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
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