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 :)