The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is _________.

Solution

Maths sir · 4 steps

Work with the total age, not the mean: one teacher swapped changes the total by exactly the age gap.

1. Total before. Mean times count gives the total. Before the change, the 25 ages add up to 25 × 40 = 1000 years.
2. Total after. There are still 25 teachers, now with mean 39. So the new total is 25 × 39 = 975 years.
3. Swap the teacher. Removing the 60-year-old and adding the new teacher of age x is the only change to the total.

   1000 - 60 + x = 975 ⇒ x = 35

4. So. The newly appointed teacher is 35 years old. As a check, the mean fell by 1 across 25 people, so the total fell by 25, and 60 - 25 = 35.

The common slip: 59. Thinking the new teacher is only 1 year younger than the one who retired, because the mean fell by 1, gives 59. The fall of 1 is spread over all 25 teachers, so the total falls by 25.

The idea to keep. When one value in a data set is replaced, convert means to totals first. The change in the total equals the new value minus the old value.

Answer from the NTA final answer key. Solution by PrepWiser.