You are watching: How many times do a clock's hands overlap in a day
Let"s first analyze the problem of overlapping clock"s hour and also minute hands.First set our imagine clock come 12:00 midnight which is the beginning time for the troubles time period.
As you watch at start time once the duration for calculation start we have an overlapping condition for clock"s hour hand and minute hand.
Now assume the the clock"s hour and also minute hands overlap because that the next time.The an initial overlapping ~ 12:00 o"clock will certainly happen in between 1 o"clock and also 2 o"clock.But we do not understand the exact time right currently at this stage of the trouble solution.
Let"s examine the over situtation now. An initial of all, clock"s hour hand only moved from position 12 to a position in between 1 and also 2. What is amazing for the equipment of this inteview question is the at the same time period, the minute hand that the clock perfect a full cycle starting from 12 to 12 and in addition traveled the same distance as hour hand in ~ the overlapping time.
Here is a illustration displaying the case visually to assist you recognize the hint for the equipment of "how plenty of times a work a clock"s hands overlap?".
We have one base mathematical equation for this trouble that will certainly lead us to solution.The minute hand is 12 times much faster than the hour hand.If you think of the moment passing from 12:00 o"clock come 1 o"clock, the minute hand travel 360 degrees. On the other hand the hour hand travels 1/12 the 360 degrees.Or in a various thinking, in a specific time period (t) minutes, the minute hand travel (360 * t) / 60 degreesBut at the exact same time the hour hand just travels 1/12 of the degree: (360 * t) / (60 * 12) degrees.
An various other equation comes from the passed hours. Every passing hour the minute hand completes a full cycle.Let"s say that the hour hand traveled (n) degrees. Therefore we deserve to say at when they overlap each other the minute hand travel (360 + n) degrees at the same time together hour hand.
So we have the right to now say the while hour hand travels (n) levels the minute hand will travel (12*n) degrees.The result that will certainly outcome native this added equation with very first mathematical equation will be together follows:
12 * n = 360 + n
Of food 360 levels is true if just 1 hour has passed.If two hours passed, climate the formula will certainly be: 12 * n = 360*2 + nSo we deserve to re-formulate the equation together follows:
12 * n = 360 * h + n
Now replace n the degree the hour hand traveled with time t, (360*t)/(60*12)
12 * (360*t)/(60*12) = 360 * h + (360*t)/(60*12)11 * (360*t)/(60*12) = 360 * h11 * t / 2 = 360 * h11 * t = 720 * h
For an initial hour we have the right to replace h v 1 and we have the right to solve the equation for an initial overlap after ~ 12:00 o"clock
t = 720 /11 = 65,45 minutes
There is an various other tricky conversion here the decimal component of the time. We need to transform it to seconds.
0,45 minutes = 45/100 minutes = 45*60/100 seconds= 27 seconds
So the very first overlap is at 65 minute 27 seconds later on which way 1:05:27
Then us can continue calculation because that the 2nd overlap of clock"s hands. This will certainly occur in between 2 o"clock and 3 o"clock.This method minute hand the the clock will travel 2 times full circle and plus the same amount as hour hand.
var t, n, temp1, temp2;var hour, minutes, seconds;for (t = 0; t n = 720 * t / 11;var n = Math.round(n * 100) / 100;hour = Math.floor(n / 60);minutes = Math.floor(n) % 60;temp1 = n * 100;temp2 = temp1 % 100;seconds = Math.floor((temp2 * 60) / 100);document.write( (t+1) + ". Time overlap wake up at " + hour + ":" + minute + ":" + secs + "");}
Within the loop, every overlap condition is calculated making use of the math equation questioned in previous section.For loop structure has mod functions and also Mart.floor() functions for converting the moment in minutes to hours:minutes:seconds display screen format.
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As a result, if you count the critical case, there room 23 times as soon as a clock"s hour and minute hand overlap in a day.If you exclude one of the midnights, we deserve to conclude that there room 22 times once a clock"s hand overlap in a day.