Ding Dong
In a day, how many times do the minute and hour hand of a clock coincide?
For clarity, consider a day to be from 1:00am to the next day 1:00 am.
The answer is 22.
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15 solutions
Ans is 22. For all those who think it should be 24, let me explain it in a simple way. In a single day, hour hand will complete 2 rotations. Now let us consider one rotation of the hour hand ie 12 hrs. Suppose hour and minute hands, both are at 12 and clock is started. Now for a complete rotation of the minute hands, hour hand will be at 1 while minute hand will be at 12. So in 1st complete rotation of the min hand, hour hand could not coincide with it. But for the next 11 rotations of the min hand, hr hand will coincide with it. So it means in 12 hrs, we can have 11 coincides of the both hands. So for the next 12 hrs of the day we will get the same, so the total coincides are 22 in a single day. Hope it will be cleared to you now.
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thanks a lot Tahir for easy explanation.
It seems to me the only way that works is if you don't count 12 Midnight at the beginning and the end of the day, in which case you have something less than a day.
good explanation
A really simple way to see this is to imagine that the two hands are racing each other around a track. Every time the minute hand 'laps' the hour hand, we have the overlaps we want.
So, we can say that the number of laps completed by the minute hand every T hours, Lm = T laps. Since there are 12hours in a full rotation of the hour hand, that hand only rotates Lh = T/12 laps.
In order for the first 'lapping' to occur, the minute hand must do one more lap than the hour hand: Lm = Lh +1, so we get T = T/12 + 1 and that tells us that the first overlap happens after T = (12/11) hours. Similarly, the 2nd lapping will occur when Lm = Lh + 2.
In general, the 'Nth' lapping will occur when Lm = Lh +N, which means every N*(12/11) hours (for N = 0,1,2,3...). In other words, it will happen approximately every 1hr5mins27secs, starting at 00:00. In 24hours, this occurs a total of 24/(12/11) = 22 times.
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Try an analog clock and see if your logic holds, I am highly doubtful. a simple problem is being made complicated when it actually is not that compplicated.
Is there a chance for which the answer is 21? If you count the 24th hour isn't that the very beginning of a new day? (counting the 00.00 hour and 24.00 hours happens to be a repetition). That example reminds me that of a week, for example. If we do as we're doing with the problem of the clock, then one week is not from Monday to Sunday or Sunday to Saturday, but Monday to Monday ot Sunday to Sunday, therefore one week has 8 days. That related to the clock problem is more than one day, leading to two days.
I appreciate some advice on this. Thanks in advance.
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If you start your day at 0000, you end it at 2359. If you start it at 0001, you end it at 2400. Either way, you can only use midnight once.
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then you have a day that is 23 hours 59 minutes long ... not a 24 hour day
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@Bill Rodawalt – Should I have said, "end it in the minute that starts at 23:59:00 and ends at 23:59:59?"
January starts on the 1st and ends on the 31st. That's still 31 days.
The operative phrase was "you can only use midnight once."
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@Brian Egedy – No ... because if you stop on second before midnight, you still do not have a 24 hour day. In order to have 24 hours you must count midnight at both ends.
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@Bill Rodawalt – Midnight to Midnight is 00:00:00 to 24:00:00, which means you're adding a full second to the calendar day. 24:00:00 on Monday is actually 00:00:00 on Tuesday, which means you have two labels for the same moment, i.e., midnight Tuesday morning.
Midnight on the 1st is not midnight on the 2nd. Midnight is a moment in time, and can only exist in one day or the other. That's akin to claiming 1 am for the previous day, or 11 pm for the following day. 12 am is one moment, and it belongs to the coming day, by convention.
One minute is sixty seconds, and those seconds are numbered from 00 to 59, because 01:00 is the following minute's first second.
One hour is sixty minutes, and those minutes are numbered from 00:00 through 59:59, because 01:00:00 is the second hour's first minute/second/moment.
One day is 24 hours, and those hours are from 00:00:00 through 23:59:59, because 00:00:00 is the first moment of the following day. The numbering system is what's important, here. There are 24 full hours from 00:00:00 to 23:59:59, because we're starting at 00, not 01. By convention, 00:00:00 belongs to the morning, not the previous evening.
If you count the seconds, you can clearly see that if the first second of the minute is 00, then the sixtieth second is 59.
If the first minute of the hour is from 00:00 to 00:59, then the sixtieth minute is 59:00 to 59:59.
If the first hour of the day is from 00:00:00 to 00:59:59, then the 24th hour of the day is 23:00:00 to 23:59:59, and there are 3600 minutes in it.
There are no minutes or seconds missing. You have to not use midnight more than once per day, or you're adding seconds to the day.
why24 is not correct
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Because hands meet roughly every 65 minutes and not 60.
Bcz both arrows don't coincide between 11 to 12 and 12 to 1 but @12 both will coincide this is bit obvious.. and for every hour they coincide once. So total=22
D'OH! I accidentally calculated for 12 hours XD I would have gotten it right too!
What do you mean by "But that would only be true if one of the hands were at rest."
And why isn't the answer 24??? Sorry ... I am really bad at visualising...
