2018 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 5 Easy

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? (2018 AMC 10A Problem, Question#5)

  • A.

    (0,4)

  • B.

    (4,5)

  • C.

    (4,6)

  • D.

    (5,6)

  • E.

    (5,\infty )

Answer:D

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

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