2016 AMC 12A Problem 4

Attempt Problem 4 of the 2016 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2016 AMC 12A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x.$ What is the value of $x?$

$50$

$60$

$75$

$90$

$100$

Answer: D

Concepts:mean median (data)mode

Difficulty rating: 1100

Solution:

The mean condition gives $\begin{gathered} \small \dfrac{60+100+x+40+50+200+90}{7}\\ =\dfrac{540+x}{7}\\ =x, \end{gathered}$ so $540+x=7x$ and $x=90.$

In nondecreasing order the data are $40, 50, 60, 90, 90, 100, 200,$ so the median is $90$ and the mode is $90,$ as required.

Thus, the correct answer is D.

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