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

#2**+13 **

@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**+10 **

that -3 tho.....I was actually trying..... I thought it was like a stupid trick question

zegroes
May 8, 2015

#5**+13 **

"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**+13 **

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

#8**+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**+9 **

**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