Difference between revisions of "2018 AMC 12A Problems/Problem 2"
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== Solution 2 == | == Solution 2 == | ||
| − | The unit value of <math>5</math>-pound rocks is <math>\$2. | + | The unit value of <math>5</math>-pound rocks is <math>\$14\div5=\$2.80</math> per pound, and the unit value of <math>4</math>-pound rocks is <math>\$11\div4=\$2.75</math> per pound. Intuitively, we wish to maximize the number of <math>5</math>-pound rocks and minimize the number of <math>1</math>-pound rocks. We have two cases: |
<ol style="margin-left: 1.5em;"> | <ol style="margin-left: 1.5em;"> | ||
<li>We get three <math>5</math>-pound rocks and three <math>1</math>-pound rocks, for a total value of <math>\$14\cdot3+\$2\cdot3=\$48.</math></li><p> | <li>We get three <math>5</math>-pound rocks and three <math>1</math>-pound rocks, for a total value of <math>\$14\cdot3+\$2\cdot3=\$48.</math></li><p> | ||
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<u><b>Remark</b></u> | <u><b>Remark</b></u> | ||
| − | Note that an upper bound of the total value is <math>\$2. | + | Note that an upper bound of the total value is <math>\$2.80\cdot18=\$50.4,</math> from which we can eliminate choices <math>\textbf{(D)}</math> and <math>\textbf{(E)}.</math> |
~Pyhm2017 (Fundamental Logic) | ~Pyhm2017 (Fundamental Logic) | ||
Revision as of 02:12, 24 August 2021
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 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)
Solution 3
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
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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
