2018 AMC 12A Problems/Problem 2
Contents
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 (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)
Solution 2
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 the
-pound rocks with the 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)
See Also
2018 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 1 |
Followed by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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