Mixing
AMC 12 2018 A Question 2
$begingroup$
While exploring a cave, Carl comes across a collection of $5$-pound rocks worth $14$ each, $4$-pound rocks worth $11$ each, and $1$-pound rocks worth $2$ each. There are at least $20$ of each size. He can carry at most $18$ pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?
I was able to solve this quickly enough without too much of a hassle, but I don't get the official solution, which states
The answer is just $3cdot 18=54$ minus the minimum number of rocks we need to make $18$ pounds, or $54-4=boxed{textbf{(C)} 50.}$ (AoPS)
How does this work? I think I'm almost getting there but I keep getting confused.
contest-math
asked Jan 16 at 4:44
jjhhjjhh
2,13911122
$endgroup$
add a comment |
$begingroup$
While exploring a cave, Carl comes across a collection of $5$-pound rocks worth $14$ each, $4$-pound rocks worth $11$ each, and $1$-pound rocks worth $2$ each. There are at least $20$ of each size. He can carry at most $18$ pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?
I was able to solve this quickly enough without too much of a hassle, but I don't get the official solution, which states
The answer is just $3cdot 18=54$ minus the minimum number of rocks we need to make $18$ pounds, or $54-4=boxed{textbf{(C)} 50.}$ (AoPS)
How does this work? I think I'm almost getting there but I keep getting confused.
contest-math
asked Jan 16 at 4:44
jjhhjjhh
2,13911122
$endgroup$
$begingroup$
What did you get and how did you get that value?
$endgroup$
– John Douma
Jan 16 at 4:49
add a comment |
$begingroup$
While exploring a cave, Carl comes across a collection of $5$-pound rocks worth $14$ each, $4$-pound rocks worth $11$ each, and $1$-pound rocks worth $2$ each. There are at least $20$ of each size. He can carry at most $18$ pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?
I was able to solve this quickly enough without too much of a hassle, but I don't get the official solution, which states
The answer is just $3cdot 18=54$ minus the minimum number of rocks we need to make $18$ pounds, or $54-4=boxed{textbf{(C)} 50.}$ (AoPS)
How does this work? I think I'm almost getting there but I keep getting confused.
contest-math
asked Jan 16 at 4:44
jjhhjjhh
2,13911122
$endgroup$
While exploring a cave, Carl comes across a collection of $5$-pound rocks worth $14$ each, $4$-pound rocks worth $11$ each, and $1$-pound rocks worth $2$ each. There are at least $20$ of each size. He can carry at most $18$ pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?
I was able to solve this quickly enough without too much of a hassle, but I don't get the official solution, which states
The answer is just $3cdot 18=54$ minus the minimum number of rocks we need to make $18$ pounds, or $54-4=boxed{textbf{(C)} 50.}$ (AoPS)
How does this work? I think I'm almost getting there but I keep getting confused.
contest-math
contest-math
asked Jan 16 at 4:44
jjhhjjhh
2,13911122
asked Jan 16 at 4:44
jjhhjjhh
2,13911122
asked Jan 16 at 4:44
jjhhjjhh
2,13911122
asked Jan 16 at 4:44
jjhhjjhh
2,13911122
asked Jan 16 at 4:44
jjhhjjhh
2,13911122
2,13911122
$begingroup$
What did you get and how did you get that value?
$endgroup$
– John Douma
Jan 16 at 4:49
add a comment |
$begingroup$
What did you get and how did you get that value?
$endgroup$
– John Douma
Jan 16 at 4:49
$begingroup$
What did you get and how did you get that value?
$endgroup$
– John Douma
Jan 16 at 4:49
$begingroup$
What did you get and how did you get that value?
$endgroup$
– John Douma
Jan 16 at 4:49
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
If a rock weighs $n$ pounds, it is worth $3n-1$ dollars.
So, if Carl gets $k$ rocks weighing $n_1, n_2, ldots, n_k$ pounds, then the total value of rocks is $3(n_1+n_2+cdots+n_k)-k$.
If Carl carries $18$ pounds of rocks out, i.e. $n_1+cdots+n_k = 18$, then the total value is $3 cdot 18 - k$, where $k$ is the number of rocks Carl carried out.
Note, I completely agree that is a poorly worded solution (assuming what you quoted was indeed their official solution).
answered Jan 16 at 4:52
JimmyK4542JimmyK4542
41.1k245107
$endgroup$
add a comment |
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1 Answer
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$begingroup$
If a rock weighs $n$ pounds, it is worth $3n-1$ dollars.
So, if Carl gets $k$ rocks weighing $n_1, n_2, ldots, n_k$ pounds, then the total value of rocks is $3(n_1+n_2+cdots+n_k)-k$.
If Carl carries $18$ pounds of rocks out, i.e. $n_1+cdots+n_k = 18$, then the total value is $3 cdot 18 - k$, where $k$ is the number of rocks Carl carried out.
Note, I completely agree that is a poorly worded solution (assuming what you quoted was indeed their official solution).
answered Jan 16 at 4:52
JimmyK4542JimmyK4542
41.1k245107
$endgroup$
add a comment |
$begingroup$
If a rock weighs $n$ pounds, it is worth $3n-1$ dollars.
So, if Carl gets $k$ rocks weighing $n_1, n_2, ldots, n_k$ pounds, then the total value of rocks is $3(n_1+n_2+cdots+n_k)-k$.
If Carl carries $18$ pounds of rocks out, i.e. $n_1+cdots+n_k = 18$, then the total value is $3 cdot 18 - k$, where $k$ is the number of rocks Carl carried out.
Note, I completely agree that is a poorly worded solution (assuming what you quoted was indeed their official solution).
answered Jan 16 at 4:52
JimmyK4542JimmyK4542
41.1k245107
$endgroup$
add a comment |
$begingroup$
If a rock weighs $n$ pounds, it is worth $3n-1$ dollars.
So, if Carl gets $k$ rocks weighing $n_1, n_2, ldots, n_k$ pounds, then the total value of rocks is $3(n_1+n_2+cdots+n_k)-k$.
If Carl carries $18$ pounds of rocks out, i.e. $n_1+cdots+n_k = 18$, then the total value is $3 cdot 18 - k$, where $k$ is the number of rocks Carl carried out.
Note, I completely agree that is a poorly worded solution (assuming what you quoted was indeed their official solution).
answered Jan 16 at 4:52
JimmyK4542JimmyK4542
41.1k245107
$endgroup$
If a rock weighs $n$ pounds, it is worth $3n-1$ dollars.
So, if Carl gets $k$ rocks weighing $n_1, n_2, ldots, n_k$ pounds, then the total value of rocks is $3(n_1+n_2+cdots+n_k)-k$.
If Carl carries $18$ pounds of rocks out, i.e. $n_1+cdots+n_k = 18$, then the total value is $3 cdot 18 - k$, where $k$ is the number of rocks Carl carried out.
Note, I completely agree that is a poorly worded solution (assuming what you quoted was indeed their official solution).
answered Jan 16 at 4:52
JimmyK4542JimmyK4542
41.1k245107
answered Jan 16 at 4:52
JimmyK4542JimmyK4542
41.1k245107
answered Jan 16 at 4:52
JimmyK4542JimmyK4542
41.1k245107
answered Jan 16 at 4:52
JimmyK4542JimmyK4542
41.1k245107
41.1k245107
add a comment |
add a comment |
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$begingroup$
What did you get and how did you get that value?
$endgroup$
– John Douma
Jan 16 at 4:49