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There are three convenience stores in Gambier. This week:

- Store I sold 80 loaves of bread, 40 quarts of milk, 16 jars of peanut butter, and 116 pounds of cold cuts.
- Store II sold 105 loaves of bread, 75 quarts of milk, 24 jars of peanut butter, and 150 pounds of cold cuts.
- Store III sold 50 loaves of bread, 40 quarts of milk, no peanut butter, and 50 pounds of cold cuts.

Complete parts (a) through (c) below.

(a) Use a [tex]$3 \times 4$[/tex] matrix to express the sales information for the three stores. Select the correct answer below.

A. [tex]\left[\begin{array}{rrrr}40 & 16 & 116 & 80 \\ 24 & 105 & 150 & 75 \\ 50 & 0 & 50 & 40\end{array}\right][/tex]

B. [tex]\left[\begin{array}{rrrr}80 & 40 & 16 & 116 \\ 105 & 75 & 24 & 150 \\ 50 & 40 & 0 & 50\end{array}\right][/tex]

C. [tex]\left[\begin{array}{rrrr}105 & 75 & 150 & 24 \\ 50 & 0 & 50 & 40\end{array}\right][/tex]

D. [tex]\left[\begin{array}{rrrr}40 & 16 & 80 & 116 \\ 105 & 24 & 150 & 75 \\ 50 & 0 & 40 & 50\end{array}\right][/tex]

Answer :

We want to create a [tex]$3 \times 4$[/tex] matrix where each row represents one store and the columns represent the following items in this order: loaves of bread, quarts of milk, jars of peanut butter, and pounds of cold cuts.

Step 1. For Store I, the sales are:
[tex]$$
\text{Bread} = 80,\quad \text{Milk} = 40,\quad \text{Peanut Butter} = 16,\quad \text{Cold Cuts} = 116.
$$[/tex]
So the first row is:
[tex]$$
[80,\ 40,\ 16,\ 116].
$$[/tex]

Step 2. For Store II, the sales are:
[tex]$$
\text{Bread} = 105,\quad \text{Milk} = 75,\quad \text{Peanut Butter} = 24,\quad \text{Cold Cuts} = 150.
$$[/tex]
So the second row is:
[tex]$$
[105,\ 75,\ 24,\ 150].
$$[/tex]

Step 3. For Store III, the sales are:
[tex]$$
\text{Bread} = 50,\quad \text{Milk} = 40,\quad \text{Peanut Butter} = 0,\quad \text{Cold Cuts} = 50.
$$[/tex]
So the third row is:
[tex]$$
[50,\ 40,\ 0,\ 50].
$$[/tex]

Step 4. Putting these rows together, we get the complete matrix:
[tex]$$
\begin{bmatrix}
80 & 40 & 16 & 116 \\
105 & 75 & 24 & 150 \\
50 & 40 & 0 & 50
\end{bmatrix}.
$$[/tex]

Step 5. Comparing with the provided options, we see that this matrix is exactly the one in Option B.

Thus, the correct answer is Option B.

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