until exactly 12:00, when the hands line up again. The hands match up 12 times in a complete cycle, including both the starting and ending positions.