How Much Water Evaporated?

Date: 06/19/2003 at 14:36:44
From: Amanda
Subject: Mixtures and percents

A substance is 99% water. Some water evaporates, leaving a substance 
that is 98% water. How much of the water evaporated?

We know that 50.5% of the water evaporates, but have no idea how to 
create a generalized equation for this problem.

Date: 06/20/2003 at 00:24:46
From: Doctor Ian
Subject: Re: Mixtures and percents

Hi Amanda,

Let's say we start with W units of water, and S units of other stuff.  
We originally have 99% water, so 

   W     99
  --- = ---
  W+S   100

Now we want to reduce the water to some fraction, F, of the original
amount.  And we want to end up with 98% water:

   FW     98
  ---- = ---
  FW+S   100

We can solve each of these equations for S:

       W     99
      --- = ---
      W+S   100


     100W = 99(W+S)

     100W = 99W + 99S

        W = 99S

      W/99 = S

and
  
      FW     98
     ---- = ---
     FW+S   100

    100FW = 98(FW+S)

    100FW = 98FW + 98S

      2FW = 98S

   2FW/98 = S
 
Two things equal to the same thing are equal to each other, so 

       W/99 = 2FW/98
 
       1/99 = 2F/98

  98/(99*2) = F

      0.495 = F

So 49.5% of the water remains, which means that 50.5% evaporated. 

Here's a similar problem. Suppose there are 198 men and 2 women in a
room. That is, the men make up 99% of the people in the room.

How many men have to leave for the percentage of men to drop to 98%?  

It's easier to see the answer if you concentrate on the women instead
of the men.  The women originally make up 1% of the people in the
room.  Later, they make up 2%.  

For 2 women to make up 2% of a room, there have to be 100 people in
the room. So there have to be 98 men. Which means 100 of the men had
to leave.

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