The Health Fare Cereal Company makes three cereals using wheat, oats, and raisins. The proportions of each cereal follow.
    	Proportion of Each
Pound of Cereal
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
Use matrix multiplication to determine the number of pounds of wheat, oats, and raisins needed to fill an order of 1460 pounds of Lite, 1640 pounds of Trim, and 2200 pounds of Health Fare.
wheat    	
 lbs
oats    	
 lbs
raisins    	
 lbs


To solve this problem, we can model the situation using matrix multiplication. We have a matrix that represents the proportions of wheat, oats, and raisins in each type of cereal and a column vector that represents the number of pounds of each type of cereal ordered.

### Step 1: Define the proportion matrix.

Let the matrix \( A \) represent the proportion of each ingredient in each cereal:

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

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

### Step 2: Define the order vector.

Let the column vector \( B \) represent the number of pounds of each cereal ordered:

\[
B = \begin{bmatrix}
1460 \\
1640 \\
2200
\end{bmatrix}
\]

This vector represents 1460 pounds of Lite, 1640 pounds of Trim, and 2200 pounds of Health Fare.

### Step 3: Multiply the proportion matrix by the order vector.

To find the total pounds of wheat, oats, and raisins needed, we multiply the matrix \( A \) by the vector \( B \):

\[
\begin{bmatrix}
0.75 & 0.25 & 0 \\
0.50 & 0.25 & 0.25 \\
0.25 & 0.50 & 0.25
\end{bmatrix}
\times
\begin{bmatrix}
1460 \\
1640 \\
2200
\end{bmatrix}
\]

### Step 4: Perform the matrix multiplication.

We calculate each element of the result by multiplying the corresponding row in the matrix \( A \) by the column vector \( B \):

\[
\text{Wheat} = (0.75 \times 1460) + (0.50 \times 1640) + (0.25 \times 2200)
\]
\[
\text{Oats} = (0.25 \times 1460) + (0.25 \times 1640) + (0.50 \times 2200)
\]
\[
\text{Raisins} = (0 \times 1460) + (0.25 \times 1640) + (0.25 \times 2200)
\]

Now let's compute these values:

\[
\text{Wheat} = 1095 + 820 + 550 = 2465 \text{ lbs}
\]
\[
\text{Oats} = 365 + 410 + 1100 = 1875 \text{ lbs}
\]
\[
\text{Raisins} = 0 + 410 + 550 = 960 \text{ lbs}
\]

### Final answer:

- **Wheat:** 2465 lbs
- **Oats:** 1875 lbs
- **Raisins:** 960 lbs

Would you like more details on any of the steps? Here are some additional questions to explore:

1. How would the solution change if the proportions of the ingredients were altered?
2. Can this matrix system be extended to include more ingredients?
3. How can matrix multiplication help in optimizing the ingredient proportions for cost efficiency?
4. What other real-world problems can be modeled using matrix multiplication like this one?
5. How would the solution change if the number of pounds of each cereal ordered was different?

**Tip:** Matrix multiplication is a powerful tool for dealing with linear systems where multiple variables (like ingredients) are combined in different proportions across multiple categories (like cereals).

The Health Fare Cereal Company makes three cereals using wheat, oats, and raisins. The proportions of each cereal follow:

Proportion of Each Pound of Cereal
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
Use matrix multiplication to determine the number of pounds of wheat, oats, and raisins needed to fill an order of 1460 pounds of Lite, 1640 pounds of Trim, and 2200 pounds of Health Fare.

Learn how to apply matrix multiplication to calculate the amount of wheat, oats, and raisins needed for a cereal order. This real-world problem involves calculating proportions of ingredients for three types of cereals: Lite, Trim, and Health Fare. Suitable for high school students in Grades 10-12.

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