# 2019 AMC 10A Problems/Problem 12

The following problem is from both the 2019 AMC 10A #12 and 2019 AMC 12A #7, so both problems redirect to this page.

## Problem

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 $12$ $1\text{s}$, $12$ $2\text{s}$, . . . , $12$ $28\text{s}$, $11$ $29\text{s}$, $11$ $30\text{s}$, and $7$ $31\text{s}$. Let $d$ be the median of the modes. Which of the following statements is true?

$\textbf{(A) } \mu < d < M \qquad\textbf{(B) } M < d < \mu \qquad\textbf{(C) } d = M =\mu \qquad\textbf{(D) } d < M < \mu \qquad\textbf{(E) } d < \mu < M$

## Solution

### Solution 1

First of all, $d$ obviously has to be smaller than $M$, since when calculating $M$, you must take into account the $29$s, $30$s, and $31$s. So we can eliminate $(B)$ and $(C)$. The median, $M$, is $16$, but the mean ($\mu$) must be smaller than $16$ since there are many fewer $29$s, $30$s, and $31$s. $d$ is less than $\mu$, because when calculating $\mu$, you include $29$, $30$, and $31$. Thus the answer is $d < \mu < M \implies \boxed{\textbf{(E)}}$

### Solution 2

Notice that there are $365$ total entries, so the median has to be the $183\text{rd}$ one. Then, realize that $12 \cdot 15$ is $180$, so $16$ has to be the median (because $16$ is from $181$ to $192$). Then, look at the modes $(1-28)$ and realize that even if you have $12$ of each, the median of those remains the same and you have $14.5$. When trying to find the mean, you realize that the mean of the first $28$ is simply the same as the median of them, which is $14.5$. Then, when you see $29$'s, $30$'s, and $31$'s, you realize that the mean has to be higher. On the other hand, since there are fewer $29$'s, $30$'s, and $31$'s than the rest of the numbers, the mean has to be lower than $16$ (the median). Then, you compare those values and you get the answer, which is $\boxed{\textbf{(E)}}$.

### Solution 3 (don't do this unless you have 30 min to spare)

Bash out each of the values. I'm not going to show work here, but you should get $M = 16$ , $d = 14.5$ , $\mu = 15.7205479$ , so the answer is obviously $\boxed{\textbf{(E)}}$.