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How many times the 29th of a month occur in 400 consecutive years?

How many times the 29th of a month occur in 400 consecutive years?

Grade:10

8 Answers

Ashwin Muralidharan IIT Madras
290 Points
12 years ago
Hi Sathyaram,
Consider a set of every four years (So as to include the leap year with Feb 29).
So in 3 years, 29 will occur 11 times, and once it would accur 12 times.
So in 4 years, there will be 3*11 + 12 = 45 days in which you will have 29th.
So in 400 years, it would be 4500 times.
Hope that helps.
All the best,
Regards,
Ashwin (IIT Madras).
Chilukuri Sai Kartik
58 Points
12 years ago
Remove february first.
So we have got 11 months every yr with 29th in each of them.
So 11x400=4400
Now we have 1 out of every 4 Februaries with 29th.
So 100 will be there in 400 yrs.
So the answer is 4400+100
So We end up with 4500
Plz approve my answer if you like...
Ashwin Muralidharan IIT Madras
290 Points
12 years ago
Hi Sathyaram,
As in every 400 years. The concept to be noted here is that, there would be 97 leap years and 303 Non-Leap years.
Any year that is a multiple of 100, will not be a leap year, unless it is also a multiple of 400.
So years like 100,200,300 are not leap years.
So 4500-3 = 4497 Days with date as 29th.
Best Regards,
Ashwin (IIT Madras).
Abhigyan Sinha
11 Points
7 years ago
Would like to have a better explanation of why 4497.
Why will 100, 200, 300 years not be leap years
Ashwin
11 Points
7 years ago
Since only a year divisible by 400 is considered as a leap year( in case of 100`s).So in the span of 400 years- only the last century will have 25 leap years, rest three centuries will have 24 leap years each. Hence 29 occurs :24×3+25= 97 times in all leap years in 400 years.Nd 29 occurs 11times in an ordinary year. So11×400= 4400 Adding both4400+97= 4497- answerI hope that helps.Regards,Ashwin(DTU)
Sushant Kumar
11 Points
5 years ago
Hey, Ashwin 
100,200 and 300 are already counted as ordinary year(not leap year) .Then why you are subtracting 3 from 4500 again.
My answer is 4500.
Please correct me if I am wrong .
Rahul
13 Points
5 years ago
Only Ashwin is correct here. 100, 200 and 300 are not considered leap years due to the actual extra time in one year that is not generally considered anywhere.
In specified terms, the earth actually makes a complete revolution in 365 days, 5 hours, 49 minutes and 16 seconds i.e, 365.2425 days.
0.2425*4=0.97 days. 
So when we add one day to a leap year in every four years, we are actually adding another 0.03 days so that it becomes a day. And in 400 years, it is adding 3 days in total.
So to compensate for this, three leap days must be taken out during that 400 year period. The Gregorian calendar reform implemented this adjustment by making an exception that there is a leap year every four years.
Now, a year divisible by hundred is not considered a leap year, unless it is also divisible by 400, and hence the answer given by Ashwin.
I’m currently preparing for IIT and hope to get into the Madras branch :)
Regards
Rahul
 
Kushagra Madhukar
askIITians Faculty 628 Points
3 years ago
Dear student,
Please find the attached solution to your question.
 
Consider a set of every four years (So as to include the leap year with Feb 29).
So in 3 years, 29 will occur 11 times, and once it would accur 12 times.
So in 4 years, there will be 3*11 + 12 = 45 days in which you will have 29th.
So in 400 years, it would be 4500 times.
 
Hope that helps.
Thanks and regards,
Kushagra

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