2019 AMC 12B Problem 7

Below is the professionally curated solution for Problem 7 of the 2019 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2019 AMC 12B solutions, or check the answer key.

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Concepts:mean median (data)casework

Difficulty rating: 1280

What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17,$ and $x$ is equal to the mean of those five numbers?

$-5$

$0$

$5$

$\dfrac{15}{4}$

$\dfrac{35}{4}$

Solution:

The mean is $\dfrac{4+6+8+17+x}{5}=\dfrac{35+x}{5}.$

If $x\le6,$ the median is $6,$ so $\dfrac{35+x}{5}=6$ gives $x=-5,$ which is consistent.

If $6\lt x\lt8,$ the median is $x,$ so $\dfrac{35+x}{5}=x$ gives $x=8.75,$ not in range. If $x\ge8,$ the median is $8,$ so $\dfrac{35+x}{5}=8$ gives $x=5,$ not in range.

The only solution is $x=-5,$ so the sum is $-5.$

Thus, A is the correct answer.

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