Counting Students

Date: 10/06/98 at 12:57:13
From: kerry
Subject: Say What?!  

The principal asked Mr. Black how many students there were in his non-
graded math class. "Well," Mr. Black replied, "3/4 of them are less 
than 16 years old, 2/3 are less than 15 years old, 12 are not yet 
14 years old, and there are twice as many between the ages of 14 
and 15 as there are between the ages of 15 and 16."

The principal murmered his thanks and went off to determine how many 
students there were altogether. After some thought he was able to 
figure out the correct mumber. Re-construct his logic.

Date: 10/06/98 at 17:28:03
From: Doctor Rick
Subject: Re: Say What?!  

Hi, Kerry. I have to tell you that I first solved this one using some
algebra, but you probably haven't started algebra yet, so I'll show 
you another way.

Let's make a diagram. Divide the class into four groups: 

   under 14, 
   14 to 15, 
   15 to 16, and 
   over 16:

   +---------+---------+---------+---------+
   |         |         |         |         |
   |   12    |         |         |         |
   |         |         |         |         |
   +---------+---------+---------+---------+
            14        15        16
    <====== 2/3 ======>
    <============ 3/4 ==========>

I put down most of the information you have: 2/3 of the students are 
in the first two boxes, 3/4 are in the first 3 boxes, and 12 students 
are in the left box. The other piece of information is that box 2 has 
twice as many as box 3.

If you answer these questions, they should lead you out of the maze 
you're lost in.

1. What fraction of the students are in box 4? (Think: what's left?)
2. What fraction are in box 3? (Subtract fractions.)
3. Can you use the last piece of information to find the fraction in 
   box 2?
4. Now what is the fraction in box 1?
5. Can you see your way out now?

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