Burger Barns three locations sell fries, burgers and soft drinks. Brant I sells 500 orders of fries, 150 burgers and 300 soft drinks on daily basis. Bran II and Bran III sell 900 and 700 orders of fries respectively, per day. Bran II sells 440 burgers and 830 soft drinks while Bran III sells 580 burgers and 1200 soft drinks each day.
(a)	Write a 3 x 3 matrix that represents daily sales for all locations.
(b)	If fries cost $0.92 per order, burger $1.54 each and soft drink $0.58 each. Write a matrix of order 1 x 3 that represents the prices.
(c)	What is the total daily income from all the locations?	(7+7+6=20)

We are asked to solve three parts of a problem related to sales data from three locations of a burger restaurant. Here's how we can approach each part:

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### (a) Write a $3 \times 3$ matrix that represents daily sales for all locations.

To represent the daily sales data, we will use a matrix where:
- Each row represents a specific branch (Branch I, II, III).
- Each column represents the type of item (Fries, Burgers, Soft Drinks).

Given the sales dataWe are asked to solve three parts of a problem related to sales data from three locations of a burger restaurant. Here's how we can approach each part:

---

### (a) Write a $3 \times 3$ matrix that represents daily sales for all locations.

To represent the daily sales data, we will use a matrix where:
- Each row represents a specific branch (Branch I, II, III).
- Each column represents the type of item (Fries, Burgers, Soft Drinks).

Given the sales 
- **Branch I**: 500 orders of fries, 150 burgers, 300 soft drinks.
- **Branch II**: 900 orders of fries, 440 burgers, 830 soft drinks.
- **Branch III**: 700 orders of fries, 580 burgers, 1200 soft drinks.

We can represent this in a matrix $ A $:

\[
A = \begin{bmatrix}
500 & 150 & 300 \\
900 & 440 & 830 \\
700 & 580 & 1200
\end{bmatrix}
\]

---

### (b) Write a $1 \times 3$ matrix that represents the prices.

The prices for each item are:
- Fries: $0.92 per order.
- Burgers: $1.54 each.
- Soft drinks: $0.58 each.

This information can be represented by a $1 \times 3$ matrix $ P $:

\[
P = \begin{bmatrix}
0.92 & 1.54 & 0.58
\end{bmatrix}
\]

---

### (c) What is the total daily income from all locations?

To find the total daily income, we multiply the sales matrix $ A $ by the prices matrix $ P $. This matrix multiplication will give us a $3 \times 1$ matrix where each element represents the total income for a specific branch. The matrix multiplication is done as follows:

\[
Income = A \times P^T
\]

Where $ P^T $ is the transpose of the prices matrix (to make it $3 \times 1$).

Let's calculate the total income next:

\[
Income = \begin{bmatrix}
500 & 150 & 300 \\
900 & 440 & 830 \\
700 & 580 & 1200
\end{bmatrix}
\times
\begin{bmatrix}
0.92 \\
1.54 \\
0.58
\end{bmatrix}
\]

Now let's perform the calculation to determine the total income for each branch.The total daily income from each branch is as follows:
- Branch I: \$865.00
- Branch II: \$1987.00
- Branch III: \$2233.20

The total daily income from all three locations is \$5085.20.

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Do you need further details or explanations? 

Here are 5 follow-up questions to expand on this topic:
1. How would the sales matrix change if a new product, such as a dessert, was introduced?
2. How would you adjust the calculation if one location started offering a different price for one of the items?
3. What happens if one of the branches offers a discount on fries during lunch hours? How would that affect the income calculation?
4. Could you calculate the total income over a week or month given these daily figures?
5. How could you calculate the average sales of each item across all branches?

**Tip:** When performing matrix multiplication, ensure that the dimensions of the matrices match appropriately for valid multiplication!

Burger Barn's three locations sell fries, burgers, and soft drinks. Brant I sells 500 orders of fries, 150 burgers, and 300 soft drinks daily. Brant II and Brant III sell 900 and 700 orders of fries, respectively, per day. Brant II sells 440 burgers and 830 soft drinks, while Brant III sells 580 burgers and 1200 soft drinks each day. (a) Write a 3x3 matrix that represents daily sales for all locations. (b) If fries cost $0.92 per order, burgers $1.54 each, and soft drinks $0.58 each, write a matrix of order 1x3 that represents the prices. (c) What is the total daily income from all the locations?

Learn how to calculate total daily income from sales data across multiple locations using matrix multiplication. This problem includes daily sales data of fries, burgers, and soft drinks from three locations of Burger Barn, price per item, and matrix multiplication to find total income. Suitable for students in Grades 10-12.

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