How many times do a clock's hands overlap in a day?

already exists.

Would you like to merge this question into it?

already exists as an alternate of this question.

Would you like to make it the primary and merge this question into it?

exists and is an alternate of .

22 times a day if you only count the minute and hour hands overlapping. The approximate times are listed below. (For the precise times, see the related question.)

2 times a day if you only count when all three hands overlap. This occurs at midnight and noon.

am 12:00
 pm 12:00

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.


So we are looking at two rotating hands. Ultimately, its just the angles we care about. Let θH represent the angle of the hours hand and θM represent the angle of the minutes hand. You could also introduce the seconds hand but that makes the problem more complicated. For now, lets assume the question only cares about the minute and hour hands. Initially we might think we are looking for:


But this doesn't take into account that if one hand has "gone around" a few times, its angle will be different from a hand in the same position that hasn't "gone around" the same number of times. So we have to modify our goal. Instead we let the angles differ by an integer multiple of 2π (360°). Let us call this arbitrary integer z. Now our condition is:


You could subtract the two angles in either order but the reason I chose to subtract hours from minutes is because it will result in positive integers which is just simpler. The minute hand goes around more times, thus its angle is bigger, thus this order of subtraction is positive. Now we have to find out how these angles depend upon the time. Let us call our time t and measure it in hours. I omit units for simplicity. The hour hand goes around a full rotation (2π) once every 12 hours. So:

θH=(2π/12) t

For those more versed in mathematics, 2π/12 is the "angular frequency" for the hour hand (usually denoted by ω).

Similarly the minute hand goes around a full rotation (2π) once every hour. So:

θM=2π t

Plugging back in:

2π t - (2π/12) t = 2πz
t - t/12 = z
(11/12) t = z

Now we are ready to solve. The two hands overlap at every solution of this equation, so we want to know the number of solutions of this equation. But remember, we want to know how many times this happens in a single day, so t cant be bigger than 24 (remember we are measuring t in hours), and technically no smaller than 0 (assuming we start our clock at 0 hours). Since t and z are proportional, each solution for z corresponds to exactly one solution for t, and accordingly exactly one solution of the equation.

Also, remember than z must be an integer. So if we wanted all the times we would just let z go from 0 (when t=0) up and solve for t and stop as soon as we passed t=24. Then of course we'd have to convert that into hour and minute format. However, we only care about the number of times this happens. So we can notice that as t increases, z is just keeping track of how many times the two hands have overlapped. When z=0 we get the first time, when z=1 we get the second time, and so on. Since t and z are directly proportional, t increases with z, thus z represents the number of times the hands have overlapped up until time t minus 1 (and starting from t=0). Since we don't want t to go past 24, we plug in 24 and solve for z which will tell us how many times this event has occurred from t=0 to t=24 (one day).

(11/12)*24 = z
22 = z

So this happens 22 times in a day. Technically this has 23 solutions (0 through 22) but the last one is for t=24 which has begun the next day. If we don't count that solution we are left with 22.■

If we want the second hand to overlap as well, we have to go a bit further. First we note that the second hand makes a full rotation once every minute, thus 60 times an hour. From this we have:

θS=(2π*60) t

We want the second and hour hands to overlap AND the minute and second hands to overlap. Those conditions can be summarized as follows, where x and y are positive integers:


Plugging in our functions of t for the θ's and solving for t we are left with:


We want our integers x and y to produce the same time (making all hands overlap at that time). So we want to set the two equations equal. Simplifying, we get;


708 and 719 are coprime (719 is prime and 708 is decomposed into 2^2*3*59). In fact 708y and 719 are coprime except for when y is an integer multiple of 719. Thus 708y/719 can only be reduced when y=719k for some integer k. In this case we have:

The first solution is when k=0. Then x=0 and t=0 corresponding to midnight. The next solution is k=1. Then x=708 and t=12 corresponding to noon. The next solution is k=2 but this corresponds to t=24 which is (midnight for) the next day and due to the direct proportionality of t and k, every k from here on up will produce t's higher than 24.

In summary, all three hands only overlap twice a day: at noon and midnight. ■

All of this assumes that the hands sweep continuously. So the math is more(?) complicated for those with fake Rolex's (or any ticking handed clocks).
6 people found this useful

How many times in a day are the hour hand and minute hand of a clock at right angles?

I just did some tallying in my head, with pencil and paper handy. In a 24 hour day, I count 44 times. Answer The continuous movement of the hour hand brings about something analogous to the "sidereal day" problem. Most of the time, there are 2 occurances per hour of 90 degrees between the hands. Bu (MORE)

How many times do the hands of a clock overlap in a day?

