14. The following twelve integers are written in ascending order:
$$
1, x, x, x, y, y, y, y, y, 8,9,11 \text {. }
$$

The mean of these twelve integers is 7. What is the median?
A 6
B 7
С 7.5
D 8
E 9
Solution
D

Because the mean of the given integers is 7 , we have
$$
1+x+x+x+y+y+y+y+y+8+9+11=12 \times 7 \text {. }
$$

We may rewrite this last equation as $1+3 x+5 y+28=84$ and therefore $3 x+5 y=55$.
It follows that $3 x=55-5 y=5(11-y)$. Therefore, because $x$ and $y$ are integers, $3 x$ is a multiple of 5 and hence $x$ is a multiple of 5 .

Because the integers are written in ascending order, $1 \leq x \leq 8$. Hence $x=5$ and therefore $15+5 y=55$. It follows that $5 y=40$ and therefore $y=8$.

Therefore the twelve integers in ascending order are $1,5,5,5,8,8,8,8,8,8,9,11$. We now see that the median is 8 .