A set S of real values consists of the following 5 elements, {4, 8, 12, 16, x} in any order. For how many values of x does the mean of set S equal the median of set S?

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Solution

The correct option is D
3

To solve this problem quickly, you can come up with likely values forxthat would make the mean equal to the median.
Three values that would make the set symmetrical are 0, 10, and 20:
{0, 4, 8, 12, 16}
{4, 8, 10, 12, 16}
{4, 8, 12, 16, 20}

To verify if there are any other possibilities for x, we need to evaluate each range of "x”

(1) If x is less than or equal to 8, then the median is equal to 8. We now set the mean equal to the median:

$\frac{40 + x}{5} = 840 + x = 40 x = 0$

(2) If x is between 8 and 12, then the median is equal to x.

$\frac{40 + x}{5} = x 40 + x = 5 x 40 = 4 x x = 10$

(3) If x is greater than 12, then the median is equal to 12.

$\frac{40 + x}{5} = 1240 + x = 60$

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