Sue Pinion, Lisa Stueve, Sandy McAdams

Cans and Bottles

#### Three neighbors named Quincy, Penny, and Rosa took part in a local recycling
drive. Each spent a Saturday afternoon collecting all of the aluminum cans
and glass bottles he or she could. At the end of the afternoon each person
counted up what he or she had gathered. They discovered that even though
Penny had collected three times as many cans as Quincy and Quincy had collected
four times as many cans as Rosa, each had collected exactly the same number
of items, and the three as a group had collected exactly as many cans as
bottles. Added together, the three had collected fewer than 200 items in
all. How many cans and bottles did each collect?

Solution:

The first attempt to solve this problem results in a labeling of unknowns
in the traditional algebraic sense. Let x= # cans collected, y= # bottles
collectes, x = y, and x + y < 200. It is also necessary to consider:
Penny's cans = 3 ( Quincy's cans) ; Quincy's bottles = 4 ( Rosa's bottles)
; and\

Penny's (cans + bottles) = Quincy's (cans + bottles) = Rosa's (cans + bottles)

This leads to a strange mix of variables that lend themselves to a spreadsheet.
The formulas were placed in the spreadsheet and "random" numbers
were selected for Quincy's cans and Rosa's bottles until the sum of the
bottles and cans were equal. We considered the possibility of another solution
with a number greater than 200, but we were unable to find one with the
same parameters.

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