2016 AMC 12A Problem 4 — Solution | LIVE by Po-Shen Loh

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2016 AMC 12A Problem 4

Below is the professionally curated solution for Problem 4 of the 2016 AMC 12A, 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 of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:mean median (data)mode

Difficulty rating: 1100

4.

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 $x?$

$50$

$60$

$75$

$90$

$100$

Solution:

The mean condition gives $\dfrac{60+100+x+40+50+200+90}{7}=\dfrac{540+x}{7}=x,$ 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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