Bottles and Cans
Molly McKee
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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, and 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 collected fewer than 200 items in all. How many cans and bottles did each collect?
bQ + cQ = QT
bP + cP = PT
bR + cR = RT
~QT = PT = RT~
bQ + bP + bR = bT
cQ + cP + cR = cT
bT = cT
bT + cT < 200
We know that 3cP = cQ and 4bQ = bR
An Introduction to the History of Mathematics
Howard Eves
6.13 Problems from the “Greek Anthology” (page 196)
Similar Problems
I was able to solve
a) How many apples are needed if 4 persons out of 6 receive 1/3, 1/8, 1/4, and 1/5, respectively, of the total number, while the fifth receives 10 apples, and 1 apple remains for the sixth person?
Let x represent the total number of apples
Then (1/3)x + (1/8)x + (1/4)x + (1/5)x + 10 + 1 = x
(109/120)x +11 = x
11 = (11/120)x
120 = x
Therefore 120 apples are needed
b) Demochares has lived a fourth of his life as a boy, a fifth of his life as a youth, a third as a man, and has spent 13 years in his dotage. How old is he?
Let x represent Demochares’ age
Then (1/4)x + (1/5)x + (1/3)x + 13 = x
(47/60)x +13 = x
13 = (13/60)x
60 = x
Therefore Demochares is 60 years old
c) After staining the chaplet of fair-eyed Justice that I might see thee, all-subduing gold, grow so much, I have nothing, for I gave forty talents under evil auspices to my friends in vain, while, O ye varied mischances of men, I see my enemies in possession of the half, the third , and the eighth of my fortune. (How many talents did the unfortunate man once possess?)
Let x represent the total number of talents
Then 40 + (1/2)x + (1/3)x + (1/8)x = x
40 + (23/24)x = x
40 = (1/24)x
960 = x
Therefore the man once possessed 960 talents
d) The 3 Graces were carrying baskets of apples, and in each was the same number. The 9 Muses met them and asked each for apples and they gave the same number to each Muse and the 9 and the 3 each had the same number. Tell me how many they gave and how they all had the same number. (This problem is indeterminate. Find the smallest permissible solution.)
Let x represent the number of apples in one of the Graces’ baskets, then 3x represents the total number of apples.
The minimum number of apples that each Grace could give to each Muse is 1, therefore each Muse now has 3 apples and each Grace has x – 9 apples.
Since the Muses and the Graces must end up with the same number of apples, the Graces must also have 3 apples in their baskets. Then,
x – 9 = 3
x = 12
Therefore each Grace began with a minimum of 12 apples
If each Grace gave away 2 apples, then
x – 9(2) = 3(2)
x – 18 = 6
x = 24
and each Grace began with 24 apples.
Therefore, 3x = 12n, where n is the number of apples each Grace gave away and x is the number of apples each Grace started with.
This is as far as I seem to be able to take this problem. Each time I think I have found the number of bottles and cans collected by each person, one aspect of the problem does work out. I may be approaching the problem in the wrong way. I am trying to solve it by using systems of equations, but perhaps entering the data into a spreadsheet will produce better results.
