Question

The mean, median, and unique mode of the positive integers $3, 4, 5, 6, 6, 7,$ and $x$ are all equal. What is the value of $x?$

$5$

$6$

$7$

$11$

$12$

Video solution:

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Written solution:

Every listed value except $6$ appears once while $6$ appears twice. If any of the other values are chosen, then we have two modes, which means we do not have a unique mode. Otherwise, the only value that shows up more than once is $6$ making that the unique mode. This also means $6$ is the mean. Since there are $7$ elements, the sum of the elements is $6\cdot 7 = 42.$ The sum is also $3+4+5+6+6+7+x$ $= 31+x$ $= 42,$ so $x = 11.$

Thus, the answer is D .

2012 AMC 8 Problem 11

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Written solution:

Problem 11 in Other Years