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
The value of -pound rocks is per pound, and the value of -pound rocks is per pound. Clearly, Carl should not carry more than three -pound rocks. Otherwise, he can replace some -pound rocks with some heavier rocks, preserving the weight but increasing the total value.
We perform casework on the number of -pound rocks Carl can carry: Clearly, the maximum value of the rocks Carl can carry is dollars.
Remark
Note that an upper bound of the total value is from which we can eliminate choices and
~Pyhm2017 (Fundamental Logic)
~MRENTHUSIASM (Reconstruction)
Solution 2
Since each rock is worth dollar less than times its weight (in pounds), 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)
Video Solution 1
~Education, the Study of Everything
See Also
2018 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 1 |
Followed by Problem 3 |
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