2018 AMC 10A Problems/Problem 5

The following problem is from both the 2018 AMC 10A #5 and 2018 AMC 12A #4, 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

For each of the false statements, we identify its corresponding true statement. Note that:

  1. $\mathrm{False}\cap\mathrm{True}=\varnothing.$
  2. $\mathrm{False}\cup\mathrm{True}=[0,\infty).$

We construct the following table: \[\begin{array}{c||c|c} & &  \\ [-2.5ex] \textbf{Hiker} & \textbf{False Statement} & \textbf{True Statement} \\ [0.5ex] \hline & & \\ [-2ex] \textbf{Alice} & [6,\infty) & [0,6) \\  & & \\ [-2.25ex] \textbf{Bob} & [0,5] & (5,\infty)  \\ & & \\ [-2.25ex] \textbf{Charlie} & [0,4] & (4,\infty) \end{array}\] Taking the intersection of the true statements, we have \[[0,6)\cap(5,\infty)\cap(4,\infty)=(5,6)\cap(4,\infty)=\boxed{\textbf{(D) } (5,6)}.\] ~MRENTHUSIASM

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 less than $5$ and greater than $6$ are included. However, since the three statements are actually false, the distance to the nearest town is one of the numbers not covered. Therefore, the answer is $\boxed{\textbf{(D) } (5,6)}$.

Video Solutions

https://youtu.be/vO-ELYmgRI8

https://youtu.be/cLJ87xJzcWI

~savannahsolver

https://youtu.be/nNEXCxWLJzc

Education, the Study of Everything

See Also

2018 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions
2018 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 3
Followed by
Problem 5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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