Leap Day Math

Every four years, during a leap year, the imperfect match between the length of a calendar year and Earth's orbit result in an interesting calendar adjustment known as a leap day. In this problem set, students calculate the difference between the calendar year and Earth's orbital period, determine how much extra time gets added to our calendar and identify which years omit leap years.

• This problem set is ideal for before, during or after leap day, but it can be used at any time of any year.
• Students can complete the problem set independently, in pairs or small groups, or together as a class, depending on student abilities.

The length of a year is based on how long it takes a planet to revolve around the Sun. Earth takes about 365.2422 days to make one revolution around the Sun. That's about six hours longer than the 365 days that we typically include in a calendar year. As a result, every four years we have about 24 extra hours that we add to the calendar at the end of February in the form of leap day.

• What would happen if Leap Days weren't inserted into the calendar every four years?
• What would happen if Leap Days weren't omitted from the calendar every 100 years?

1. Earth revolves around the Sun in approximately 365.2422 days. How long is that in days, hours, minutes and seconds?
   
   0.242 * 24 hours = 5.8128 hours
   0.8128 * 60 minutes = 48.768 minutes
   0.768 * 60 seconds = 46.08 seconds
   Answer: 365 days, 5 hours, 48 minutes, 46 seconds
   
2. After the Earth orbits the Sun four times, how many extra hours, minutes and seconds have built up that need to be balanced out by a leap year?
   
   4 * 0.2422 days = 0.9688 days
   0.9688 * 24 hours = 23.2512 hours
   0.2512 * 60 minutes = 15.072 minutes
   0.072 * 60 seconds = 4.32 seconds
   Answer: 23 hours, 15 minutes, 4 seconds
   
3. What is the difference in hours, minutes and seconds between what is added to the calendar every four years with leap day and what is actually needed?
   
   1 day minus 23 hours, 15 minutes, 4 seconds
   Count on from 23 hours, 15 minutes, 4 seconds to get to 24 hours.
   Answer: 44 minutes, 56 seconds
   
4. With the extra, unneeded time that has been added to the calendar every four-year leap cycle, how much extra time would be added to the calendar every 400 years?
   
   44 minutes, 56 seconds * 100 = 4400 minutes, 5600 seconds
   5600 seconds / 60 = 93.33 minutes
   4400 minutes + 93.33 minutes = 4493.33 minutes
   4493.33 minutes / 60 / 24 = 3.12 days
   Answer: 3.12 days
   
5. Because of the extra time that gets added to the calendar over a 400-year span, years that are divisible by 100 omit leap day unless they are also divisible by 400. Which years in the past 500 years and the next 500 years have omitted leap day?
   
   Answer: 1700, 1800, 1900, 2100, 2200, 2300, 2500