Age of individuals, given median and average

Question

Average age among three people is 6 where everyone is of different ages. Median age is 8. How old are the people?

I see that

$\frac{x_1+x_2+x_3}{3}=6 \Longleftrightarrow x_1+x_2+x_3 = 18$ and that $x_2 = 8$ so that I get a system of equations

$\begin{equation}\begin{cases}x_1+x_2+x_3 = 18\\x_2=8 \end{cases}\end{equation}$

But now I have three unknowns and two equations, I which doesn't seem solvable. I seem to be missing an equation, right?. A relationship I am not seeing.

Answer 1

Without loss of generality, assume that $x_1<x_2<x_3$ (we use strictly less than here because they are different ages). Next, because $x_2=8$, $$x_1+8+x_3=18$$ $$x_1=10-x_3$$ $$x_1<10-8$$ Since somebody cant be 0 years old, we have that $0<x_1<2$, so that $x_1=1$ and $x_3=9$, so the ages are 1, 8, and 9