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The problem requires determining the number of pounds of wheat, oats, and raisins needed to fulfill an order of different amounts of three types of cereals using matrix multiplication.

### Given Information:
We have three types of cereals—Lite, Trim, and Health Fare—each made with a specific proportion of wheat, oats, and raisins.

| Cereal      | Wheat | Oats  | Raisins |
|-------------|-------|-------|---------|
| **Lite**    | 0.75  | 0.25  | 0       |
| **Trim**    | 0.50  | 0.25  | 0.25    |
| **Health Fare** | 0.25  | 0.50  | 0.25    |

We also know the following order quantities:
- 1300 pounds of Lite,
- 1820 pounds of Trim, and
- 2080 pounds of Health Fare.

### Step-by-Step Matrix Multiplication Approach:

1. **Set up the matrix for proportions** (rows for each cereal, columns for ingredients):
   \[
   A = \begin{pmatrix}
   0.75 & 0.25 & 0 \\
   0.50 & 0.25 & 0.25 \\
   0.25 & 0.50 & 0.25
   \end{pmatrix}
   \]
   Here, each row represents a cereal (Lite, Trim, Health Fare) and each column represents the proportions of wheat, oats, and raisins.

2. **Set up the matrix for the order quantities**:
   \[
   B = \begin{pmatrix}
   1300 \\
   1820 \\
   2080
   \end{pmatrix}
   \]
   This matrix contains the order quantities for Lite, Trim, and Health Fare respectively.

3. **Multiply matrix A by matrix B** to determine the total amount of each ingredient (wheat, oats, raisins):
   \[
   A \times B = \begin{pmatrix}
   0.75 & 0.25 & 0 \\
   0.50 & 0.25 & 0.25 \\
   0.25 & 0.50 & 0.25
   \end{pmatrix}
   \times
   \begin{pmatrix}
   1300 \\
   1820 \\
   2080
   \end{pmatrix}
   \]

Let's calculate the result:

- **Wheat**: \( (0.75 \times 1300) + (0.50 \times 1820) + (0.25 \times 2080) \)
- **Oats**: \( (0.25 \times 1300) + (0.25 \times 1820) + (0.50 \times 2080) \)
- **Raisins**: \( (0 \times 1300) + (0.25 \times 1820) + (0.25 \times 2080) \)

### Calculations:
- Wheat = \( 975 + 910 + 520 = 2405 \) pounds.
- Oats = \( 325 + 455 + 1040 = 1820 \) pounds.
- Raisins = \( 0 + 455 + 520 = 975 \) pounds.

### Final Answer:
- **Wheat**: 2405 pounds.
- **Oats**: 1820 pounds.
- **Raisins**: 975 pounds.

Would you like further details on matrix multiplication or have any other questions?

### Follow-up Questions:
1. How can matrix algebra be used to solve real-life problems in manufacturing?
2. What is the advantage of using matrices for such calculations instead of manual addition?
3. Can matrix operations be used in other areas of business, such as logistics?
4. How can we extend this method for a more complex cereal company with more ingredients?
5. What would happen if the proportion of ingredients changed after a few months?

### Tip:
When performing matrix multiplication, always ensure the dimensions of the matrices are compatible: the number of columns in the first matrix must equal the number of rows in the second.

Use matrix multiplication to determine the number of pounds of wheat, oats, and raisins needed to fill an order of 1300 pounds of Lite, 1820 pounds of Trim, and 2080 pounds of Health Fare.

Learn how to use matrix multiplication to determine the total amount of wheat, oats, and raisins needed for an order of Lite, Trim, and Health Fare cereals. This problem covers key matrix concepts and linear algebra, suitable for high school students.

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