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:

1. We get three -pound rocks and three -pound rocks, for a total value of
2. 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 the least upper bound of the total value is dollars, from which we can eliminate choices and

~Pyhm2017 (Fundamental Logic)

~MRENTHUSIASM (Reconstruction)

See Also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.