Every number in a list is replaced by 100 minus that number. If the average of the original list was 38, what is the average of the new list?

Whatever is done to every number is done to the average as well, so the new average is 100 - 38 = 62. (With $n$ numbers the new total is $100n$ minus the old total, and dividing by $n$ gives 100 minus the old average - the count never matters.) 38 assumes the average is untouched, which would need the change to cancel out, and it does not. 138 adds where the rule subtracts. 50 is the halfway point of 0 and 100, as though every such swap dragged the average to the middle; it does that only for a list already averaging 50.