2019 AMC 10A Problem 12

Problem 12 of 25IntermediateAlgebra

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

copies of

1,

12

copies of

2,

and so on through

12

copies of

28,

then

11

copies of

29,

11

copies of

30,

and

7

copies of

31.

Let

d

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

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Solution

The modes are all the integers from

1

through

28,

so their median is

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

There are

365

entries, so

M

is the

183

rd number. The dates from

1

through

15

occupy

15 \cdot 12 = 180

positions, so

M=16.

The sum of all dates is

\begin{aligned}12(1+\cdots+28)&+11(29+30)\\&+7\cdot31=5738.\end{aligned}

Hence

\mu=\dfrac{5738}{365}\approx15.72.

Therefore

d \lt \mu \lt M.

Thus, E is the correct answer.