Question

Sharon is making greeting cards, which she will sell by the box at an arts fair. She paid $50 for a booth at the fair, and the materials for each box of cards cost $6. She will sell the cards for $8 per box of cards. At some point, she will sell enough cards so that her sales cover her expenditures. How much will the sales and expenditures be? Write a system of equations, graph them, and type the solution.

Answer 1

The sales and expenditure will be equal to $200 each with the solutions to the system of equations as x = 25, y = 200.

The fixed cost for the fair booth = $50

The unit cost of materials for each box of cards = $6

The selling price per box of cards = $8

The contribution margin per box of cards = $2 ($8 - $6)

To break even (make her sales cover her expenditures) = Fixed Cost/Contribution margin per unit

= 25 ($50/$2)

Let the number of boxes of cards sold = x

Let the total sales = y

The system of equations is:

y = 8x (equation for total sales)

y = 50 + 6x (equation for total expenditures)

To solve the system of equations, we can graph the two equations and find the point of intersection.

The graph is as follows:

The first equation, y = 8x, is a straight line with a slope of 8, passing through the origin.

The second equation, y = 50 + 6x, is also a straight line with a y-intercept of 50 and a slope of 6.

When these two equations are graphed, the intersection occurs at the point (25, 200), which concurrently solves the system of equations.

The solution to the system of equations is x = 25, y = 200.

At this break-even point:

The total sales = $200 ($8 x 25)

The total expenditures = $200 ($50 + 6(25).

Thus, Sharon will need to sell 25 boxes of cards for her sales to cover her expenditures.