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 (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

NOT DONE YET. WILL FINISH UP. NO EDIT PLEASE.

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 that Carl can carry out:

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

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