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How to calculate the total no. of days in 70 years (or any other no. of years) considering that this period also includes leap years?

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3 Answers

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It depends on which 70 years. There might be anywhere from 16 through 18 leap years. For example, the 70-year period from the beginning of 1901 through the end of 1970 contains 17 leap years (1904, 08, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68), but the period from the beginning of 1904 through the end of 1973 contains 18 leap years (the same as the other, plus 1972).

To get 16 leap years, one observes that (for example) there was an exception in 1900, which was not a leap year, so that the period from 1897 through 1966 contains only 16 leap years. (1904 through 1964, the one in 1900 having been skipped.)

The solution to the general problem is something like this: Let $Y$ be the number of years. Calculate $Y\cdot 365$ and then add the leap days. You can estimate the leap days by computing $Y\div 4$ and rounding off, but to get an exact count the simplest thing to do is to check the years one at a time. How you check the years depends on the calendar: in the Julian calendar, 1900 was a leap year, but in the Gregorian calendar, it was not.

In a different calendar, the answer will be completely different. For example, in the Islamic calendar, the common year has not 365 but 355 days, and the leap year has 356 days; the Hebrew calendar is different again.

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The average number of days in 70 years is $365.2425 \times 70 = 25 566.975$.

A more accurate answer depends on which 70 years you mean. The basic calculation is $365 \times 70 = 25 550$ plus the number of leap days (February 29), so the question now becomes how many leap days there are.

Recall that the Gregorian leap year rule is:

• Years divisible by 4 are leap years, except that
• Years divisible by 100 are not leap years, except that
• Years divisible by 400 are leap years.

This makes a cycle of 97 leap years every 400 years (which is how I arrived at the $\frac{97}{400} = .2425$ fractional part of the average number of days in a year).

Within each 400-year cycle, the number of days between a date (other than Feb. 29) in the year $Y$ and the same date in the year $Y + 70$ is:

• 25 566 days (16 leap days), for 102/400 years.
• 25 567 days (17 leap days), for 206/400 years.
• 25 568 days (18 leap days), for 92/400 years.

If you're working within a period like 1901-2099 when there are no 100-year leap-year exceptions, then the exact number of days can be calculated simply by considering $Y$ modulo 4:

• If it's 0 or 3, there are 18 leap days between 1 Jan $Y$ and 1 Jan $Y + 70$.
• If it's 1 or 2, there are 17 leap days.

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Actually, in the Gregorian calendar, it is a little more complicated than that! For example, the year 1600, though divisible by 4, was NOT a leap year because it is also divisible by 400. The year 2000, on the other hand, WAS a leap year because it was also divisible by 4000. (It was the first time that particular provision was used.)

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