Bottles and Cans
by
Angie Head, Beth Richichi, and Teisha Wright
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 bottles 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:
Before we began to solve this problem, we labeled the unknowns. We let x
= # cans and y = # bottles. From the information given in the problem, we
also know that x = y and x + y < 200. Also given are the following facts:
the number of Penny's cans = 3 times the number of Quincy's cans; the number
of Quincy's bottles = 4 times the number of Rosa's bottles; and Penny's
(cans + bottles) = Quincy's (cans + bottles) = Rosa's (cans + bottles).
From this we have,
P(x) = 3Q(x), Q(y) = 4R(y), and P(x+y) = Q(x+y) = R(x+y).
Since we have several variables, we decided to compute this on a spreadsheet.
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