To solve the problem, we need to find the number of students in a group given the mean age of the students and how it changes when a teacher's age is added. Let's break this down step-by-step. ### Step 1: Understand the Information Given 1. **Mean age of students**: 15 years 2. **Teacher's age**: 45 years 3. **New mean age (with teacher)**: 18 years ### Step 2: Set Up the Equations Let \( n \) be the number of students in the group. - The total age of the students can be calculated using the mean: \[ \text{Total age of students} = \text{Mean age} \times \text{Number of students} = 15n \] - When the teacher's age is added, the total age becomes: \[ \text{Total age with teacher} = 15n + 45 \] - The new mean age with the teacher included is given as 18 years. The total number of individuals (students + teacher) is \( n + 1 \). Therefore, we can express the new mean as: \[ \text{New mean} = \frac{\text{Total age with teacher}}{\text{Total number of individuals}} = \frac{15n + 45}{n + 1} \] ### Step 3: Set Up the Equation for the New Mean We know that the new mean is 18, so we can set up the equation: \[ \frac{15n + 45}{n + 1} = 18 \] ### Step 4: Solve the Equation To eliminate the fraction, we can multiply both sides by \( n + 1 \): \[ 15n + 45 = 18(n + 1) \] Expanding the right side: \[ 15n + 45 = 18n + 18 \] Now, we can rearrange the equation to isolate \( n \): \[ 15n + 45 - 18n - 18 = 0 \] \[ -3n + 27 = 0 \] \[ 3n = 27 \] \[ n = 9 \] ### Step 5: Conclusion The number of students in the group is \( n = 9 \). ### Step 6: Verify the Solution To ensure our solution is correct, we can check the calculations: - Total age of students: \( 15 \times 9 = 135 \) - Total age with teacher: \( 135 + 45 = 180 \) - Total number of individuals: \( 9 + 1 = 10 \) - New mean: \( \frac{180}{10} = 18 \) Since the new mean is indeed 18, our solution is verified. ### Step 7: Analyze the Options - **Option A: 9** - This is the correct answer. - **Option B: 7** - Incorrect. If there were 7 students, the total age would be \( 15 \times 7 = 105 \), and the new mean would not equal 18. - **Option C: 42** - Incorrect. This would imply an unrealistically high total age for the students, leading to a new mean far exceeding 18. - **Option D: 15** - Incorrect. Similar to option C, this would also lead to a new mean that does not match the given information. ### Revision Summary - The mean age of students is calculated using the formula: \( \text{Mean} = \frac{\text{Total age}}{\text{Number of individuals}} \). - When adding a new individual (the teacher), the total age and number of individuals must be updated to find the new mean. - Rearranging equations and isolating variables is key to solving for unknowns. - Always verify your solution by plugging it back into the original conditions of the problem.