2019 AMC 12A Problem 7

Problem 7 of 25EasierAlgebra

Melanie computes the mean

\mu,

the median

M,

and the modes of the

365

values that are the dates in the months of

2019.

Thus her data consist of

12

1

12

2

\ldots,

12

28

11

29

11

30

s, and

7

31

s. Let

d

be the median of the modes. Which of the following statements is true?

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Solution

The values

1

through

28

each appear

12

times and are the modes, so

d = \dfrac{14 + 15}{2} = 14.5.

The

183

rd of the

365

ordered values is the median. Values

1

through

15

fill the first

180

positions, so position

183

16;

thus

M = 16.

The total of all values is

12(1 + \cdots + 28)

+ 11(29 + 30)

+ 7 \cdot 31 = 5738,

\mu = \dfrac{5738}{365} \approx 15.7.

Therefore

d \lt \mu \lt M.

Thus, the correct answer is E.