Touché Toiletries, Inc., has developed an addition to its Lizardman Cologne line tentatively branded Ode d’Toade Cologne. Unit variable costs are 45 cents for a 3-ounce bottle, and heavy advertising expenditures in the first year would result in total fixed costs of $900,000. Ode d’Toade Cologne is priced at $7.50 for a 3-ounce bottle.
How many bottles of Ode d’Toade must be sold to break even?
Unit variable costs on the product are costs proportional to the number of items produced. Whereas, fixed costs are one-off costs independent of the number of items produced.
In the case of Touche Toiletries, Inc. the unit variable costs are 45 cents ( = $0.45) per item produced (3-ounce bottle of cologne), and the fixed costs in the first year are $900,000. The product is sold at $7.50 per bottle.
To break even, the company must match profit from sales to the fixed costs of $900,000.
Now, for each item produced (3-ounce bottle of cologne), the profit P(b) from sale is, by subtracting manufacturing cost from the market price, given by
P(b) = (7.50 - 0.45) x b = 7.05 x b
where b is the number of bottles made (which note are all assumed to be gone on to be sold). This is a linear equation linking the profit to the number of items produced by the company.
We need to solve the equation to find the value of b that leads to an output of 900,000 for the profit P(b). So we set the equation to the profit required to break even, that is, set
P(b) = 900,000
The value of b that satisfies this is the number of bottles that the company need to make and sell to break even. Setting P(b) = 900,000 implies that
7.05 x b = 900,000, giving further that
b = 900,000 divided by 7.05 = 127659.6 which is approximately 127,660 bottles of cologne.
The company need to make and sell 127,660 bottles of cologne to break even.