The average age of the 4 doctors at a small clinic is 41 years. One of them, aged 56, leaves and a younger doctor takes his place, bringing the average age down to 38 years. How old is the newcomer?

The clinic still has 4 doctors, so a 3-year fall in the average means the total of the ages fell by $4\times 3=12$ years. The only change was the swap, so the newcomer is 56 - 12 = 44 years old. (Totals: 164 before, 152 after; $152-(164-56)=44$.) 53 subtracts the 3 years once, forgetting that four people share the average. 68 adds the 12 years, but a younger doctor is needed for the average to fall. 38 is the new average, not the newcomer's age.