A Better Approach (with reasoning) There are 2 cases depending on the working of the watch. Case 1: the movement of the second, minute and hour hands are continuous (not step-wise or click-based) Answer is: the hour and minute hands overlap every hour. . Case 2 (very unusual): the hour hand (MORE)

How many time hour hand and minutes hand cross each other in 1 day or 24 hours?

23 times.. The time interval between each 'meeting' of the minute and hour hands is 65minutes and 27.272727(recurring) seconds. There are number of ways of working this out. But my method allows you to calculate this mentally.. it takes 60 minutes for a minutes hand to complete 360 degrees. ie 36 (MORE)

Imagine an analog clock set to 12 o'clock Note that the hour and minute hands overlap. How many times each day do both the hour and minute hands overlap How would you determine the exact times?

The hands do not overlap between 11 to 12, so only 22 overlaps per day. ---------------------------------. 1 of 2: Cjcarr2000 answer: --------------------------------- It occurs 12 times a day Overlap Time = Hour : (Hour * 5) 12:00, 1:05, 2:10, 3:15, 4:20, etc. -------------------- (MORE)

How many times do you have to wash your hands for your hands to be clean and for how long?

Answer Ideally, you should wash your hands at least two times for around two minutes. Then, dry with a clean paper towel and dispose of in into a bin. sing the national anthem 8 times Anytime you come in contact with any object orenvironment that may have germs, you should wash your hands. This (MORE)

How many times in a day the hands of a clock are straight?

the hands of a clock are straight 1)when they overlap & face the same direction & . 2)when the overlap & face opposite directions. this hapns evry 65 min(approx)for each case . =>in 1 day 24 hrs=> 24x60 min . therefore each case hapns (24x60)/65 times each day................= 22(approx). t (MORE)

How many times in a day do the hour hand and minute hand point in the same direction?

22 times in a 24 hour day. At midnight (the start of the day) the minute hand and hour hand are both pointing at 12. It will happen every 65 minutes 27.27 seconds. Here are the times: . 12:00:00 midnight . 1:05:27 . 2:10:55 . 3:16:22 . 4:21:49 . 5:27:16 . 6:32:44 . 7:38:11 . 8:43:38 . 9: (MORE)

What is overlapping?

an overlapping run is when one player has the ball and the other player cuts out to the outside and overlaps and the player who has the ball passes and shoot

What times of the day do the hour hand and minute hand of a clock form right angles?

This happens every 32 minutes, 43.64 seconds, starting at 12:16:22. Here are the times: 12:16:22, 12:49:05, 01:21:49, 01:54:33, 02:27:16, 03:00:00, 03:32:44, 04:05:27, 04:38:11, 05:10:55, 05:43:38, 06:16:22, 06:49:05, 07:21:49, 07:54:33, 08:27:16, 09:00:00, 09:32:44, 10:05:27, 10:38:11, 11:10:55 (MORE)

What is the speed of a clock's second hand?

A clock's second hand makes one complete revolution each minute. Thus, by definition, it is rotating at one revolution per minute or one RPM. That's its "rotational velocity" and it is the same no matter how big or small the clock might be. The actual velocity that the tip of the second hand might t (MORE)

How many times in one day does a clock hands form a 180 degree angle?

12.30; 1.35; 2.40; 3.45; 4.50; 5.55; 6.00; 7.05; 8.10; 9.15; 10. 20; 11.25 - AM 12.30; 1.35; 2.40; 3.45; 4.50; 5.55; 6.00; 7.05; 8.10; 9.15; 10. 20; 11.25 - PM Answer: 24 The answer, 24, is correct; however, the times stated are not correct, they are only approximate. For example, 12.30 is clo (MORE)

What time between 2 and 3 o' clock do the minute hand and the hour hand overlap each other?

If the hands start off together at midnight, say, then the minute hand must catch up with the hour hand 11 times before they both end up on top of each other again at noon. Hence between one and two o'clock the time when the hands are in the same place is 60/11 = 5.4545... minutes past the hour, b (MORE)

How many times can you wash your hands per day?

As many times as you want...BUT no matter how long or how many times you wash your hands, it will always have germs and never be completely clean. The average time is 20 seconds. The average amount for ME is 5 times.

How many times do the hands of a clock coincide in a day?

The hands of a clock coincide 11 times in every 12 hours (Since between 11 and 1, they coincide only once, i.e., at 12 o'clock). 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 (MORE)

How many times a day does a clock's hands overlap?