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The minute hand crosses the hour hand 22 times rather than 24 times. This is because the hour hand is constantly going clockwise and hence it is trying to move away from the minute hand
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Which are the 2 occurrences/2 hours that the minute hand doesn't cross the hour hand (e.g. 1pm or sth??)
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@Fan Zhang – That happens in the eleventh and the twenty-third hour
i didnt understand please explain it again
Nice solution :)
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There was a similar problem which taught me this approach: how many times a day are the two hands of the clock perpendicular to each other.
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is the answer (of perpendicular problem) is 24..??
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@Prashant Saxena – No. Hint: that situation arises (almost) twice an hour.
Brilliant,
Thanks buddy. U have explained it in a really simplified manner. :)
Hey. I dont know if this line of thinking is accurate but here it goes. If you think about two people running at the same speed, if they look at eachother it's as if they were stopped. They would always see each other faces the same way as if they were stopped. So that is relative speed. The speed of the clock hands vary, so to get a relative speed I did the difference between the faster one and then slower one so that it is as if the slower one was stopped. So, 360°/h - 15°/h = 345°/hour. Considering that the day starts at 1am, the hour one has an advance of 30° so, in 24 hours, the degrees the new min hand (345°/h) would do is 345x24 = 8280. Since there is a difference of 30°, then we take that ... 8280-30 = 8190. Then, to see the number of times they cross we do: 8190/360 and we get 22.9. So 22 xD Again, don't know if this is right :)
Nice solution.
For those thinking 23 and using midnight as your 23rd instance, note the clarification in the problem that the start and stop time should be 1am. Therefore our first overlap occurs at 1:05 and once our day ends at 1:00, we are 5 mins away from our 23rd overlap. A nice way to avoid the confusion that can arise at midnight.
how it is?
This is a wrong solution In 12 hours the minute hand rotates 12 times and will pass over the hour hand 12 times. Which means in 12 hour durations we have 12 meetings. In 24 hours it is repeated and the answer is 24.
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But the hour hand rotates too. As a result the number of collisions is reduced.
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how is the answer of (overlapping problem) came 24....? after one complete cycle(i.e. after 12 hours),the clock will repeat itself in the similar way for the next 12 hours....so the answer should b 24.... isn't it.??
am 12:00 1:05 2:11 3:16 4:22 5:27 6:33 7:38 8:44 9:49 10:55 pm 12:00 1:05 2:11 3:16 4:22 5:27 6:33 7:38 8:44 9:49 10:55
great this answer gives the real insight to the answer
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but we don't get to a 24 hour day until the hands hit 12 am again where in that infinitesimal moment in time it is both the end of one day and the beginning of the next ... so the answer should be 23
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Midnight on one day can't also be midnight on the next. Our definition of time doesn't allow a moment, however infinitesimal, to be a part of two days. It's not the moment that one day transfers into the next. There is a "final moment of Tuesday" followed by a "first moment of Wednesday". By convention, the final moment of one day is during hour 23:59:59, and the first moment of the next day is the first moment of 00:00:00.
You can't discuss mathematics if you can't agree on the definitions of terms.
They overlap 22 times in a day. Proof:
in 1 hour, minute hand turns 1 time => 360°, so its angular speed is:
ωm = 360/60 = 6 deg/min;
in 12 hours, hour hand turn 1 time, in 1 hour it turns 1/12 time =>
(360/12) = 30°, so its angulatr speed is:
ωh = 30/60 = 0.5 deg/min;
if t is the time wIth t = 0, at 00:00, the angle of minute hands is:
θm = ωm*t = 6t deg;
the angle of hour hands is:
θh = ωh*t = 0.5t deg;
they overlap when:
θm - θh = k360 deg, k ∈ ℕ ∪ {0};
(6 - 0.5)t = k360; tk = k360/5.5 = 65.(45)k min;
tk = 65.(45)k min; (times when minut and hour hands overlap),
so they overlap every To = 65.(45) min = 1h, 5min, 27.(27)s;
tk must be: tk ≤ 24*60 min; (1 day = Td)
tk ≤ 1440 min; so the hand overlap N = ceil(Td/To) times,
N = ceil(1440/65.(45)) = ceil(22) = 22 times
I will exlain the answer in a simple way, the hour hand doesn't meet the minute hand at exactly the same time like at for the time 2am or pm the time they meet is at 2:11 not 2:10 because minute hand has to travel 10 minutes to reach the 2 position, so within that ten minutes the hour hand would also move away from the 2 position. for the position 5 it is 5:27 not 5:25 for 6 it is 6:33 not 6: 30 for 7 it is 7:38 not 7:35 finally, for 10, we would think that the crossover happens at 10:50 but the minute hand has almost travelled for an hour so meeting will be very near to 10:55 not 10:50. soon after 10:55 it will 11 o clock. for the position 11 there will be no cross over or it can be said that for position 11 the cross over happens at 12 o clock because the minute hand has travelled exactly for one hour so the hour hand would have moved from the 11th position to the 12rh position. This eliminates one cross over that we imagined on the 11 position.
