You are watching: How many times a day do the hands on a clock overlap
Let"s very first analyze the difficulty of overlapping clock"s hour and minute hands.First set our imaginary clock come 12:00 midnight i beg your pardon is the beginning time because that the difficulties time period.
As you see at start time once the duration for calculation start we have an overlapping problem for clock"s hour hand and also minute hand.
Now assume the the clock"s hour and minute hands overlap for the next time.The very first overlapping after 12:00 o"clock will certainly happen in between 1 o"clock and also 2 o"clock.But we do not recognize the precise time right currently at this stage of the problem solution.
Let"s research the above situtation now. An initial of all, clock"s hour hand only relocated from position 12 come a position between 1 and also 2. What is exciting for the solution of this inteview concern is the at the exact same time period, the minute hand the the clock perfect a complete cycle starting from 12 come 12 and additionally traveled the same distance as hour hand in ~ the overlapping time.
Here is a drawing displaying the instance visually to assist you understand the hint for the solution of "how numerous times a work a clock"s hands overlap?".
We have actually one base mathematical equation because that this trouble that will lead united state to solution.The minute hand is 12 times faster than the hour hand.If girlfriend think of the time passing indigenous 12:00 o"clock come 1 o"clock, the minute hand travel 360 degrees. Meanwhile the hour hand travel 1/12 of 360 degrees.Or in a various thinking, in a certain time duration (t) minutes, the minute hand travels (360 * t) / 60 degreesBut at the same time the hour hand only travels 1/12 of that degree: (360 * t) / (60 * 12) degrees.
An other equation originates from the pass hours. Each passing hour the minute hand completes a full cycle.Let"s say the the hour hand travel (n) degrees. So we have the right to say at when they overlap each various other the minute hand travel (360 + n) degrees at the same time as hour hand.
So we have the right to now say that while hour hand travels (n) levels the minute hand will travel (12*n) degrees.The result that will outcome native this added equation with an initial mathematical equation will certainly be together follows:
12 * n = 360 + n
Of food 360 degrees is true if just 1 hour has actually passed.If two hours passed, then the formula will certainly be: 12 * n = 360*2 + nSo we deserve to re-formulate the equation as follows:
12 * n = 360 * h + n
Now replace n the degree the hour hand traveled over 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 can replace h with 1 and we can solve the equation for an initial overlap after 12:00 o"clock
t = 720 /11 = 65,45 minutes
There is an other tricky conversion right here the decimal component of the time. We require to convert it come seconds.
0,45 minutes = 45/100 minute = 45*60/100 seconds= 27 seconds
So the first overlap is at 65 minutes 27 seconds later on which means 1:05:27
Then us can continue calculation for the second overlap of clock"s hands. This will certainly occur in between 2 o"clock and also 3 o"clock.This means minute hand that the clock will travel 2 times complete circle and plus the very 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 occurs at " + hour + ":" + minute + ":" + secs + "");}
Within the loop, each overlap condition is calculated making use of the mathematical equation debated in ahead section.For loop structure consists of mod functions and also Mart.floor() attributes for converting the moment in minutes to hours:minutes:seconds display format.
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As a result, if you count the last case, there room 23 times as soon as a clock"s hour and also minute hand overlap in a day.If friend exclude among the midnights, we deserve to conclude the there space 22 times once a clock"s hand overlap in a day.