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 do you compare mortgage rates?

Mortgage rates fluctuate day-by-day based on a variety of variables. A comparison of rates between markets and states are even more difficult when there are point differentials.

Where can you get a list of bank mortgage rates?

You can get a list of bank mortgage rates directly from the bank that you want to work with. You can also get the current mortgage rate from a realtor, but these rates will change periodically.

What is the average bank mortgage rate?

Current mortgage rates are around 3.75%. You will need to check with your local banks for more specific information, since rates vary based on area, amount of loan, and amount of down payment.

What is a reliable mortgage rate calculator?

A reliable mortgage rate calculator lets you know exactly how much money you can borrow against your house. It will also take into account how long of a term you are planning on signing for.

How do you calculate your mortgage rate?

You can use a clculator to add or takeawayor times ,shares and other maths. to add these them.write it on a bit of papper to see if the mortgageis high. you always check your bank.before you buy new mortgage.

What do you need to calculate your mortgage and the rates?

First you will need a copy of your income, bills, and mortgage itself. Next you will need to find the rates specific to your location. This can be done by seeing a mortgage broker or watching you local news.

What is the current rate of California mortgage?

In California one can get 30 year fixed loan from 4.125% or a 15 year fixed loan from 3.250%. Percentages may change depending on amount of loan, down payment, bank you are loaning from, annual income, and other factors.

What mortgage rates does Chase Mortgage offer?

Chase Mortgage offers competitive mortgage rates. You can complete a simple calculator and find your own personal rates. For example a 30 year fixed loan would be 4.625, a 15 year fixed loan would be 3.625, and so on.

What is different about fixed rate mortgages?

The most important difference is the interest rate on the note remains the same through the term of the loan, instead of a common loan where the interest rate may adjust.

What kind of home mortgage rates are there?

There are different rates depending on the institution you decide to go with. A mortgage broker can instruct you on what the rates are if you contact them.

What is the best rate of home mortgages?

There is no set best rate for home mortgages. You should shop around and see what offers you receive from different lenders and find out which is the best for you out of all of them.

What does the lowest mortgage rates mean?

The lowest mortgage rates refer to when a mortgagor approves a all time low rate to be paid by the mortgagee. This basically means that you'll be paying less out of your pocket each month.

How can you adjust and refinance your mortgage rates?

You can adjust and refinance your mortgage rate by shopping around for the best rates in your area. By checking the current rates of local banks narrow down the lowest rate. After that visit the bank and talk to their loan officer about refinancing possibilities.

What refinance mortgage rates do you qualify for?

The only way to find out is to check with your local bank or lender! They will use your income, assets, liabilities and credit score to determine your rate. The better these items look, the lower your rate will be.

What are reasonable refinance rates for a mortgage?

Reasonable refinance rates for a mortgage is quite complex. It actually depends on what type of mortgage you wish to lock into. For example, a 5 year variable rate can start as low as 2.5%, where as a 5 year fixed rate can start as low as 3.79%.

What are mortgage interest rates at woolwich?

The mortgage rates at Woolwich can vary depending on the type of loan you get. For example, for a 2 year fixed remortgage, the interest rate is 3.5%. For a 2 year fixed Barclay's loyalty mortgage, it is 3.6% APR. A 5 year capped rate mortgage has an interest rate of 3.3% APR.

What are the interest rates with a pnc mortgage?

For PNC mortgage, it will all depend on how much money you're paying. The more money you are taking (as a loan) the higher your overall intrest rate will become.

What are typical house mortgage rates?

At this time typical mortgage rates are around 4.465% for a 30 year fixed loan. Though as we all know can vary depending on location and credit rating.

What is the average mortgage loan rate?

As of August 23, 2011, the average 30 year fixed mortgage rate is 4.55%. Consumers also may be interested to know that a 15 year fixed mortgage currently yields an average national rate of just 3.83%.

Are mortgage rates still in the news?

Mortgage rates are often in the news. The recent recession and continuing fears about the economy mean that changes in mortgage rates are a hot topic for reporters.

Who has the lowest mortgage rate currently?

"According to JD Powers, Speedtrack Loans and Quicken have the lowest mortage rates currently with the major players such as Bank of American and Citibank coming next in line."

Where can you find the lowest mortgage rate?

Mortgage rates change on a daily basis so it is best for you to search around. You can typically find low rates through your own bank or credit union. There are several websites that will allow you to compare rates from numerous mortgage companies.

Why si there second mortgage rates?

"Second mortgage rates are for people who already have a first mortgage out and need the money for bills. Or, sometimes if there is an emergency and they don't have the money to cover it, they will take a second mortgage out."

What are the home mortgage rates in Chicago?

The home mortgage rates in Chicago vary from area to area and are dependent upon the type of home you are thinking of owning or already own. You can contact a mortgage representative to learn more about home mortgage rates in your specific area.

What is the average mortgage rate in Texas?

"The average mortgage rate ranges from 3.2 to 4.1% in Texas, depending on the mortgage company you choose and your current credit score. Also, it may vary with the type and area of property you are looking at."

