The mean (average) of the four integers $78,83,82$, and $x$ is 80 . Which one of the following statements is true?

(A) $x$ is 2 greater than the mean

(B) $x$ is 1 less than the mean

(C) $x$ is 2 less than the mean

(D) $x$ is 3 less than the mean

(E) $x$ is equal to the mean
Solution 1

Since 78 is 2 less than 80 and 82 is 2 greater than 80 , the mean of 78 and 82 is 80 .

Since the mean of all four integers is 80 , then the mean of 83 and $x$ must also equal 80 .

The integer 83 is 3 greater than 80 , and so $x$ must be 3 less than 80 .

That is, $x=80-3=77$.

(We may check that the mean of $78,83,82$, and 77 is indeed 80 .)

Solution 2

Since the mean of the four integers is 80 , then the sum of the four integers is $4 \times 80=320$.

Since the sum of 78,83 and 82 is 243 , then $x=320-243=77$.

Therefore, $x$ is 77 which is 3 less than the mean 80 .

ANSWER: (D)