I know that the midpoint of a segment is the average of the endpoints, so I can use that formula to find the other endpoint. Let $(x,y)$ be the other endpoint, then I have $\left(\frac{8+x}{2}, \frac{0+y}{2}\right) = (6,-10)$. I can equate the corresponding components and solve for $x$ and $y$. For the x-coordinate, I have $\frac{8+x}{2} = 6$, so multiplying both sides by $2$, I get $8+x = 12$. Subtracting $8$ from both sides, I get $x = 4$. For the y-coordinate, I have $\frac{0+y}{2} = -10$, so multiplying both sides by $2$, I get $0+y = -20$. Subtracting $0$ from both sides, I get $y = -20$. So the other endpoint is $(4,-20)$. To find the sum of the coordinates, I just add $4$ and $-20$. The sum is $-16$. 
The point with coordinates $(6,-10)$ is the midpoint of the segment with one endpoint at $(8,0)$. Find the sum of the coordinates of the other endpoint.