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:

We start with the fundamental relation for average (mean): $$\text{Mean}=\frac{\text{Sum of observations}}{\text{Number of observations}}.$$

Initially, there are 25 teachers and their mean age is 40 years. So, using the above relation, we find the total of their ages.

$$\text{Initial total age}= \text{Mean} \times \text{Number of teachers}=40 \times 25 = 1000.$$

One teacher, whose age is 60 years, retires. Hence, the sum of ages is reduced by 60.

$$\text{Total age after retirement}=1000 - 60 = 940.$$

A new teacher is appointed. Let his age be $$x$$ years. After this appointment, the number of teachers is again 25 and their new mean age is 39 years. Applying the mean formula once more,

$$\text{New total age}= \text{New mean} \times \text{Number of teachers}=39 \times 25=975.$$

This new total age also equals the sum just after retirement plus the age of the newly appointed teacher, so

$$940 + x = 975.$$

Solving for $$x$$, we subtract 940 from both sides:

$$x = 975 - 940 = 35.$$

Thus, the age of the newly appointed teacher is $$35$$ years.

Hence, the correct answer is Option A.