The average age of the 24 students in a class is 13 years. When the class teacher's age is included, the average goes up by exactly 1 year. The teacher's age (in years) is:

The 24 students total $24\times 13=312$ years. With the teacher there are 25 people averaging 14 years, a total of 350, so the teacher is 350 − 312 = 38. (Shortcut: the teacher must carry the new average of 14 for himself plus 1 extra year for each of the 24 students, i.e. 14 + 24 = 38. Answering 37 uses the old average, 13 + 24.)