A set of 9 numbers averages 50. The first number is then increased by 6, the second by 12, the third by 18, and so on, each increase being 6 more than the one before, up to the ninth number, which is increased by 54. What is the average of the 9 numbers now?

The nine increases are 6, 12, 18, ... , 54, which add to $6\times (1+2+...+9)=6\times 45=270$. The old total was $9\times 50=450$, the new total is 720, and 720/9 = 80. Quicker: the increases themselves average $(6+54)/2=30$, and adding an average of 30 to the numbers adds 30 to their average, giving 50 + 30 = 80. 56 adds only the smallest increase, 6, as though every number went up by the same amount. 104 adds only the largest, 54. 320 adds the whole 270 to the average instead of to the total.