Fractions and School Enrollments

Date: 02/26/2003 at 21:34:20
From: Leontia
Subject: Fractions

The sum of 1/2, 1/3, and 1/4 of the enrollment of the Business School 
is exactly the enrollment of the Language School. The sum of 1/5, 1/6, 
1/7, and 1/8 of the enrollment of the Business School is exactly that 
of the Design School. What are the enrollments of these three schools, 
assuming no school has more than 1000 pupils?

Date: 02/27/2003 at 10:23:58
From: Doctor Ian
Subject: Re: Fractions

Hi Leontina,

The first thing you can do is add the fractions.  If I tell you that
the sum of 1/2 and 1/4 of all Republicans support a particular policy, 
then you can conclude that

  1/2 + 1/4 = 2/4 + 1/4 = 3/4

of all Republicans support it, right? Same thing here. So you want to 
add 1/2, 1/3, and 1/4 to find out what fraction of the enrollment of 
the Business School is equal to the enrollment of the Language School.  
That will give you an equation like

  11/17 B = L

where B stands for the enrollment of the Business School, L stands for
the enrollment of the Language School, and 11/17 is a fraction that I
just made up. If you do the same thing again for the Design School,
you'll end up with a second equation that will look like

  13/24 B = D

Now, note that the enrollments must be integers. You can't enroll part 
of a student.

If B and L are both going to be integers, then B had better be a
multiple of 17. And if B and D are both going to be integers, then B
had better be a multiple of 24. So B would have to be a multiple of
both 17 and 24, which narrows things down considerably, doesn't it? 

Again, note that I made up these fractions. You have to get the real
ones yourself, by adding the fractions in the problem. If you're not
sure how to add fractions, take a look at these answers from the Dr. 
Math archives:

   Adding Fractions
   http://mathforum.org/library/drmath/view/60691.html 
   http://mathforum.org/library/drmath/view/61948.html 

Is this enough to get started? 

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

Date: 02/27/2003 at 21:18:42
From: Leontia
Subject: Thank you (Fractions)

Dear Dr Math,

Thanks a lot for helping to clear my doubts in Maths again. 
Thanks for taking the time to answer my question. Thank you!