I am not convinced of all the above argument. Speak about half a day the hour will do one rotation and the minute will do 12 rotation they will meet 12 times. The first one is at 12 and so on.
In the second half of the day they will meet again at 12 and do also another 11 meetings. So in total 24. I hope this will cpnvince you.
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for 12 hours they meet 11 times for 24 hours they meet 22 times! think abut it @0.0AM they meet once. again at 12 the min hand movie little bit forward so they meet only 11 times in 12 hours. again @24 hours they meet but it is consider as next day so 22 is the correct answer
Albert einstein's here :p.. ans is correct i.e. 22
for 12 hours they meet 11 times for 24 hours they meet 22 times! think abut it @0.0AM they meet once. again at 12 the min hand movie little bit forward so they meet only 11 times in 12 hours. again @24 hours they meet but it is consider as next day so 22 is the correct answer
The approximate times are listed below.
am 12:00 1:05 2:11 3:16 4:22 5:27 6:33 7:38 8:44 9:49 10:55 pm 12:00 1:05 2:11 3:16 4:22 5:27 6:33 7:38 8:44 9:49 10:55
Assume the hour hand moves x tick marks from 12 o'clock until the first time the hands meet. So the minute hand moves 60+x tick mark in this interval. On the other hand, the minute hand moves 60 tick marks while the hour hand moves 5 tick marks every hour. So the minute hand moves (60/5)x = 12x tick marks in this interval. Solving 60+x = 12x, we obtain x = 60/11. Since the hands move at constant angular speeds, they meet each other every (60/11)/5 = 12/11 hours. In other words, they meet 24/(12/11) = 22 times in a day. Since the hands form a right angle two times between two consecutive meetings of the hands, As a result, the hands form a right angle 2*22 = 44 times in a day.
wow..!! GOT IT
the hands of a clock overlap 22 times a day. Thus the hands of the clock overlap at 12:00, ~1:05, ~2:10, ~3:15, ~4:20, ~5:25, ~6:30, ~7:35, ~8:40, ~9:45, ~10:50. Note that there is no ~11:55. This becomes 12:00.
It coincides once in every hour but in 12 hours in coincides 11 times because in coincides just once between 11 and 1 at 12.So in 24 hours it will coincide 22 times.
The min. hand which goes 360 degree in 1 hr., goes ahead of the hr. hand which goes only 30 degree in the same time i.e., in 1hr. Obviously, there is no chance to happen the 1st coincide by 1 hr. Now, suppose it get 1st coincide in T hr. then, by condition, the angle (30T degree) gone by the hr. hand equals that ( (T-1)multiplied by 360 degree ) of the Min. hand. That is 30T= (T-1)*360 which gives T = 12/11 hr. Now, the No. of coincides = 24/ (12/11) = 22
let's divide the time intervals as 1-2,2-3,.....11-12,12-1. one can visibly note that when the minute and and hour hand completes the 12-1 time interval the doesn't coincide. therefor for 12ve hours they coincide 11 times and for 24 hours 22 times
Just don't count the first and the last hour because time already went passed one second and so on when you started counting minus first last 2-x/24=22
Since we have 24 hours in a day, we can solve the above equation for N
24 = 24/12 + N 24 = 2 + N N = 22
In T hours, the minute hand completes T laps. In the same amount of time, the hour hand completes T/12 laps.
The first time the minute and hour hands overlap, the minute hand would have completed 1 lap more than the hour hand. So we have T = T/12 + 1. This implies that the first overlap happens after T = 12/11 hours (~1:05 am). Similarly, the second time they overlap, the minute hand would have completed two more laps than the hour hand. So for N overlaps, we have T = T/12 + N.
Since we have 24 hours in a day, we can solve the above equation for N
24 = 24/12 + N 24 = 2 + N N = 22
Thus, the hands of a clock overlap 22 times a day. Thus the hands of the clock overlap at 12:00, ~1:05, ~2:10, ~3:15, ~4:20, ~5:25, ~6:30, ~7:35, ~8:40, ~9:45, ~10:50. Note that there is no ~11:55. This becomes 12:00.
Right ans is 22 because when we calculate in 12 hr we get 11 .so 11×2=22
Time taken to meet once is given by [ relative angular velocity w *t=2 pi ] implies (2 pi/1h -2 pi/12h) *t =2 pi =>t=12/11; in 24 h no of times hour hand meet minute hand is 24/t=24/(12/11)=22
practically we know that for every hour of the day the minute hand should meet the hours hand at least once, this is known by even illitrate s also.Thus, the hands of a clock overlap 22 times a day by completing 24 laps. Thus the hands of the clock overlap at 12:00, 1:05, 2:10, 3:15, 4:20, 5:25, 6:30, 7:35, 8:40, 9:45, 10:50. Note that there is no 11:55.This becomes 12:00
Note that for every full rotation of the minute hand, it must meet the hour hand at least once. But that would only be true if one of the hands were at rest.
The hour hand does a complete rotation twice a day while the minute hand does a complete rotation 24 times a day.
Relative to the hour hand, the minute hand makes 22 complete rotations and hence makes 22 coincidences.