Question

The Make-It-Good manufacturing company produces three products, Widgets, Mingets, and Tringles.

During a given year they plan to produce a total of 13,000 units of these products.

The per unit production costs for Widgets, Mingets, and Tringles are $4, $5, and $7 respectively.

The per unit profit for the Widgets, Mingets, and Tringles is $1, $2, and $3 respectively.

If the production costs are to be $70,000 and the desired profit is $27,000, how many of each product should the company produce?

Set up the solution: define variables and determine the equations. Then, solve the system of equations using any valid method, and answer the question asked in the problem.

Answer 1

Define variables and make the system of equations:

Let w - count of Widgets.

Let m - count of Mingets.

Let t - count of Tringes.

Then

4w - costs of Widgets,

     5m - costs of Mingets,

     7t - costs of Tringes.

And w - profit for Widgets

     2m - profit for Mingets

     3t - profit for Tringes.

$$\begin{cases} m + t + w = 13000 \\ 5m + 7t + 4w = 70000 \\ 2m + 3t + w = 2700 \end{cases}$$

Solve it:

$$\begin{pmatrix} 1 & 1 & 1 \\ 5 & 7 & 4 \\ 2 & 3 & 1 \end{pmatrix} \begin{pmatrix} 13000 \\ 70000 \\ 27000 \end{pmatrix}$$

$$\begin{pmatrix} 1 & 1 & 1 \\ 0 & 2 & -1 \\ 0 & 1 & -1 \end{pmatrix} \begin{pmatrix} 13000 \\ 5000 \\ 1000 \end{pmatrix}$$

$$\begin{pmatrix} 1 & 3 & 0 \\ 0 & 2 & -1 \\ 0 & -1 & 0 \end{pmatrix} \begin{pmatrix} 18000 \\ 5000 \\ -4000 \end{pmatrix}$$

$$\begin{pmatrix} 1 & 3 & 0 \\ 0 & 2 & -1 \\ 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} 18000 \\ 5000 \\ 4000 \end{pmatrix}$$

$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} 6000 \\ -3000 \\ 4000 \end{pmatrix}$$

so m = 6000, w = 3000, t = 4000

The Answer:

Tringles - 3000 items

Widgets - 6000 items

Mingets - 4000 items