1. So someone said to me "If a clock has its little hand just past 8 and the big hand at 8:10, how long before the two hands are in the same position?" I thought it would be a little more than 30.5 minutes since the big hand would move 60 times faster than the little hand. This would have made the time of the holding of the hands 8:40.5

I asked a friend and he took a complex approach. I don't know where he got the 240 degrees from, as I did consider the 360 degrees of a circle as also an approach but the hands would have been 180 degrees apart I would have thought.

He came up with this:

"May,

t=0; x=240 + t
y=60 + 60*t

y moves 60 times faster than x; hence the multiplier; the two values describe intitial locations in degrees when t=0.

x=y, solve for t

Since x=y when both hands are aligned both equations are equivalent.

240 + t = 60 + 60*t

59 * t = 180

t = 180/59 = 3.05

Solve for x = 240 + t = 243.05

Solve for y = 60 + 60 * 3.05 = 243.05 degrees, equivalent to 8:40.50 or 8:40 and 30 seconds."

I don't know how he came up with his answer but it did match mine. He is btw far further knowlegable and educated than I in math and Algebra than I am but he always respects my ways of viewing as well.  2. Oh crap. I think I know how he got 240. the little hand is the start at 8:40. 40 minutes is 2/3 of an hour. 2/3 of 360 (degrees in a circle) is 240.  3. Originally Posted by mayflow Oh crap. I think I know how he got 240. the little hand is the start at 8:40. 40 minutes is 2/3 of an hour. 2/3 of 360 (degrees in a circle) is 240.
The angle is not either at 8:10 nor at 8:40. Carry on.  4. y=60 represents the minute hand of the clock. It is ten minutes past 8. This represents ten of 60 minutes in the hour which is 1/6 of an hour. 1/6 of 360 degrees of a circle is 60, just as 2/3 of a circle is 240.  5. Your posted question is : "If a clock has its little hand just past 8 and the big hand at 8:10, how long before the two hands are in the same position?"

I surmise that you mean hour hand for little hand and you mean minute hand for big hand.

On a standard 12 hour clock it takes 12 hours for the hour and minute hand to return to their starting position.
Obviously the hour hand has to go full circle once and the minute hand to go full circle 12 times.

8:10 AM + 12 hours = 8:10 PM
8:10 PM + 12 hours = 8:10 AM  6. Mayflow, it looks like you did take the same approach, your friends approach was not more complex just written in maths. For bonus points anyone when will the hands next be lined up?  7. Originally Posted by pikpobedy Your posted question is : "If a clock has its little hand just past 8 and the big hand at 8:10, how long before the two hands are in the same position?"

I surmise that you mean hour hand for little hand and you mean minute hand for big hand.

On a standard 12 hour clock it takes 12 hours for the hour and minute hand to return to their starting position.
Obviously the hour hand has to go full circle once and the minute hand to go full circle 12 times.

8:10 AM + 12 hours = 8:10 PM
8:10 PM + 12 hours = 8:10 AM
True, but the question was probably not expressed by me properly. What I meant to ask is when will the hands be next in the same position on the clock? I guess when will the minute hand cross over the hour hand? They are moving at a rate of 60 to one. It would be really helpful if I coud draw it.  8. Originally Posted by mayflow y=60 represents the minute hand of the clock. It is ten minutes past 8. This represents ten of 60 minutes in the hour which is 1/6 of an hour. 1/6 of 360 degrees of a circle is 60, just as 2/3 of a circle is 240.
Repeating the same error over and over doesn't make it right. This kind of attitude precludes you from learning.  9. Originally Posted by Jilan Mayflow, it looks like you did take the same approach, your friends approach was not more complex just written in maths. For bonus points anyone when will the hands next be lined up?
First, I think even though we got the exact same answers. I think both my friend and I were off a little, and I think it is related to a clock being divided by 12 major units each being divided by 5 minor units. I looked at my alarm clock and the hands met around 8:44. They will probably meet around 9:49 next go round, but I don't know why yet. My friend has much math training and acuity and I am only a high school grad at this point. He has a double E degree, but he must have made a similar mistake as to my mistake. I will see if he can figure out where his math went wrong (I do believe now we were both close but not totally accurate) - I just tend to reason things out due to relationships and linear base 10 thinking and percentages - I think our mistakes were based somehow on a 10-12 relationship.  10. Originally Posted by pikpobedy Your posted question is : "If a clock has its little hand just past 8 and the big hand at 8:10, how long before the two hands are in the same position?"

I surmise that you mean hour hand for little hand and you mean minute hand for big hand.

On a standard 12 hour clock it takes 12 hours for the hour and minute hand to return to their starting position.
Obviously the hour hand has to go full circle once and the minute hand to go full circle 12 times.

