# Difference between revisions of "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 , the median , and the modes of the values that are the dates in the months of . Thus her data consist of , , . . . , , , , and . Let be the median of the modes. Which of the following statements is true?

## Solution 1

First of all, obviously has to be smaller than , since when calculating , we must take into account the s, s, and s. So we can eliminate choices and . Since there are total entries, the median, , must be the one, at which point we note that is , so has to be the median (because is between and ). Now, the mean, , must be smaller than , since there are many fewer s, s, and s. is less than , because when calculating , we would include , , and . Thus the answer is .

## Solution 2

As in Solution 1, we find that the median is . Then, looking at the modes , we realize that even if we were to have of each, their median would remain the same, being . As for the mean, we note that the mean of the first is simply the same as the median of them, which is . Hence, since we in fact have 's, 's, and 's, the mean has to be higher than . On the other hand, since there are fewer 's, 's, and 's than the rest of the numbers, the mean has to be lower than (the median). By comparing these values, the answer is .

## Solution 3 (direct calculation)

We can solve this problem simply by carefully calculating each of the values, which turn out to be , , and . Thus the answer is .
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