It's really important for businesses to make a profit! Otherwise, they won't be around for very long. So businesses should know their break-even point, the amount of money they need to take in to cover all their expenses. At this point there is no profit or loss, and their income is equal to their expenses.
To find this point, we use break-even analysis. Break-even analysis looks at the graphs of cost and revenue together to determine where they cross. This point of intersection is the break-even point where income equals expenses.
We can then use the graphs to determine where profit and loss occurs.
Here is an example of a break-even analysis for a single day of operation for the company Lovely Lemonade.
The horizontal axis is the number of drinks, or units, they sell. The expenses line (red) starts off higher than the income line (green). So for a low number of units sold, Lovely Lemonade will lose money. For example, if they only sell $1000$1000 units, the green line tells us they earn $\$200$$200, the red line tells us they spend $\$600$$600, and so overall they make $\$200-\$600=-\$400$$200−$600=−$400. The negative sign means overall they lose $\$400$$400.
But eventually, the income line becomes higher than the expenses line, so for a high number of units sold, Lovely Lemonade will make money. For example, if they sell $5000$5000 units, the graph tells us that they will earn $\$1100$$1100 but only spend $\$1000$$1000. Overall they make $\$1100-\$1000=\$100$$1100−$1000=$100, and the positive sign means a profit of $\$100$$100.
The important point is when the two lines meet - this is the break-even point. In this example we can see that selling $4000$4000 units means they don't make any profit, but they don't lose any money either - both their income and their expenses are $\$900$$900. Reaching this amount should be an important first goal for Lovely Lemonade every single day!
We can also use break-even analysis to compare two different pricing plans for the same product or service to see which one offers the best deal for a particular situation.
Here's an example comparison between two energy plans provided by rival companies Thorgate and Callisto:
In this scenario the break-even point at $\left(40,10\right)$(40,10) tells us that both plans charge $\$10$$10 for $40$40 kWh. For energy amounts less than the break-even point we can see that Thorgate is cheaper, and for energy amounts more than the break-even point Callisto is cheaper. Knowing how much energy someone plans on using can then determine the best plan for them.
Practice questions
Question 1
Consider the following phone plans:
GO SMALL plan: This plan has a $\$30$$30 monthly base charge and charges $90$90 cents per minute for all calls.
GO MEDIUM plan: This plan has a $\$38$$38 monthly base charge and then charges $70$70 cents per minute for all calls.
Complete the following table of values for various total monthly call times for the two plans:
Call time (in minutes) Total cost for GO SMALL plan Total cost for GO MEDIUM plan $30$30 $\editable{}$ $\editable{}$ $40$40 $\editable{}$ $\editable{}$ $50$50 $\editable{}$ $\editable{}$ $60$60 $\editable{}$ $\editable{}$ Sketch the graph of the two plans.
Loading Graph...Determine how many minutes of calls results in the same monthly bill for both plans.
Question 2
The cost for a sporting goods manufacturer to make bats is $\$60$$60 per bat plus a fixed setup cost of $\$400$$400. The bats will sell for $\$85$$85 each.
Write an equation in the form $y=mx+b$y=mx+b to represent the cost $y$y of manufacturing $x$x bats.
Write an equation in the form $y=mx$y=mx to represent the revenue $y$y generated from the sale of $x$x bats.
The two equations are plotted on the graph.
Loading Graph...What are the coordinates of the break even point?
$\left(x,y\right)=\left(\editable{},\editable{}\right)$(x,y)=(,)
Fill in the blanks to complete the statement:
When $\editable{}$ bats are produced and sold, both cost and revenue are $\editable{}$ dollars.
Question 3
The two equations $y=5x+800$y=5x+800 and $y=7x$y=7x represent the cost and revenue functions of Jim's Haberdashery respectively. The value of $x$x is the number of clients he sees in a month.
Draw the graphs of both equations on the axes below.
Loading Graph...How much does each client cost Jim?
How much revenue does each client earn Jim?
What does the point $\left(400,2800\right)$(400,2800) represent? Select all the correct options.
The point at which Jim's Haberdashery maximises their profit.
AThe break even point for Jim's Haberdashery.
BThe point at which Jim's Haberdashery has its revenue equal to its costs.
CThe point at which Jim's Haberdashery shuts down operations.
D