Counting Days of the Week

Date: 02/02/99 at 22:26:08
From: Joel
Subject: Math

If today is Thursday, what day would it be 120 days from now?

I have figured it using different months to start and then counting to 
120 days. I feel the answer could be different each time depending on 
the number of days in each month. I sure would like to figure out this 
problem mathematically. Thank you.

Date: 02/03/99 at 11:16:28
From: Doctor Erin
Subject: Re: Math

Hi Joel!

This is an interesting problem. I understand why you think that the 
answer might depend on the number of days in the month. Whether this is 
true depends on what exactly the question means. If "what day it would 
be" means what date it would be, like the 13th or the 22nd, then it 
certainly would depend on the months.

However, if we assume "what day would it be" means what day of the 
week, like Monday or Friday, then the month (the number of days in the 
month) does not matter.  

For example: if the 28th of October is a Wednesday, 9 days later would 
be the 6th of November, which is a Friday. In the same way, if the 28th 
of April is a Wednesday, 9 days after that is the 7th of May, which is 
also a Friday. This shows you that the number date depends on the 
month, but the day of the week doesn't, even though April and October 
have different numbers of days. We know that nine days after any 
Wednesday will be a Friday. Try a few more examples to make sure you 
understand how this works.

You were right, though; there is an easier way to do this problem than 
counting:

Think about any Thursday. How many days is it until the next Thursday? 
I hope you got seven. Now, seven more days is a Thursday again.  Check 
this out several more times. You will see that 7 days, 14 days, 21 
days, ... after the first Thursday are all Thursdays as well! All we 
are doing here is counting by weeks at a time instead of days. Notice 
that 14 = 7*2, 21 = 7*3, and this continues for 7*4, 7*5, etc.  

So, if we want to find what day of the week it will be 120 days after a 
Thursday, we can simplify this by "counting" the Thursdays (or weeks) 
between the original one and the day we are looking for. We would do 
this by dividing the number of days we are counting by 7 to find the 
number of weeks, and then using the remainder to count to the day we 
are looking for.

I will solve a similar problem to make sure you are on the right track 
for your question. For example, if we wanted to find what day of the 
week it would be 45 days after a Tuesday, this is how we would do it:

        45/7 = 6 (remainder 3)

This means we are counting 6 weeks and 3 days from the Tuesday. Six 
weeks from the Tuesday will still be a Tuesday, and if we count forward 
three extra days, we find that it is a Friday. Now we know that 45 days 
after any Tuesday is a Friday.

You can use this example to find out what day it will be 120 days after 
any Thursday.

Good luck Joel, and write back if you have any questions.

- Doctor Erin, The Math Forum
  http://mathforum.org/dr.math/