The mean age group of some students is 15years. When the age of a teacher, 45 years old, is added to the ages…

1. Analyze the question

We are given the mean age of a group of students. When a teacher (45 years old) is added to this group, the mean age increases. We need to find the original number of students.

2. Explain the concept

The formula for the mean is:

From this, we can derive:

3. Show the solution

Let $n$ be the number of students. Let $S$ be the sum of the ages of the students.

Step 1: Express the initial state The mean age of the students is 15.

Step 2: Express the state after adding the teacher When the teacher (45 years old) is added, the new sum of ages is $S + 45$, and the new number of people is $n + 1$. The new mean is 18.

Step 3: Solve for $n$ Substitute $S = 15n$ into the equation:

Multiply both sides by $(n + 1)$:

Rearrange the equation to group the $n$ terms:

4. Identify the answer

The number of students in the group is 9. This matches option B.

5. Tip

When dealing with "mean" problems involving adding a new person, always use the "Sum = Mean × Count" approach. It prevents confusion when the number of people in the denominator changes. Also, always check if your answer makes sense: since the teacher is much older than the average, adding them should increase the mean, which is exactly what happened here (from 15 to 18).