Question

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 was 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 $d?$

$(0, 4)$

$(4, 5)$

$(4, 6)$

$(5, 6)$

$(5, \infty)$

Answer: D

Concepts:logical deduction inequality

Difficulty rating: 1200

Solution:

Negating the three false statements gives $d \lt 6,$ $d \gt 5,$ and $d \gt 4.$ The intersection of these conditions is $5 \lt d \lt 6,$ that is, the interval $(5, 6).$

Thus, the correct answer is D.

2018 AMC 12A Problem 4

Answer: D

Difficulty rating: 1200

Solution:

Problem 4 in Other Years