Emilia writes down the numbers 5, $x$ and 9. Valentin calculates the mean (average) of each pair of these numbers and obtains 7,10 and 12. The value of $x$ is
(A) 5
(B) 15
(C) 3
(D) 25
(E) 1
Since the average of 5 and 9 is $\frac{5+9}{2}=7$, then the averages of 5 and $x$ and of $x$ and 9 must be 10 and 12 .

In other words, $\frac{5+x}{2}$ and $\frac{x+9}{2}$ are equal to 10 and 12 in some order.

Adding these, we obtain $\frac{5+x}{2}+\frac{x+9}{2}=10+12$ or $\frac{14+2 x}{2}=22$ and so $7+x=22$ or $x=15$. (We could have also noted that $\frac{5+x}{2}<\frac{x+9}{2}$ since $5<9$, and so $\frac{x+9}{2}=12$.)

ANswer: (B)