Been Clockin' Around ?
Consider a normal clock. If we measure the angle between the two hands the hour hand and the minute hand from clockwise of hour hand ( for example if it is 1: 30 pm then the angle between the hour and minute hand will be considered from right of hour hand that is clockwise from hour hand ) and supposing that at any particular time when any of the two hands that is the hour and the minute hand are not pointing exactly to any of the twelve markings in the the clock rather when they both point between any two adjacent markings and subtend a angle of p degrees such that p is a natural number then let the time be x : y ( x hours and y minutes ) then surely y would not be a natural number rather it would be a fraction ( if and only if we measure it very accurately and precisely ). Let y = a / b where a could be any natural number but b would always be constant. What is the value of b?
The answer is 11.
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1 solution
Clearly the angle subtended by any two adjacent markings is 30 degrees. So the hour hand moves 30 degrees every 60 minutes therefore it would move 1 / 2 degrees every minute. Similarly the minute hand moves 360 degrees every 60 minutes so it would move 6 degrees every minute . Now as mentioned the minute and hour hand should point between any two adjacent markings and not at exactly at any particular marking also given that they should make a angle of p degrees where p is a natural number. So now if we take at particular time x : y when the hour and minute hand make a angle of p degrees then we say and make a equation as follows- 30 - y/2 + 30q + 6y - 30r = p Because it is given to us that the we would count the angle clockwise from the hour hand so if it is y minutes then the hour hand would move x/2 degrees and 30 - x/2 give us the angle between the nearest marking clockwise to hour hand and the hour hand. We add 30q because q is number of markings that lie between the hour hand and the minute and we multiply 30 two give the total angle subtended by those markings. Next we add 6x - 30 because 6x represent the total angle from the marking that represents 12 : 00 clock and the minute hand and 6x - 30r give the angle between the minute hand and the nearest marking anticlockwise from minute hand with r being the total no. of markings between marking that represents 12 : 00 clock and the minute hand . Adding all these gives us the the angle between the hour hand and the minute hand which we took p . Simplifying gives us- 11y/2 + 30(1+ q - r)=p which finally leads us to y = p - 30 (1+ q - r)/11 with the denominator being constant 11.