2018 AMC 12A Problem 2

Question

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?

48

49

50

51

52

Compare the value per pound:

\$2.80,

\$2.75,

and

\$2

for the three rock sizes

Filling the full

18

pounds matters more than always grabbing the best per-pound rock; try two

5

-pound and two

4

-pound rocks

The rocks are worth $

2.80,

$

2.75,

and $

2

per pound, respectively. For each possible number of

5

-pound rocks, use as many

4

-pound rocks as fit and fill any leftover capacity with

1

-pound rocks. Taking

0,1,2,

or

3

of the

5

-pound rocks gives maximum values $

48,

$

49,

$

50,

and $

48,

respectively.

Thus two

5

-pound and two

4

-pound rocks use all

18

pounds, and the maximum value is $

50.

Thus, the correct answer is C.

Problem 2 in other years