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### How Much Popcorn did Paul Sell?

```
Date: 01/28/2002 at 08:50:16
From: Melissa
Subject: How much popcorn did he sell?

Paul made \$44.14 selling 27 items (beer and popcorn). If he made \$1.22
selling popcorn and \$2.62 selling beer, how many boxes of popcorn were
sold?

What I did...

I said that x(1.22) + n(2.62) = 44.14

Then I said that x+n = 27

Then I divided 27 by 2 because he sold two items, and I got 13.5

So then I said that he couldn't sell half an item, so I made the
pocorn 14 and the beer 13. 14(1.22) = 17.08 and 13(2.62) = 34.06 and
17.08+34.06 = 51.14, but that's too much because he only made \$44.14.

Help, please.
```

```
Date: 01/28/2002 at 10:04:39
From: Doctor Ian
Subject: Re: How much popcorn did he sell?

Hi Melissa,

You've set up exactly the right equations, although, can I make a
suggestion?  Whenever you're working with dollars and cents, consider
working with pennies, since then everything will be an integer.
Integers are generally the easiest kind of number to work with.

Note that you _could_ find the answer by trying other values for x
and n. For example, when you set x = 14 and n = 13, you end up with
too much money. Since beer is the more expensive item, you could get a
smaller total by selling fewer beers and more popcorn, so you might
set x = 15 and n = 12:

15(122) + 12(262) = 4974

which is still too big, but we're getting closer. If we keep
increasing x and reducing n, we'll eventually get to the answer:

14(122) + 13(262) = 5114

15(122) + 12(262) = 4974

16(122) + 11(262) = 4834
.
.
__(122) + __(262) = 4414

But a reasonable question to ask is: Is there an easier way to find
the answer? (One way of thinking about math is this: It's the study of
finding easier ways to get answers. Which is why everyone places so
much emphasis on it.)

Let's look at your equations:

122x + 262n = 4414

x +    n = 27

Dividing 27 by two would give you the average number of items sold,
but that's not really what you're after. What you want to find is
_exactly_ how many of each type that he sold.

To do that, you need to find a set of values for x and n that make
both equations true at the same time.  If that concept doesn't make
sense to you, you might want to take a look at

The Idea behind Simultaneous Equations
http://mathforum.org/dr.math/problems/laura.10.14.01.html

If the concept makes sense, but you just don't know how to solve the
equations, take a look at

Substitution and Elimination
http://mathforum.org/dr.math/problems/kathryn.01.13.02.html

Does this help?

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

```
Date: 01/28/2002 at 13:30:10
From: Melissa
Subject: How much popcorn did he sell?

Thanks!

19 popcorn and 8 beers :)
```
Associated Topics:
High School Basic Algebra
Middle School Algebra
Middle School Word Problems

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