AMC10 2012 10A, Problem 8

The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?

Solution: Let the three numbers be equal to $a$, $b$, and $c$. We can now write three equations: $a+b=12$ $b+c=17$ $a+c=19$ Adding these equations together, we get that $2(a+b+c)=48$ and $a+b+c=24$ Substituting the original equations into this one, we find $c+12=24$ $a+17=24$ $b+19=24$ Therefore, our numbers are 12, 7, and 5. The middle number is $\fbox{7}$

Answer: 7