2022 AMC 10A Problems/Problem 8

Problem

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 $X$ ?

$\textbf{(A) } 10 \qquad \textbf{(B) } 26 \qquad \textbf{(C) } 32 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 40$

Solution

Case 1: the mean is $5$

$X = 5 \cdot 6 - 20 = 10$ .

Case 2: the mean is $7$

$X = 7 \cdot 6 - 20 = 22$ .

Case 3: the mean is $X$

$\frac{20+X}{6} = X \Rightarrow X=4$ .

Hence, adding up the cases, the answer is $10+22+4=\boxed{\textbf{(D) }36}$ .

~MrThinker

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2022 AMC 10A Problems/Problem 8

Problem

Solution