Question

-One revolution of the Sun by the Earth requires 365 days, 5 hours, 48 minutes and 49.7 seconds. the 5 hours, 48 minutes and 49.7 second is approximated to 1/4 of a day. Every 4th year, one full day is added( leap year). How much of an error in minutes and seconds does this approximation produce every 4 years?( This time difference is corrected at some later date.)

-Find the result when the largest 4 digit number, all of whose digits are different, is subtracted from the smallest 5-digits number, all of whose digits are different .

Answer 1

first: what is the difference in 5h48'49.7" and 6hr?
answer: 11'10.3" check that. In four years, it is four times that.

second. smallest 5 digit number dealing with digits 0,1,2,3,4, and leading digit not zero: 10234
largest 4 digit number: 9876

Answer 2

5 hrs, 48 min, 49.7 sec
= 5/24 + 48/(24x60) + 49.7/(24x60x60)
= .2422442 of a day error

in 4 years that would be .9689676
so when we add a day we have a difference of
1 - .9689676 = .0310324 of a day off every 4 years

which is equal to 44 minutes and 41.2 seconds

BTW, the current rule is this:
If the year is divisible by 4, then you have a leap year UNLESS, the year was a century whose first 2 digits were not divisible by 4, then you did not have a leap year, otherwise the century would also be a leap year

Thus in 1900 we did not have a leap year
but in 2000 we did.

There is something called "Zeller's Congruence"

http://en.wikipedia.org/wiki/Zeller's_congruence

which uses the above rules to calculate what day of the week any particular date is. I used to assign my computer science class to make up a program than has as input the date of your birthday, and it was to tell you if that was a Monday, Tuesday, etc.