Step 1: Understanding the Concept:
The concept of "Average" (specifically the Arithmetic Mean) represents a central value of a set of data. It is calculated by dividing the total sum of all observations by the number of observations.
When a new entity is added to a group, the total count increases, and the sum of values increases by the value of that new entity.
If the average increases after adding a new person, it implies that the new person's age is higher than the original average.
Conversely, if the average had decreased, the new person would be younger than the original average.
The simplest way to solve this is to calculate the total "Age Sum" before and after the teacher joins and find the difference.
Step 2: Key Formula or Approach:
1. \(\text{Sum of Values} = \text{Average} \times \text{Total Number of Individuals}\)
2. \(\text{Teacher's Age} = \text{New Total Sum} - \text{Original Total Sum}\)
Step 3: Detailed Explanation:
Let's analyze the problem in two distinct states:
Initial State (Before Teacher Joins):
There are \(5\) students in the class.
The average age is \(18\) years.
Total age of these 5 students = \(5 \times 18 = 90\) years.
Final State (After Teacher Joins):
Now, the teacher joins the group, so the total number of people becomes \(6\) (\(5 \text{ students} + 1 \text{ teacher}\)).
The new average age for the entire group is given as \(22\) years.
Total age of all 6 people = \(6 \times 22 = 132\) years.
Calculating the Teacher's Age:
The difference between the total age of the group with the teacher and without the teacher must be the teacher's individual age.
\[ \text{Teacher's Age} = 132 - 90 \]
\[ \text{Teacher's Age} = 42 \text{ years} \]
Alternative Logic (Deviation Method):
To join a group with an average of \(22\), the teacher must "bring" enough years to match that average.
Furthermore, the teacher's presence raised the average age of the other 5 students from \(18\) to \(22\) (an increase of \(4\) years per student).
Total increase required for the students = \(5 \times 4 = 20\) years.
Therefore, Teacher's age = \(\text{Target average} + \text{Total increase given to students} = 22 + 20 = 42\) years.
Step 4: Final Answer:
The age of the teacher is 42 years.