The average salary of all staff of a school is Rs 10,000. The average salary of 20 teaching staff is Rs 12,000 and that of non-teaching staff is Rs 5000. the number of non-teaching staff will be:

Options

A. 7
B. 8
C. 10
D. 12

Answer

To find the number of non-teaching staff in the school, we can follow these steps:

### Step 1: Define the Variables
Let the number of non-teaching staff be \( X \).

### Step 2: Calculate Total Salaries
- The average salary of all staff is Rs 10,000.
- The average salary of 20 teaching staff is Rs 12,000.
- The average salary of non-teaching staff is Rs 5,000.

### Step 3: Calculate Total Salary of Teaching Staff
The total salary of the 20 teaching staff can be calculated as:
\[ \text{Total salary of teaching staff} = \text{Number of teaching staff} \times \text{Average salary of teaching staff} = 20 \times 12,000 = 240,000 \text{ Rs} \]

### Step 4: Calculate Total Salary of Non-Teaching Staff
The total salary of non-teaching staff can be expressed as:
\[ \text{Total salary of non-teaching staff} = X \times 5,000 \]

### Step 5: Calculate Total Salary of All Staff
The total salary of all staff (teaching + non-teaching) can be expressed as:
\[ \text{Total salary of all staff} = \text{Total salary of teaching staff} + \text{Total salary of non-teaching staff} = 240,000 + 5,000X \]

### Step 6: Set Up the Average Salary Equation
The average salary of all staff is given by:
\[ \text{Average salary} = \frac{\text{Total salary of all staff}}{\text{Total number of staff}} = \frac{240,000 + 5,000X}{20 + X} \]
We know that the average salary is Rs 10,000, so we can set up the equation:
\[ 10,000 = \frac{240,000 + 5,000X}{20 + X} \]

### Step 7: Cross-Multiply to Solve for \( X \)
Cross-multiplying gives:
\[ 10,000(20 + X) = 240,000 + 5,000X \]
Expanding this, we get:
\[ 200,000 + 10,000X = 240,000 + 5,000X \]

### Step 8: Rearranging the Equation
Rearranging the equation to isolate \( X \):
\[ 10,000X - 5,000X = 240,000 - 200,000 \]
\[ 5,000X = 40,000 \]

### Step 9: Solve for \( X \)
Dividing both sides by 5,000 gives:
\[ X = \frac{40,000}{5,000} = 8 \]

### Conclusion
The number of non-teaching staff is \( 8 \).

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