Question

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 now the mean age of the teachers in this school is $39$ years, then find the age (in years) of the newly appointed teacher?
A. 40
B. 30
C. 25
D. 35

Answer 1

Hint: From the question, we have to find the age of newly appointed teachers. First, we have to express the given data in the mathematical form and use the mean (average) formula to get the required age of the newly appointed teacher.

Formula used: \[{\text{Mean = }}\dfrac{{{\text{Total of all items}}}}{{{\text{Number of items}}}}\].
\[ \Rightarrow {\text{Total of all items = Mean}} \times {\text{Number of items}}\]
 Let the total sum of ages of teachers be ${\text{x}}$.

Complete step-by-step solution:
By the given, the mean age of $25$ teachers in a school is $40$ years.
Therefore, thus the number of teachers is $25$ and the mean of $25$ teachers is $40$.
Now, we are going to find the total sum of age teachers by using the mean formula.
\[{\text{Mean = }}\dfrac{{{\text{Total number of all items}}}}{{{\text{Number of items}}}}\]
\[ \Rightarrow {\text{Total number of all items = Mean}} \times {\text{Number of items}}\]
Substituting the values,
$ \Rightarrow {\text{x}} = 40 \times 25$
Multiplying we get,
$ \Rightarrow {\text{x}} = 1000$.
Thus, the total sum of ages of teachers is $1000$
Since, a new teacher appointed in place of a teacher whose age is $60$
Then,
The sum of teacher ages after the teacher get retired $ = {\text{x}} - 60$
$ \Rightarrow 1000 - 60$
$ \Rightarrow 940$
Now, the mean age of the teachers in this school is $39$ years
The new sum of ages of teacher $ = 39 \times 25 = 975$
We can get the age of the newly appointed teacher by subtracting the sum of the teacher ages after the teacher gets retired from the new sum of ages of the teacher.
The sum of teacher ages after the teacher get retired $ = 940$
The new sum of ages of teacher $ = 975$
Hence,
The age of the newly appointed teacher is $ = 975 - 940 = 35$

$\therefore $ The age of the newly appointed teacher is $35$ years.

Note: The mean is the meaning of an average. The process of sharing out equally is the basis of average. We call the averaged or quantity as the arithmetic average (or arithmetic mean or simply average or mean).
Data handling is an art. Sometimes the raw data (data as they are) will not be useful to get the required information. In order to get proper useful information, we have to process very important useful information from the given data.