8:10 AM + 12 hours = 8:10 PM
8:10 PM + 12 hours = 8:10 AM
12 hours is a good answer but it is not the earliest time. The problem is asking for the hands to make the same angle, this happens much faster, ever hour and change. The equation that gives the general answer about "when" this happens is: , so, the same angle gets repeated every: So, the situation repeats every , 23 times a day.  11. Online, most say 22 times per day. This if rounded is a 17% (22/24) differential which fits into my theory of 10/12 or 5/6 which is (if rounded) = 83%. Which is about the ratio of my and my friend's answers of 8:40.5 to my observed on my alarm clock 8:44 or so.  12. Originally Posted by mayflow Online, most say 22 times per day. This if rounded is a 17% (22/24) differential
But , it isn't, the correct number is 24/23.

which fits into my theory of 10/12 or 5/6 which is (if rounded) = 83%.

Which is about the ratio of my and my friend's answers of 8:40.5 to my observed on my alarm clock 8:44 or so.
Science doesn't care what online "says"  13. But science might care that I have a mechanical alarm clock and if I move the hands in their 60:1 ratio, they do cross exactly 11 times in 12 hours. This is a ratio of 1.0909090909... This means to me that the hands of the clock will meet about every 65.4545.... seconds. There is some sort of relationship here as if 1.0909090909.. were multiplied to both my and my friend's independent equations here the hands would have synched at about 8:44187875 which as far as I can tell from observing this on my alarm clock, is correct.  14. Originally Posted by mayflow But science might care that I have a mechanical alarm clock and if I move the hands in their 60:1 ratio, they do cross exactly 11 times in 12 hours.
They cross 11 times in the first 12 hours BUT there is time left. So, in the next 12+ hours they cross 12 times for a total of 23, not 22.

This is a ratio of 1.0909090909
Nope, for the ones who can do their own calculations, the ratio is 24/23=1.04347826087.  15. 12 hours is the correct answer for the question you put.

Otherwise you must reframe the question to the more astute question that is seen in many puzzle books.

Albeit, I wondered briefly (slightly briefly) why such a simple question was asked. But there are all sorts of posters and questions are often asked upsidedown, backwards and insideout.

Start by phrasing and framing your puzzle correctly.

And if you are asking questions as puzzles to amuse people... say so outright at the outset.
If you are asking as homework help or because you do not know how to answer it.... say so.
Otherwise your come backs may cause confusion, frustration and irritation, as one can find in this thread.
Enuff for today.  16. Originally Posted by pikpobedy 12 hours is the correct answer for the question you put.

Otherwise you must reframe the question to the more astute question that is seen in many puzzle books.

Start by phrasing and framing your puzzle correctly.

And if you are asking questions as puzzles to amuse people... say so outright at the outset.
If you are asking as homework help or because you do not know how to answer it.... say so.
Otherwise your come backs may cause confusion, frustration and irritation, as one can find in this thread.
Enuff for today.
You are asking too much of mayflow, she cannot do that, things are muddled in her mind. It is for the others to figure out what she's about, this is how it has always been and it will not change.  17. Originally Posted by x0x The angle is not either at 8:10 nor at 8:40. Carry on.
This was I think his mistake. I think his formulas were correct. I am not going to try them out tonight as my computer has a problem and I lost my excel progam, and it will let me download things like open office, but will not let me install them. I REALLY do NOT know the odd stuff I get when I quote you " " Anyway I think if he changes 240 degrees for the correct starting angle he will probably be correct.

I solved it a different way.

When the hands are aligned at 12:00 one travels 360 degrees in 1 hour. The other at 1/6 the speed travels 30 degrees. (360 degrees in an hour = 360 degrees/60 minutes=1/6) The one traveled at 360-330 so the ratio of 360 to 330 - is 12/11. This is 1.0909 per adinfinitum. If you multiply this times the hours and you will see the precise time the hands will cross. I will see if I can do a spreadsheet tomorrow from work (don't tell my boss 'cause he will just say - well May is working on expanding May's mind and that is always beneficial and useful)

Also, if you access a clock like this and actually watch when the hands match, they will match just as I predict. The spreadsheet will show that the next time 12 o'clock comes around the hands willbe in unison at 12:00 and this is pretty much observable by anyone, and they will meet 11 times precisely along the way. I think my friend's approach will to this as well, but we will see. I am quite sure it could be done by geometry and angles as well, but I suck at geometry. I can learn it though, I think, so if someone has more of an approach at figuring this out by the angles, it probably would have something to do with the angles in parallelograms somehow.  Posting Permissions
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