# 2018 AMC 10A Problems/Problem 5

The following problem is from both the 2018 AMC 12A #4 and 2018 AMC 10A #5, so both problems redirect to this page.

## Problem

Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let $d$ be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of $d$? $\textbf{(A) } (0,4) \qquad \textbf{(B) } (4,5) \qquad \textbf{(C) } (4,6) \qquad \textbf{(D) } (5,6) \qquad \textbf{(E) } (5,\infty)$

## Solution 1

From Alice and Bob, we know that $5 < d < 6.$ From Charlie, we know that $4 < d.$ We take the intersection of these two intervals to yield $\boxed{\textbf{(D) } (5,6)}$, because the nearest town is between 5 and 6 miles away.

## Solution 2

Think of the distances as if they are on a number line. Alice claims that $d > 6$, Bob says $d < 5$, while Charlie thinks $d < 4$. This means that all possible numbers before $5$ and after $6$ are included. But since the three statements are actually false, the distance to the nearest town is one of the numbers not covered, which yields the interval $\boxed{\textbf{(D) } (5,6)}$.

## Video Solutions

~savannahsolver

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