Question

A online merchant, SchoolStuff sells pens for 40¢, pencils for 40¢ and erasers for 10¢. Office World sells erasers for 6¢ and Desks R Us sells erasers for 3¢. Find the least expensive place to order 500 pens, 300 pencils and 150 erasers.

Use matrix multiplication to find your answer. Show the matrices used (you may use a calculator for the calculations)

Answer 1

Using matrix multiplication, the total cost matrix for purchasing 500 pens, 300 pencils, and 150 erasers from SchoolStuff, Office World, and Desks R Us is $25,000, $3,000, and $1,500, respectively. Desks R Us is the least expensive option.

Let's denote the prices of the items at each store in a matrix:

`\[ \text{Price Matrix} = \begin{bmatrix} 40 & 40 & 10 \\ 6 & 0 & 0 \\ 3 & 0 & 0 \end{bmatrix} \]`

Now, let's denote the quantities of pens, pencils, and erasers in a matrix:

`\[ \text{Quantity Matrix} = \begin{bmatrix} 500 \\ 300 \\ 150 \end{bmatrix} \]`

The total cost can be found by multiplying the Price Matrix by the Quantity Matrix:

`\[ \text{Total Cost Matrix} = \text{Price Matrix} * \text{Quantity Matrix} \]`

Performing the matrix multiplication:

`\[ \text{Total Cost Matrix} = \begin{bmatrix} 40 & 40 & 10 \\ 6 & 0 & 0 \\ 3 & 0 & 0 \end{bmatrix} * \begin{bmatrix} 500 \\ 300 \\ 150 \end{bmatrix} \]\[ \text{Total Cost Matrix} = \begin{bmatrix} (40 * 500) + (40 * 300) + (10 * 150) \\ (6 * 500) + (0 * 300) + (0 * 150) \\ (3 * 500) + (0 * 300) + (0 * 150) \end{bmatrix} \]\[ \text{Total Cost Matrix} = \begin{bmatrix} 25000 \\ 3000 \\ 1500 \end{bmatrix} \]`

Now, we can compare the total costs for each store. The least expensive place is the one with the minimum value in the Total Cost Matrix. In this case, Office World has the least expensive erasers at $3000.