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To solve this problem using matrix multiplication, we will define the proportions and quantities in matrix form.

### Step 1: Define the Proportions Matrix
The given proportions of wheat, oats, and raisins in each type of cereal are:

\[
P = \begin{bmatrix} 0.75 & 0.25 & 0 \\ 0.50 & 0.25 & 0.25 \\ 0.25 & 0.50 & 0.25 \end{bmatrix}
\]

Each row of matrix \( P \) corresponds to a cereal (Lite, Trim, Health Fare), and each column corresponds to the proportion of wheat, oats, and raisins, respectively.

### Step 2: Define the Quantities Matrix
The quantities of each cereal needed are given as:

\[
Q = \begin{bmatrix} 1300 \\ 1820 \\ 2080 \end{bmatrix}
\]

This matrix \( Q \) represents the total pounds of Lite, Trim, and Health Fare cereals, respectively.

### Step 3: Calculate the Total Pounds of Each Ingredient
Let \( X = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \), where \( x_1 \), \( x_2 \), and \( x_3 \) represent the total pounds of wheat, oats, and raisins needed.

The relationship between \( P \), \( Q \), and \( X \) is given by the matrix equation:

\[
P \times X = Q
\]

Thus, we solve for \( X \):

\[
X = P^{-1} \times Q
\]

Let's calculate this in detail to find the amount of each ingredient needed.The solution indicates that to meet the given order, the Health Fare Cereal Company needs the following amounts of each ingredient:

- **Wheat**: 1,040 pounds
- **Oats**: 2,080 pounds
- **Raisins**: 3,120 pounds

This calculation ensures that the given proportions in the cereals are maintained. Do you want further details or have any questions?

### Related Questions
1. How would the ingredient amounts change if the order size for Lite cereal increased by 10%?
2. What happens to the quantities of wheat, oats, and raisins if the proportions in one of the cereals are modified?
3. How can we solve this problem if the matrix \( P \) is not invertible?
4. How would the solution be impacted if a new cereal is introduced with a different set of proportions?
5. Can we use a linear programming approach to minimize the cost of ingredients for a given order?

**Tip**: Matrix multiplication can be used to model and solve various practical problems, such as optimizing ingredient usage or scheduling production tasks.

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 calculate the pounds of wheat, oats, and raisins required to fill an order of 1300 pounds of Lite, 1820 pounds of Trim, and 2080 pounds of Health Fare cereal. Understand the proportions of ingredients in different cereals and apply linear algebra to solve the system.

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