You are watching: How many times a day clock hands overlap
Let"s first analyze the trouble of overlapping clock"s hour and minute hands.First set our imaginary clock come 12:00 midnight which is the beginning time for the difficulties time period.
As you watch at begin time when the duration for calculation starts we have an overlapping condition for clock"s hour hand and minute hand.
Now assume that the clock"s hour and also minute hand overlap because that the following time.The very first overlapping ~ 12:00 o"clock will certainly happen between 1 o"clock and 2 o"clock.But we do not recognize the exact time right now at this phase of the problem solution.
Let"s examine the above situtation now. Very first of all, clock"s hour hand only relocated from place 12 come a position between 1 and also 2. What is interesting for the solution of this inteview concern is that at the same time period, the minute hand that the clock completed a complete cycle starting from 12 come 12 and in addition traveled the same distance as hour hand at the overlapping time.
Here is a drawing displaying the situation visually to assist you understand the hint for the solution of "how plenty of times a work a clock"s hand overlap?".
We have one basic mathematical equation for this difficulty that will certainly lead united state to solution.The minute hand is 12 times faster than the hour hand.If you think of the time passing from 12:00 o"clock to 1 o"clock, the minute hand travels 360 degrees. Meanwhile the hour hand travels 1/12 that 360 degrees.Or in a various thinking, in a specific time duration (t) minutes, the minute hand travel (360 * t) / 60 degreesBut in ~ the exact same time the hour hand only travels 1/12 of the degree: (360 * t) / (60 * 12) degrees.
An other equation comes from the pass hours. Every passing hour the minute hand completes a complete cycle.Let"s say the the hour hand traveled (n) degrees. For this reason we have the right to say at as soon as they overlap each various 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 travel (n) degrees the minute hand will take trip (12*n) degrees.The an outcome that will outcome indigenous this additional equation with an initial mathematical equation will be as follows:
12 * n = 360 + n
Of course 360 levels is true if just 1 hour has passed.If two hours passed, then 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 change n the level 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 very first hour we can replace h with 1 and we deserve to 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 part of the time. We need to convert it to seconds.
0,45 minute = 45/100 minute = 45*60/100 seconds= 27 seconds
So the first overlap is at 65 minutes 27 seconds later which way 1:05:27
Then we can proceed calculation because that the 2nd overlap that clock"s hands. This will occur in between 2 o"clock and also 3 o"clock.This method minute hand the the clock will travel 2 times complete circle and plus the exact same amount together 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 + ":" + minutes + ":" + seconds + "");}
Within the loop, each overlap condition is calculated utilizing the mathematics equation discussed in ahead section.For loop structure includes mod functions and Mart.floor() features for converting the moment in minute to hours:minutes:seconds display format.
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As a result, if you counting the critical case, there room 23 times once a clock"s hour and also minute hands overlap in a day.If girlfriend exclude one of the midnights, we have the right to conclude the there room 22 times once a clock"s hand overlap in a day.