A company sells two types of gift baskets: Standard and Premium. The sales at the company'… — ACT

A company sells two types of gift baskets: Standard and Premium. The sales at the company's North and South branches are represented by matrix $S$ :

S = \begin{bmatrix} 12 & 8 \\ 15 & 10 \end{bmatrix}

where the rows represent the North and South branches, respectively, and the columns represent the number of Standard and Premium baskets sold, respectively. The cost of the items inside each basket is represented by matrix $C$ :

C = \begin{bmatrix} x & 24 \\ 30 & y \end{bmatrix}

where the rows represent Standard and Premium baskets, respectively, and the columns represent the cost of food items and the cost of gift items (in dollars), respectively. If the total food item cost at the North branch is $600$ and the total gift item cost at the South branch is $760$ , what is the value of $x + y$ ?

Answer: 70

Answer

The value of

x + y

is 70.

To find the total costs, the sales matrix

S

is multiplied by the unit cost matrix

C

. The product matrix

R = SC

is a

2 \times 2

matrix where the entry in Row 1, Column 1 represents the total food cost at the North branch, and the entry in Row 2, Column 2 represents the total gift cost at the South branch. Setting up the equations

12x + 240 = 600

and

360 + 10y = 760

yields

x = 30

and

y = 40

, which sum to 70.

Step-by-Step Solution

Set up the matrix multiplication

R = SC

to represent the total costs.

R = \begin{bmatrix} 12x + 240 & 288 + 8y \\ 15x + 300 & 360 + 10y \end{bmatrix}

The product of the sales matrix and the unit cost matrix yields the total cost matrix, where rows represent the branches and columns represent the cost categories.

Identify the expression for the total food item cost at the North branch and solve for

x

x = 30

The total food item cost at the North branch is the entry in Row 1, Column 1 of the product matrix, which is

12x + 8(30) = 12x + 240

. Setting this equal to

600

gives

12x + 240 = 600

, which simplifies to

12x = 360

, so

x = 30

Identify the expression for the total gift item cost at the South branch and solve for

y

y = 40

The total gift item cost at the South branch is the entry in Row 2, Column 2 of the product matrix, which is

15(24) + 10y = 360 + 10y

. Setting this equal to

760

gives

360 + 10y = 760

, which simplifies to

10y = 400

, so

y = 40

Calculate the sum of the variables

x

and

y

x + y = 70

Substitute

x = 30

and

y = 40

into the expression

x + y

to find the final answer.

Key Concept

Matrix multiplication and translating real-world scenarios into matrix equations.

Estimated Time:3m 0s

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