2022 AMC 12A Problem 6 — Solution | LIVE by Po-Shen Loh

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2022 AMC 12A Problem 6

Attempt Problem 6 of the 2022 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 2022 AMC 12A solutions, or check the answer key.

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

6.

A data set consists of $6$ (not distinct) positive integers: $1,$ $7,$ $5,$ $2,$ $5,$ and $X.$ The average (arithmetic mean) of the $6$ numbers equals a value in the data set. What is the sum of all positive values of $X?$

$10$

$26$

$32$

$36$

$40$

Answer: D

Concepts:mean casework

Difficulty rating: 1270

Small Hint:

The five known numbers sum to $20,$ so the mean is $\dfrac{20+X}{6}$

Big Hint:

The mean must equal one of $1,2,5,7,$ or $X;$ only some give positive $X$

Solution:

The known numbers sum to $20,$ so the mean is $\dfrac{20+X}{6},$ which must equal an element of the set.

Setting it to $5$ gives $X=10;$ to $7$ gives $X=22;$ and to $X$ itself gives $20+X=6X,$ so $X=4.$ Values $1$ and $2$ give negative $X.$

The positive values are $10,22,4,$ summing to $36.$

Thus, the correct answer is D.

Problem 6 in Other Years

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