To raise money for an orphanage, students of three schools A, B and C organized an exhibition in their locality, where they sold paper bags, scrap books and pastel sheets made by them using recycled paper, at the rate of Rs. 20, Rs. 15 and Rs. 5 per unit respectively. School A sold 25 paper bags, 10 scrap books and 30 pastel sheets, School B sold 20 paper bag, 15 scrap book and 30 pastel-sheets While school C sold 25 paper bags, 18 scrap books and 35 pastel sheets. Using matrices, find the total amount raised by each school.

Answer

To solve the problem of calculating the total amount raised by each school using matrices, we will follow these steps:

### Step 1: Represent the data in matrix form

We have three schools (A, B, and C) and three items (paper bags, scrap books, and pastel sheets) with their respective sales.

Let's represent the sales data in a matrix \( X \):

\[ X = \begin{bmatrix} 25 & 10 & 30 \\ 20 & 15 & 30 \\ 25 & 18 & 35 \end{bmatrix} \]

Where:
- The first row represents the sales from School A.
- The second row represents the sales from School B.
- The third row represents the sales from School C.

### Step 2: Create a price matrix

Next, we will create a price matrix \( Y \) that represents the price of each item:

\[ Y = \begin{bmatrix} 20 \\ 15 \\ 5 \end{bmatrix} \]

Where:
- The first element is the price of paper bags (Rs. 20).
- The second element is the price of scrap books (Rs. 15).
- The third element is the price of pastel sheets (Rs. 5).

### Step 3: Multiply the matrices

To find the total amount raised by each school, we need to multiply the price matrix \( Y \) by the sales matrix \( X \). The multiplication will be done as follows:

\[ \text{Total Amount} = Y^T \cdot X \]

Where \( Y^T \) is the transpose of matrix \( Y \):

\[ Y^T = \begin{bmatrix} 20 & 15 & 5 \end{bmatrix} \]

Now we perform the multiplication:

\[ \text{Total Amount} = \begin{bmatrix} 20 & 15 & 5 \end{bmatrix} \cdot \begin{bmatrix} 25 & 10 & 30 \\ 20 & 15 & 30 \\ 25 & 18 & 35 \end{bmatrix} \]

Calculating each element:

1. For School A:
\[ 20 \times 25 + 15 \times 10 + 5 \times 30 = 500 + 150 + 150 = 800 \]

2. For School B:
\[ 20 \times 20 + 15 \times 15 + 5 \times 30 = 400 + 225 + 150 = 775 \]

3. For School C:
\[ 20 \times 25 + 15 \times 18 + 5 \times 35 = 500 + 270 + 175 = 945 \]

Thus, the total amounts raised by each school are:
- School A: Rs. 800
- School B: Rs. 775
- School C: Rs. 945

### Step 4: Calculate the total amount raised by all schools

Now, we can find the total amount raised by all schools combined:

\[ \text{Total Amount Raised} = 800 + 775 + 945 = 2520 \]

### Final Result

The total amount raised by each school is:
- School A: Rs. 800
- School B: Rs. 775
- School C: Rs. 945

The total amount raised by all schools combined is Rs. 2520.

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