At noon, the hour and minute hands of a clock point in the same direct

Question

At noon, the hour and minute hands of a clock point in the same direction. Determine, to the nearest second, the last time before noon that the hour and minute hands point in the same direction.

Gwydion · Answer

I'll give it a shot:
When starting from noon the hands first overlap at shortly past one, then shortly past two and so on. Count how often they overlap, it will be 11 times. From this we can draw that the first exact time they overlap will be 60min + 5min + 1/11*5min.
The question asks for the last time before noon that the hands point in the same direction so just multiply the numbers by 10 and you get the minutes: (60min + 5min + 1/11*5min)*10. Convert this into hours, minutes and seconds and you have the answer: 10:54:33.

Hope this is faster than the way you are solving this?!

AKProdigy87 · Answer

Let H = degrees from '12' at which the Hour hand resides
Let M = degrees from '12' at which the Minute hand resides

Let x and y represent the current time in hours and minutes respectively (i.e. 3:45 -> x = 3, y = 45)

H = 30x + y/2
M = 6y

We want the hour hand and minute hand to overlap. Therefore:

H = M
30x + y/2 = 6y
30x = (11/2)y
y = (60/11)x

Intuitively, subbing in x = 11 will result in a time of 12:00. Therefore, set x = 10.

y = (60/11)*10
= 600/11
= 54 + 6/11
=~ 54 + 33/60

Therefore,  10:54:33  is the last time before noon that the hour and minute hands point in the same direction.

HarvardSean · Answer

Thanks Gwydion that is faster. Also thanks AK that is the same way i did.

Wadree · Answer

For minute hand to move every 12 units..... the hour hand moves 1 unit. Now imagine an analog clock at noon where minute and hour hand points same way. Then imagine the minute hand moving anti clockwise a whole hour(60 units) and return to the same position pointing at 12.... so the hour hand will move anti clockwise (from previous position of pointing at 12) for 5 units and reach 11. So now its exactly 11 am and the hands haven't met at the same position yet but they will meet if they move anti clockwise one more time. Now imagine the round clock is worth 3600 sec=3600 units and minute hand is at 12 and hour hand is at 11(300 units behind minute hand). So their difference is 300 units. For every 12 units the minute hand moves back.... the hour hand moves back 1 unit..... in other words for every 12 seconds the minute hand goes back..... their difference reduces by 12-1=11 units. So 300 difference will be reduced if minute hand moves back (300/11)x12=327.272727 seconds≈327 seconds≈5 minute 27 second. So the minute and hour hands will meet again if time goes back another 5 minute 27 second from 11 am and reach 10:54:33 time.

At noon, the hour and minute hands of a clock point in the same direct

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At noon, the hour and minute hands of a clock point in the same direct

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