Clock John
John is a little boy who has recently learnt how to measure angles using a protactor. Once, he said to his mother, "When I went to play the hands of clock were making a angle of 170 degrees and when I returned they were making a angle of 60 degrees. How Long did John went to play in minutes?(upto correct 2 decimal places) Details- The angle will be measured from right of hour hand. He went between 12 :15 and 12:45 came back between 1:15 and 1:30 Note - Sexagesimal angle measurement
The answer is 45.45.
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1 solution
Clearly the angle subtended by any two adjacent measures (here measure refers to the 12 different markings on a clock) is 30 degrees . The hour hand moves 30 degrees every 60 minutes so 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 . Given when John went it was between 12:15 and 12:45 . Clearly the minute hand would be somewhere between the 6th and 7th measure then only the hour hand and minute one could make a angle of 170 degrees. So we make a equation as follows if we take minutes after 12 hours be x. 30 -x/2 + 150 +6x- 180 = 170 We do so because the the hour hand would have moved x/2 degrees from 12th measure in x minutes . But because we are measuring the angle from right of hour hand so we take 30 -x/2 . We add 150 degrees because there would be 5 measures between the hour hand and minute hand. We then add 6x - 180 to get the angle between the minute hand the 6th measure . This should be equal to 170 so we write = 170. This comes to x = 340/11. So we get that John went at 12 hours and 340/11 minutes. Similarly we can make another equation to determine the time at which he came back as- 30 - x/2 +30 +6x -90 = 60 Which comes to x = 180/11 So we can get the total time he played would be clearly equal to - 60 - 340/11 + 180/11 = 500/11 minutes = 45.45 minutes AMAZING MATHEMATICS!