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### Counting Days of the Week

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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.
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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/
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Associated Topics:
Elementary Division
Elementary Number Sense/About Numbers
Elementary Puzzles
Middle School Division
Middle School Number Sense/About Numbers
Middle School Puzzles

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