2013 AMC 12A Problem 16

Question

A,

B,

and

C

are three piles of rocks. The mean weight of the rocks in

A

is

40

pounds, the mean weight of the rocks in

B

is

50

pounds, the mean weight of the rocks in the combined piles

A

and

B

is

43

pounds, and the mean weight of the rocks in the combined piles

A

and

C

is

44

pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles

B

and

C?

55

56

57

58

59

Answer: E

Concepts:mean optimization

Difficulty rating: 1980

Small hint:

Let

a, b, c

be the counts; the

A,B

mean

43

gives

7b = 3a

Big hint:

Write the

B,C

mean in terms of

k

and

c

using

a = 7k,\ b = 3k

and the

A,C

mean

44

Solution:

Let

a, b, c

be the numbers of rocks in the piles. From

\dfrac{40a + 50b}{a + b} = 43,

we get

7b = 3a,

so

a = 7k

and

b = 3k.

Let

\mu_{BC}

be the mean of

B

and

C.

Using the

A, C

mean

44

to express

\mu_C = \dfrac{28k + 44c}{c},

we find

\mu_{BC} = \dfrac{178k + 44c}{3k + c},

so

(\mu_{BC} - 44)c = (178 - 3\mu_{BC})k.

Since

B

is heavier than

A,

the mean of

B

and

C

exceeds

44,

forcing

178 - 3\mu_{BC} \gt 0,

i.e.

\mu_{BC} \lt \tfrac{178}{3} = 59\tfrac13.

The value

59

is attainable by taking

k=15c;

then pile

C

has mean

464

and the displayed formula gives

\mu_{BC}=59.

Thus the greatest integer mean is

59.

Thus, the correct answer is E.

2013 AMC 12A Problem 16

Answer: E

Difficulty rating: 1980

Small hint:

Big hint:

Solution:

Problem 16 in other years