Expert: Steve Holleran - 6/7/2007

Question
If water costs .50 cents a bottle, soda pop cost $3.00 a bottle, and beer costs $10.00 a bottle. How many of each do need to get exactly 100 bottles costing exactly $100.

Answer
Hi Luke,

Yeah, I can see where this would be annoying.  The cause of the annoyance is that there are three variables but only two conditions.  Normally, you would need as many conditions as there are variables.

However, here we have something else to help us out.

First, let's set up two equations:

Let w = # bottles water
   s = # bottles soda
   b = # bottles beer (this is pretty expensive beer!!!)

Okay, then as far as the total number of bottles go, we have:

            w + s + b = 100   (I)

Then, we also have some value information:

       0.50w + 3s + 10b = 100 (II)

Okay, now let's multiply equation (II) by 2

        1w + 6s + 20b = 200

And subtract equation (I) from it:

    (II) :   1w + 6s + 20b = 200

    -(I) :  -w  -s    -b   = 100

Gives              5s + 19b = 100

Now normally, we'd have a problem, but here, we are helped out by the fact that s and b have to be positive whole numbers.  If you focus on the 19b term, whatever b is, when you multiply it by 19, and subtract from 100, the number left must be divisible by 5 in order for s to be a whole number.

You can go through all the possibilities, if you like:

19 * 1 = 19        19 * 2 = 38 , etc, but I think you'll see that only 19 * 5 = 95 gives us an answer so that the remainder from 100 is divisible by 5.

So, then if b = 5, 5s + 95 = 100  means s = 1.

Then since s = 1 and b = 5, w = 100 - (5+1) = 94.

This would suggest 94 bottles of water
                   5 bottles of beer
                   1 bottle of soda.

Check:   94(.50) + 1(3.00) + 5(10.00)

      =   47.00 + 3.00  + 50.00 = 100.00

I hope my explanation was clear to you.

Steve Holleran