Eight numbers average 27. Two of them, 19 and 35, are taken out and two different numbers are put in their place, and the average of the eight stays at 27. What must the two new numbers add up to?

The count is still 8 and the average is still 27, so the total is still $8\times 27=216$. The only thing that changed is that 19 + 35 = 54 left and two new numbers arrived, so those two must bring back exactly 54. Note that their individual values are not fixed - 20 and 34 would do as well as 26 and 28 - only their total is. 27 is the average itself, which would be the answer only if a single number had been swapped, not two. 216 is the total of all eight numbers. 16 is 35 - 19, the difference between the two numbers removed, which says nothing about how much weight they carried away.