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# How many times do the two hands of a clock point in the same direction between 6:00 am and 6:00 pm of a single day?

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How many times do the two hands of a clock point in the same direction between 6:00 am and 6:00 pm of a single day?

civonamzuk  May 8, 2015

#1
+4150
+14

is it 1 time tecnically?

zegroes  May 8, 2015
#1
+4150
+14

is it 1 time tecnically?

zegroes  May 8, 2015
#2
+1068
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@zegroes:/  The minute hand moves through 12 times faster than the hour hand. In the 'x' number of times, both of them, for a split second,  point in the same direction. This obviously occurs more than once.

civonamzuk  May 8, 2015
#3
+4150
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that -3 tho.....I was actually trying..... I thought it was like a stupid trick question

zegroes  May 8, 2015
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+87333
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"Between" 6AM  and 6PM  it occurs 11 times

1 time between  6AM and 7AM              1 time between   1PM and 2 PM

1 time between  7AM and 8AM              1 time between   2PM and 3 PM

1 time between  8AM and 9AM              1 time between    3PM and 4PM

1 time between  9AM and 10AM            1 time between    4PM and 5 PM

1 time between  10AM and 11AM          1 time between    5PM and 6PM

1 time  {at  noon}

CPhill  May 9, 2015
#6
+519
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I think I can do it in algebra.

Let,s say that two hands of a clock are two men and having a race,and the total distance of the race is 1

A men run 1/60 of the race in one minute ,so he run 1/*(60*60)=1/3600 of race in one second

and other men run 1/12 of the race in one hour ,so he run 1/720 in one minute and run 1/43200 in one second Now,suppose the clock are moving form 6:00 am to 7:00 am.

so the second men (hands） are starting at the half way of the distance -1/2

and If after x second ,the two men meet (in the same direction)

then we can set an equation as

$${\frac{{\mathtt{1}}{\mathtt{\,\times\,}}{\mathtt{x}}}{{\mathtt{43\,200}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}} = {\frac{{\mathtt{1}}{\mathtt{\,\times\,}}{\mathtt{x}}}{{\mathtt{3\,600}}}}$$

$${\frac{{\mathtt{1}}}{{\mathtt{2}}}} = {\frac{{\mathtt{11}}{\mathtt{\,\times\,}}{\mathtt{x}}}{{\mathtt{43\,200}}}} \Rightarrow {\mathtt{x}} = {\frac{{\mathtt{21\,600}}}{{\mathtt{11}}}} \Rightarrow {\mathtt{x}} = {\mathtt{1\,963.636\: \!363\: \!636\: \!363\: \!636\: \!4}}$$

1963.6363636363636364s*$${\frac{{\mathtt{1}}{min}}{{\mathtt{60}}{s}}}$$=32.7272727272727273min

(32.7272727272727273min-32min)*$${\frac{{\mathtt{60}}{s}}{{\mathtt{1}}{min}}}$$=43.636363636363638s

so if the clock move from 6:00 am to 7:00 am, the two hands will meet between 6:32 43 and 6:32 44

to calculate when the two hands in the same direction, just put other value instead of 1/2 into the formula

hopefully this help.

fiora  May 9, 2015
#7
+87333
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I kinda' see what you did here, fiora.......and it does look impressive....I'm just not sure that that much math is requred.....but....who knows???....maybe I'm wrong......!!!

BTW...I still gave you 3 points.....!!!

CPhill  May 9, 2015
#8
+26753
+14

Here's a graphical way of looking at it:

The bighand and littlehand scale indicates minutes past the hour

Alan  May 9, 2015
#9
+92806
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This turned out to be a really great question.   Thanks Civonamzuk

Our youngest mathematicians could have a good chance of working this out (as  CPhill shows you - thanks Chris) yet we are shown so many other ways as well.

Thanks Fiora for you algebraic method. Did you actually get an answer with that?

Alan I really like your graph.  The graph itself is easy to understand (for me anyway) the first mod fuction for the minute hand is okay but I don't think I would have gotten the second mod function.

Modulus arithmetic - just something else I am not very good at.  ://

Melody  May 9, 2015
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+92806
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Melody  May 10, 2015
#11
+92806
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I have included this question in the Puzzle Thread

Melody  May 10, 2015