Quandaries and Queries


Name: Craig

Who is asking: Parent
Level: Secondary


A dance studio charges $80 per student for a series of 2 hour lessons. The studio's costs are $30 per hour for the instructor, $15 per lesson for the room rental, and $3 per student for miscellaneous expenses. If x is the number of students enrolled in the class, express the studio's profit P(x) in terms of x. Find the profit if 10 students are enrolled in the class.

I would like to know how to set this up so that I can help my son.



Hi Craig,

The dance studio charges each student $80 for a 2 hour lesson, but it doesn't get to keep the entire $80, it has some expenses. The profit is the income (what the students pay) minus the expenses. There are three expenses.

  1. $30 per hour for the instructor

  2. $15 per lesson for the room rental

  3. $3 per student for miscellaneous expenses

If there are x students then the income is $80 per student so the total income is $80  x. The expenses are:

  1. The way I read this is that the instructor charges $30 per hour, regardless of the number of students. Since this is a 2 hour lesson the instructor charges $60.

  2. Rental of the room for 2 hours costs $15.

  3. The miscellaneous expenses are $3 per student hence, if there are x students, then the total miscellaneous expenses are $3  x.

Since the profit, P(x), is (income) - (expenses)

P(x) = $80  x - ($60 + $15 + $3  x)


Thank you for the help, but I have one further question. My son has a formula that goes like this:

Profit = x(m-nx)-(a+bx) with x = the number is students
a= fixed costs
bx= variable costs

the above I see in the answer, but I am not sure about the m-nx.

He says it has something to do with supply and demand. I can see where the equation you posted is good for the profit, but is there anything more you can tell us regarding the above information.

Thank you again.

Hi Craig,

It really bothers me to see mathematics presented as a collection of formulas.

The m - nx might represent supply and demand, or some kind of discount. Maybe, to try to attract more students, the dance studio trys to get customers to encourage their friends to join with the following fee structure.

The basic cost per student is $80 but every time another student joins, every student gets a $2 discount. Thus if there is only 1 student the cost is $80. If there are two students then each pays $78, if there are three then each pays $76,... With this plan each student pays $82 - $2  x, where x is the number of students. Hence the profit would be P(x) = x  ($82 - $2  x) - ($60 + $15 + $3  x)


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