The average age of the 5 members of a music club is 34 years. A new member joins and the average age of the club drops to 31 years. How old is the new member?

The five members total $5\times 34=170$ years and the six now total $6\times 31=186$ years, so the newcomer is 186 - 170 = 16 years old. The shortcut: the newcomer must sit at the new average of 31 and also make good the 3-year drop for each of the 5 who were already there, so $31-5\times 3=16$. 13 uses the new count of 6 for that repair rather than the 5 existing members. 19 subtracts the 15 years from the OLD average of 34. 31 is the club's new average, not anyone's age.