Answer is: the hour and minute hands overlap every 65 minutes. Interviewers often expect this answer because they do not think accurately. The exact times are: 0000 (12:00 AM) 0105 (01:05 AM) 0211 (02:11 AM) 0316 (03:16 AM) 0422 (04:22 AM) 0527 (05:27 AM) 0633 (06:33 AM) 0738 (07:3 (MORE)

What is mean by valve timing overlap?

Valve timing overlap is the time when both exhaust and intake valves are open most engines with catalytic converters require valve overlap in order to send a small amount of raw fuel/ air mix to the converter's. An "open cam" has valve overlap a "closed" cam does not

How many time hour hand and minutes hand cross?

22 times. hour hand meets minute hand each hour. Example : they meet at about 1h6, 2h17,... ( it's not exactly). But the 11th hour, they don't meet any times. So in a round of hour hand, it meets minute hand only 11 times and 22 times in a day

How can the days overlap in the new zodiac signs?

So many people are wondering! First, I'd like to say that if you are reading your horoscope, you will still read your original sign. If you were born a Pisces, you are still a Pisces; your sign is determined by the position of the sun the moment you were born. Since you can't change the sky or y (MORE)

Should there be a term like Overlapping Time?

…trying to do two or more things at a time! It saves time but can be a muddle.For example -He has been Overlapping Time (OT) - driving and talking over the mobile. can feel more 'Space of Time' by physiologically or pharmacologically slowing the heart rate! P.S. - Wrote on 16.3.08 to bb (MORE)

How many times a day does the hands of a clock form a straight angle?

Eight times in a day the hands of a clock form straight angle. At first they form straight angle when the hour amd minute hand are on 3 and 9 in noon and night. Second, when the hour amd minute hand are on 9 and 3 in morning and night. Third when the hour amd minute hand are on 12 and 6 At last whe (MORE)

How many times a day does a person need to do hand washing?

A person does not "need" to wash their hands on a daily basis, but rather "should" wash their hands whenever circumstances dictate. These instances include: . when your hands are dirty . before eating or touching food (like if you're helping cook or bake, for example) . after using the bat (MORE)

How many times in a day do the two hands in a clock are at right angle to each other?

I'm pretty sure the answer is 44 times (22 times per revolution of the hour hand). . 00:16 . 00:49 . 01:21 . 01:54 . 02:27 . 03:00 . 03:32 . 04:05 . 04:38 . 05:11 . 05:43 . 06:16 . 06:49 . 07:21 . 07:54 . 08:27 . 09:00 . 09:32 . 10:05 . 10:38 . 11:11 . 11:43 . 12:16 . 12:4 (MORE)

How many times does the minute hand of a clock overlap with the hour hand from 10am to 12am?

23 times in all. Note that from 11:00 to 11:59 (am or pm) the hands can never overlap. Thus from 10am to 11:59am, the hands will overlap just once at around 10:54am. The hands will overlap again at exactly 12:00pm (noon). And from 12:01pm to 12:am (midnight) the hands will overlap another 11 times. (MORE)

What is overlap?

verb verb: overlap ; 3rd personpresent: overlaps ; past tense: overlapped ; pastparticiple: overlapped ; gerund or present participle: overlapping ˌōvərˈlap/. 1 . . extend over so as to cover partly.. "the canopy overlaps the house roof at oneend". cover part of the same area (MORE)

How many times a day do the hour hand and minute hand of the clock lie in a straight line?

During the course of a full 24 hour day, the hour and minute hands of an analog clock lay in a straight line 22 times: 11 times in the morning and 11 times in the afternoon/evening, as follows: 12:32AM, 01:38AM, 02:43AM, 03:49AM, 04:54AM, 06:00AM, 07:05AM, 08:10AM, 09:16AM, 10:21AM, 11:27AM (MORE)

How many times to wash hands?

It is depends on us. Whenever we felt our hands are not clean or wewere picked up some dirty stuffs, after that we should wash ourhands.

How many times from one day midnight to the next midnight do the hands of a clock form perfect right angles?

Twice an hour, so in twenty-four hours, forty-eight times. Work it out with me: . 12:00 AM to 1:00 AM - at 12:15 and 12:45 . 1:00 AM to 2:00 AM - at 1:20 and 1:50 . 2:00 AM to 3:00 AM - at 2:25 and 2:55 . 3:00 AM to 4:00 AM - at 3:00 and 3:30 . 4:00 AM to 5:00 AM - at 4:05 and 4:35 . 5:00 AM (MORE)