The break-even point

Most businesses have two kinds of costs: fixed costs, paid even if nothing is sold, and a cost per unit, paid for every unit made.

Yoav opens a juice stall at the school fair.
Renting the stall and the blender costs 600 ₪.
The fruit and the cup of each juice cost 4 ₪.
Yoav sells each cup of juice for 10 ₪.

When $\displaystyle x$ is the number of cups Yoav sells:

• The total cost: $\displaystyle {C}={600+4x}$ – a linear model: a fixed cost and a cost per unit.
• The revenue: $\displaystyle {R}={10x}$.

$\displaystyle F$ – the fixed costs, $\displaystyle c$ – the cost of one unit.

On the graph, $\displaystyle x$ is the number of cups and $\displaystyle y$ the shekels: the revenue $\displaystyle {y}={10x}$, and the cost $\displaystyle {y}={4x+600}$ (the dashed line).

• The cost line starts at 600: the fixed costs are paid for 0 cups too.
• The revenue line starts at 0, but rises faster: its slope is 10, the cost’s 4.
• To the left of the point $\displaystyle A$ the cost is higher than the revenue – a loss. To its right the revenue is higher – a profit.
• At the point $\displaystyle A$, at 100 cups, the revenue and the cost are equal, 1000 ₪: no profit and no loss.

This point is the break-even point: the number of units at which the revenue equals the cost.

At the break-even point the revenue equals the cost, so an equation finds it.

At Yoav’s stall:

• Subtract $\displaystyle 4x$ from both sides: $\displaystyle {6x}={600}$.
• Divide by 6: $\displaystyle {x}={100}$ cups.

On every cup Yoav gets 10 ₪ and spends 4 ₪, so every cup covers 6 ₪ of the fixed costs. To cover 600 ₪ takes $\displaystyle {600\div6}={100}$ cups.

The profit is a linear model too: the revenue minus the cost.

• The slope, 6, is the profit from every extra cup.
• At 0 cups the profit is $\displaystyle -600$: a loss of all the fixed costs.
• The line meets the $\displaystyle x$-axis at the break-even point, at 100 cups.
• At 150 cups: $\displaystyle {6\times150-600}={300}$ ₪ profit.

Liat prints T-shirts with her drawings.
The printing machine cost 2400 ₪.
The shirt and the paint cost 15 ₪ for every shirt.
Liat sells each shirt for 45 ₪.
On the graph, $\displaystyle x$ is the number of shirts; the revenue $\displaystyle {y}={45x}$, and the cost $\displaystyle {y}={15x+2400}$ (the dashed line).
Mark the break-even point on the graph.
Solve the equation, and check that you got the point you marked.

Liat wants to make a profit of 1500 ₪.
Her profit is $\displaystyle {45x-(2400+15x)}={30x-2400}$.
Write an equation and solve it: how many shirts must she sell?

Year 10 is putting on a play.
Renting the hall and the costumes costs 4500 ₪ in all.
For every member of the audience a programme is printed, for 2 ₪.
A ticket for the play costs 20 ₪.

A café sells a sandwich for 28 ₪.
The café’s fixed costs are 3600 ₪ a month, and making a sandwich costs 12 ₪.
$\displaystyle x$ is the number of sandwiches in a month.
Write the formula of the revenue and the formula of the cost on the graph.
Write an equation and solve it: how many sandwiches a month are the break-even point?
The café sells about 300 sandwiches a month. How much profit does it make?