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### 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/
```
Associated Topics:
Middle School Ratio and Proportion
Middle School Word Problems

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