Let p and q be positive integers such that (5)/(9)\lt(p)/(q)\lt(4)/(7) and q is as small as possible. What is q-p?

2018 AMC 12B Problem 17 — Solution

2018 AMC 12B Problem 17

Below is the professionally curated solution for Problem 17 of the 2018 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2018 AMC 12B solutions, or check the answer key.

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Concepts:fraction inequality bounding to limit cases

Difficulty rating: 2090

17.

Let $p$ and $q$ be positive integers such that $\dfrac{5}{9}\lt\dfrac{p}{q}\lt\dfrac{4}{7}$ and $q$ is as small as possible. What is $q-p?$

$7$

$11$

$13$

$17$

$19$

Solution:

From $\tfrac59\lt\tfrac pq$ we get $9p-5q\ge1,$ and from $\tfrac pq\lt\tfrac47$ we get $4q-7p\ge1.$ Now $\dfrac{1}{63}=\dfrac47-\dfrac59=\dfrac{4q-7p}{7q}+\dfrac{9p-5q}{9q}\ge\dfrac{1}{7q}+\dfrac{1}{9q}=\dfrac{16}{63q}.$

Hence $q\ge16.$ With $q=16,$ the fraction $\tfrac{9}{16}$ lies strictly between $\tfrac59$ and $\tfrac47,$ so $p=9$ and $q-p=16-9=7.$

Thus, the correct answer is A.

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