2018 AMC 12A Problems/Problem 2
Problem
While exploring a cave, Carl comes across a collection of -pound rocks worth
each,
-pound rocks worth
each, and
-pound rocks worth
each. There are at least
of each size. He can carry at most
pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?
Solution 1
Since each rock is worth dollar less than
times its weight, the answer is just
minus the minimum number of rocks we need to make
pounds. Note that we need at least
rocks (two
-pound rocks and two
-pound rocks) to make
pounds, so the answer is
~Kevindujin (Solution)
~MRENTHUSIASM (Revision)
Solution 2
The ratio of dollar per pound is greatest for the pound rock, then the
pound, lastly the
pound. So we should take two
pound rocks and two
pound rocks. The total value, in dollars, is
~steakfails
Solution 3
The unit value of -pound rocks is
per pound, and the unit value of
-pound rocks is
per pound. Intuitively, we wish to maximize the number of
-pound rocks and minimize the number of
-pound rocks. We have two cases:
- We get three
-pound rocks and three
-pound rocks, for a total value of
- We get two
-pound rocks and two
-pound rocks, for a total value of
Clearly, Case 2 produces the maximum total value. So, the answer is
Remark
Note that an upper bound of the total value is from which we can eliminate choices
and
~Pyhm2017 (Fundamental Logic)
~MRENTHUSIASM (Reconstruction)
See Also
2018 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 1 |
Followed by Problem 3 |
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All AMC 12 Problems and Solutions |
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