# 2013 AMC 8 Problems/Problem 19

## Problem

Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, 'I didn't get the lowest score in our class,' and Bridget adds, 'I didn't get the highest score.' What is the ranking of the three girls from highest to lowest?

$\textbf{(A)}\ \text{Hannah, Cassie, Bridget} \qquad \textbf{(B)}\ \text{Hannah, Bridget, Cassie} \\ \qquad \textbf{(C)}\ \text{Cassie, Bridget, Hannah} \qquad \textbf{(D)}\ \text{Cassie, Hannah, Bridget} \\ \qquad \textbf{(E)}\ \text{Bridget, Cassie, Hannah}$

## Solution

If Hannah did better than Cassie, there would be no way she could know for sure that she didn't get the lowest score in the class. Therefore, Hannah did worse than Cassie. Similarly, if Hannah did worse than Bridget, there is no way Bridget could have known that she didn't get the highest in the class. Therefore, Hannah did better than Bridget, so our order is $\boxed{\textbf{(D) Cassie, Hannah, Bridget}}$

Note: It could be said that Cassie did better than Hannah, and Bridget knew this, and she had the same score as Hannah, so Bridget didn't get the highest score. This may be a reason that MAA didn't put this answer choice in.

## Video Solution

https://youtu.be/CqmAwiiMeBQ ~savannahsolver

## Video Solution 2

https://youtu.be/U3VrL9yKh3g. Soo, DRMS, NM