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Filling Perfume Bottles

```
Date: 03/11/97 at 19:53:57
From: David
Subject: Problem Solving

Dear Dr. Math,

This is the problem I am stuck on:

Three girls have a 24-ounce bottle of perfume.  They want to divide
the perfume equally between them.  They have three empty bottles.  One
holds five ounces, another holds eleven ounces, and the third holds
thirteen ounces.  How can the girls divide the perfume so that each of
them has eight ounces in a bottle?

I don't know where to start.

Thanks, David
```

```
Date: 03/12/97 at 12:15:51
From: Doctor Wilkinson
Subject: Re: Problem Solving

Let's see what we can do. All we've got to work with is the four
bottles: the one that holds 5 ounces, the one that holds 11, the one
that holds 13, and the original bottle containing 24 ounces.  All we
can do is pour from one bottle to another, either the entire contents
or as much as will fit. (I'm just trying to make the rules a little
clearer.)  Now let's make a little table to keep track of where we
are.  I'll make a heading for each bottle, and then underneath I'll
put how much perfume is in each bottle at each step.

Since all the perfume is in the original 24-ounce bottle to begin
with, to start out with we have:

5       11      13      24
-       --      --      --

0        0       0      24

We've got to do something and there are only three choices.  We can
pour as much perfume as will fit into one of the three empty bottles.
Now we're looking to get 8 ounces somewhere.  If the 13-ounce bottle
is full, we can fill the 5-ounce bottle from it, and that will leave
8 ounces. That would be progress. So let's fill the 13-ounce
bottle from the 24-ounce bottle, and then fill the 5-ounce bottle from
the 13-ounce bottle.  That adds two rows to our table:

5       11      13      24
-       --      --      --

0       0       0       24      starting position

0       0       13      11      24->13

5       0        8      11      13->5

Now the trick of pouring from the 13-ounce bottle to the 5-ounce
bottle worked pretty well, so I'd like to use it again.  So let's get
the 8 ounces out of the 13-ounce bottle so I can use it again:

5       11      13      24
-       --      --      --

5        8       0      11      13->11

I need to empty the 5-ounce bottle so I can use it:

5       11      13      24
-       --      --      --

0        8       5      11      5->13

and refill the 13-ounce bottle:

5       11      13      24
-       --      --      --

0        8      13       3      24->13

Can you see how to finish this now?

-Doctor Wilkinson,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
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