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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