Ten numbers have an average of 46. Five of the numbers are each increased by 8 and the other five are each decreased by 4. What is the average of the 10 numbers now?

The total was $10\times 46=460$. The rises add $5\times 8=40$ and the falls take away $5\times 4=20$, a net gain of 20, so the new total is 480 and the new average is 480/10 = 48. Equivalently the average of the changes is $(8-4)/2=2$, giving 46 + 2 = 48. 66 adds the whole net gain of 20 to the AVERAGE instead of to the total. 54 applies the +8 to every number and 42 applies the -4 to every number; each change touches only half the set.