A finance company has offices located in ewery division,
every didtrict and every taluka in a certain state in India.
Assume that there are five divisions, thirty districts and
200 talukas in the state. Each office has one head clerk,
one cashier, one clerk and one peon. A divisional office
has, in addition, one office superntendent, two clerks,
one typist and one poen. A district office, has in
addition, one clerk and one peon. The basic monthly
salaries are as follows :
Office superintendernt Rs 500, Head clerk Rs 200, cashier Rs 175, clerks and typist
Rs 150 and peon Rs 100. Using matrix motation find
the total basic monthly salary bill of each kind of office

Explanation

To find the total basic monthly salary bill of each kind of office using matrix notation, we will follow these steps:

### Step 1: Define the Salaries
We first define the basic monthly salaries for each position:
- Office Superintendent (OS): Rs 500
- Head Clerk (HC): Rs 200
- Cashier (C): Rs 175
- Clerk (CL): Rs 150
- Typist (T): Rs 150
- Peon (P): Rs 100

### Step 2: Define the Structure of Each Office
Next, we define how many of each position exists in each type of office:
- **Divisional Office (DO)**:
- OS: 1
- HC: 1
- C: 1
- CL: 3 (1 + 2 additional)
- T: 1
- P: 1
- **District Office (DTO)**:
- OS: 0
- HC: 1
- C: 1
- CL: 2 (1 + 1 additional)
- T: 0
- P: 2
- **Taluka Office (TO)**:
- OS: 0
- HC: 1
- C: 1
- CL: 1
- T: 0
- P: 1

### Step 3: Create the Matrix Representation
We can represent the number of employees in each type of office in a matrix format:

\[ \text{Office Matrix} = \begin{bmatrix} 1 & 1 & 1 & 3 & 1 & 1 \\ % Divisional Office 0 & 1 & 1 & 2 & 0 & 2 \\ % District Office 0 & 1 & 1 & 1 & 0 & 1 % Taluka Office \end{bmatrix} \]

### Step 4: Create the Salary Matrix
Next, we create a salary matrix for each position:

\[ \text{Salary Matrix} = \begin{bmatrix} 500 \\ % OS 200 \\ % HC 175 \\ % C 150 \\ % CL 150 \\ % T 100 % P \end{bmatrix} \]

### Step 5: Multiply the Matrices
Now, we multiply the Office Matrix by the Salary Matrix to find the total salaries for each type of office.

\[ \text{Total Salary} = \text{Office Matrix} \times \text{Salary Matrix} \]

Calculating each row:
1. **Divisional Office**:
\[ 1 \times 500 + 1 \times 200 + 1 \times 175 + 3 \times 150 + 1 \times 150 + 1 \times 100 = 500 + 200 + 175 + 450 + 150 + 100 = 1575 \]

2. **District Office**:
\[ 0 \times 500 + 1 \times 200 + 1 \times 175 + 2 \times 150 + 0 \times 150 + 2 \times 100 = 0 + 200 + 175 + 300 + 0 + 200 = 875 \]

3. **Taluka Office**:
\[ 0 \times 500 + 1 \times 200 + 1 \times 175 + 1 \times 150 + 0 \times 150 + 1 \times 100 = 0 + 200 + 175 + 150 + 0 + 100 = 625 \]

### Step 6: Total Monthly Salary Bill
Now, we need to calculate the total monthly salary bill for all offices:
- There are 5 Divisional Offices, 30 District Offices, and 200 Taluka Offices.

Calculating the total:
\[ \text{Total Salary Bill} = (1575 \times 5) + (875 \times 30) + (625 \times 200) \]

Calculating each term:
1. \(1575 \times 5 = 7875\)
2. \(875 \times 30 = 26250\)
3. \(625 \times 200 = 125000\)

Adding these together:
\[ \text{Total Salary Bill} = 7875 + 26250 + 125000 = 153125 \]

### Final Answer
The total basic monthly salary bill of each kind of office is Rs 153125.

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