Step 1: Understanding the Question:
This problem is a classic Averages question involving the addition of a new member to a group. When a person with a different value (age) joins a group, they pull the group's average toward their own value. If the average increases, the new person must be older than the current average. We are given the data for 15 students and the change caused by the teacher's arrival. The goal is to calculate the teacher's specific age.
Step 2: Key Formulas and approach:
The most direct approach is the "Sum Difference" method.
1. Calculate the total age of the original group: $\text{Total 1} = \text{Average 1} \times \text{Count 1}$.
2. Calculate the total age of the new group: $\text{Total 2} = \text{Average 2} \times \text{Count 2}$.
3. The teacher's age is simply $(\text{Total 2} - \text{Total 1})$.
Alternatively, one can use the "Excess Method," which looks at how much extra the teacher brings to raise everyone's average by 1 year.
Step 3: Detailed Explanation:
We first find the total combined age of the 15 students. Since their average is 18, the total is $15 \times 18 = 270$ years.
When the teacher is added, the total number of people in the group becomes $15 + 1 = 16$.
The problem states the average increases by 1, so the new average for all 16 people is $18 + 1 = 19$ years.
Now, we find the total combined age of these 16 people: $16 \times 19 = 304$ years.
To isolate the teacher's age, we subtract the students' total age from this new grand total: $304 - 270 = 34$ years.
Looking at it another way: The teacher must provide 18 years just to match the old average, plus an additional 1 year for every person now in the room (16 people) to raise the average. So, $18 + 16 = 34$.
Both methods lead to the same result, confirming that the teacher's age must be 34 to satisfy the given conditions.
Step 4: Final Answer:
The teacher's age is 34 years.