# 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...

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? **

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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.**

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