Five Bags of Nuts

Date: 04/17/2002 at 16:33:43
From: Candy De Loera
Subject: The Bag of Nuts 

George Crackham put five paper bags on the breakfast table. On being 
asked what they contained, he said:

"Well, I have put a hundred nuts in these five bags. In the first and 
second there are altogether fifty-two nuts; in the second and third 
there are forty-three; in the third and fourth, thirty-four; in the 
fourth and fifth, thirty."

How many nuts are there in each bag?

Date: 04/17/2002 at 17:05:21
From: Doctor Ian
Subject: Re: The Bag of Nuts 

Hi Candy,

Here are the bags, with the combined sums:

               43          30
            _________   _________
           /         \ /         \
     [ a ] [ b ] [ c ] [ d ] [ e ]
     \_________/ \_________/
         52           34

Translating to equations, we get

  a + b             = 52

      b + c         = 43

          c + d     = 34

              d + e = 30

That looks like a mess, doesn't it? However, let's look at that fourth 
equation. We can rewrite it this way:

  d = 30 - e

Why do we care? Because this means that we can substitute for d in the 
third equation:

           c + d = 34

    c + (30 - e) = 34

               c = 34 - (30 - e)

                 = 34 - 30 + e

                 = 4 + e

And now we can substitute for c in the second equation, and if you 
keep going, eventually you can find all the other variables in terms 
of e. 

How does that help you? Remember, all the bags add up to 100 nuts.  So 
that gives us one final equation:

  a + b + c + d + e = 100

If you substitute in this equation for all the other variables, e.g., 

  a + b + [4 + e] + [30 - e] + e = 100

you'll end up with a single equation with one variable: e.  Solve 
for e, and you can find the value of all the other variables.  

Does this help? 

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

Date: 04/19/2002 at 18:59:28
From: Candy De Loera
Subject: The Bag of Nuts 

Dear Dr. Math,

Yes, the equation did help. The only thing that was used to solve the 
equation was a lot of substitution and simple algebra. It was quite 
useful because I saw how you interpreted the word problem into an 
equation that I could apply to other similsr problems in the future.
Thank you very much. :)

Sincerely,
Candy De Loera