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.