In a school , the average age of students is 6 years , and the average age of 12 teachers is 40 years . If the average age of the combined group of all the teachers and students is 7 years , then the number of student is :

Options

A. 396
B. 400
C. 408
D. 416

Answer

To solve the problem step by step, we will use the information given about the average ages of students and teachers, and the combined average age of both groups.

### Step 1: Define Variables
Let:
- \( n \) = number of students
- The average age of students = 6 years
- The average age of teachers = 40 years
- Number of teachers = 12
- Combined average age of students and teachers = 7 years

### Step 2: Calculate Total Ages
Calculate the total age of the teachers:
\[ \text{Total age of teachers} = \text{Average age of teachers} \times \text{Number of teachers} = 40 \times 12 = 480 \text{ years} \]

Calculate the total age of the students:
\[ \text{Total age of students} = \text{Average age of students} \times \text{Number of students} = 6n \]

### Step 3: Set Up the Equation for Combined Average
The combined average age of students and teachers is given as 7 years. The total number of individuals (students + teachers) is \( n + 12 \). Therefore, we can set up the equation:
\[ \text{Combined average} = \frac{\text{Total age of students} + \text{Total age of teachers}}{\text{Total number of individuals}} \]
Substituting the known values:
\[ 7 = \frac{6n + 480}{n + 12} \]

### Step 4: Clear the Fraction
Multiply both sides by \( n + 12 \) to eliminate the fraction:
\[ 7(n + 12) = 6n + 480 \]

### Step 5: Expand and Rearrange the Equation
Expanding the left side:
\[ 7n + 84 = 6n + 480 \]
Now, rearranging the equation to isolate \( n \):
\[ 7n - 6n = 480 - 84 \]
\[ n = 396 \]

### Conclusion
The number of students \( n \) is 396.

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