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### 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.

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
```
Associated Topics:
Middle School Algebra
Middle School Word Problems

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