for
Shanti L. Howard and Audrea K. Bankston
PROBLEM: Bottles and Cans
Three neighbors named Quincy, Penny, and Rosa took part in
a local recycling drive. Each spent a Saturday afteroonm collecing
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?
|
Names |
Cans |
Bottles |
|
Rosa |
c + 3b
6 + 3(7)
6 + 21 = 27
|
b
c + 4b - b
=c + 3b
b = 7
|
|
Penny |
3c
3(6) =18
|
4b - 2c
c + 4b - 3c
4b -2c
4(7) - 2(6)
28 - 12 =16
|
|
Quincy |
c
c = 6
|
4b
c + 4b
4(7) = 28
|
R(cans + bottles)
= P(cans + bottles) = Q(cans + bottles)
R(y + b) = P(3c
+ x) = Q(c + 4b)
R (cans) + P (cans) + Quincy (cans)
R(3b + c) + P(3c) +Q(c) =
R(b) + P(4b - 2c) + Q(4b)
c = b
R(cans) c + 3b..............R(bottles)
b
P(cans) 3c...............P(bottles)4b-2c
Q(cans) c..................Q(bottles)
4b
3b +c +3c +c = b
+ 4b - 2c + 4b
2c + 3b +5c = 9b
- 2c +2c
7c + 3b = 3b = 9b
-3b
7c = 6b
so c has to be to
6 and b has to be equal to 7.
The final answer:
They collected a total of 34 items each.