What are the interest rates for Amerisave mortgages?

"Currently interest rates (as of 08/30/2011) for Amerisave mortgages range from 3.750 (3.923 APR) for a 30 year fixed to 1.875 (3.127 APR) for a 3-year ARM (Adjustable Rate Mortgage); rates for FHA loans are from 3.625 (4.365 APR) for a 30 year fixed rate loan to 3.000 (3.659) to a 15-year fixed loa ( Full Answer )

What are the benefits of second mortgage rates?

"There are many benefits when having a second mortgage. A few are that you can get bigger loans because your house is the loan security, you also start to build up home equity, and maybe pay off some bills or use the money however you need too!"

What can happpen with an adjustable- rate mortgage?

The most obvious thing that can happen is that your interest rate can rise and fall either monthly, quarterly, annually or in otherwise determined sets of years (5 years, 10 years etc). This can occur from the inception of the loan or can occur after an initial fixed rate period expires. These rate ( Full Answer )

What are the rates for loans with Halifax Mortgages?

The rates of mortgages are running 3.48% right now. If choosing Halifax Mortgages one will have to pay a little higher rate to 3.78% if they have a low credit score it will be a little higher.

What is the mortgage rate calculator used for?

A mortgage rate calculator will take a person's mortgage loan amount and the interest rate associated with the loan and give you an estimated payment rate. Normally, an estimated monthly payment rate.

How can the best mortgage rates be found?

The lowest rates for an array of the financial tools can be found at the bankrate website. Local credit unions and banks are also a great source for finding competitive rates.

What is the average rate for a mortgage loan?

The average rate for a mortgage loan would depend on the rates for the day that you are inquiring. Rates may change on a daily basis. Today's average rate is 3.25%.

What are the typical home mortgage rates?

The mortgage rates you pay are defined by what type of mortgage you have. If you have a fixed rate mortgage, you'll pay the interest rate which existed when you signed the deal. A tracker rate mortgage will track to the current base rate in the economy.

What is the average house mortgage rate?

There are many factors that can play into your house mortgage rate such as age and credit history as well as the size of your loan. On average, a mortgage will run you about 3% to 4.5%

What exactly is a mortgage competitive rate?

A mortgage competitive rate is one that is in line with what the market will bear. For example, if the going rate for a 30 year fixed loan is 4.5%, then a competitive rate would be one that is very close to that percent (4.4% to 4.6%).

What are the mortgage rates in the UK?

Current mortgage rates in the UK range from 1.75% offered by HSBC to as high as 4.49% which is offered by Santander. Most of these rates are initially low and then go up 1-3 percentage points over the course of the loan period.

What is the interest rate on a Vanderbilt mortgage?

Vanderbilt specializes in helping with mortgage financing for those with low, slow, poor or bad credit so their rates are extremely high. The average person looking to finance a home for $100,000.00 will end up repaying roughly $400.000.00 over the term of the mortgage. Their rates are anywhere fr ( Full Answer )

Where can mortgage rates be found in Jacksonville?

Mortgage rates can be found in Jacksonville on the websites Zillow and Trulia. They have all the rates and real time quotes. You can customize your quote by filling out a short form with your information.

What are the current mortgage interest rates?

Current mortgage rates vary depending on the mortgage company and the length of the mortgage. As of today the average rate for a 30 year fixed term mortgage is 3.61%. A 15 year fixed term mortgage is 2.85%.

What are typical Colorado mortgage rates?

The average price for a house in Colorado is substantially higher than that of the average for the entire United States. Some of the available mortgages in Colorado include fixed rate mortgages, adjustable rate mortgages and home equity loans. The average for a 30 year fixed mortgage is 3.65% while ( Full Answer )

What are average mortgage rates in the US?

According to the website Bankrate, the average mortgage rate in the United State for the last twenty years is 3,65%. According to Ycharts, said average is of 3,42%.

What is the average California mortgage rate?

The average California mortgage rate is as low as 3.250 percent for a 30 year term. That rate is from "loan safe" and is one day old. Other rates were found claiming that the average was 3.43 percent, which even though it is slightly higher, will make a big difference over the years.

What are the benefits of a fixed rate mortgage?

The homeowners know almost to the penny what they owe, it is very secure, and they feel good.Interest rate is low, various benefits. Some may not choose this because their rate/plan/mortgage is better or they do not like this one.

What is the concept of jumbo mortgage rate about?

The concept of a jumbo mortgage rate is about having a larger rate so that you can pay it off faster. In the long run however, you will be paying more money to the bank.

What mortgage rates are charged by RBC?

The RBC website offers a mortgage rate calculator and one could also get the information by contacting the Royal Bank at their toll free number, at 1-800-769-2511.

What are the advantages of fixed rate mortgages?

The biggest advantage of a fixed rate mortgage is that you never have to worry about your interest rate being raised. This helps you to budget since you always know what your payment will be.

What is the current prime mortgage rate?

There are many ways one might learn the current prime mortgage rate. The most reputable resource one might use to find current rates is one's local financial institution.