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So, I got this question: Essentially, a person wants to open an art venue, but the cost is $\$400$ and the foodservice she's working with is charging $\$3.75$ per person for food and drink. The task is to create an equation that evenly splits the cost of the venue between tickets. My equation was $3.75x + \frac {400}x$

A friend is insisting that it's $\frac {3.75x +400}{x}$ Who is correct? I thought that his equation can't be correct because dividing whatever $x$ is by itself would just be one, and then $3.75$ would be multiplied by $1$.

lulu
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Alexander The Great Hamilton L
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  • Note: the formatting system used by this site interprets dollar signs as code, so using them to denote currency confuses the system. That's why your post was so garbled. – lulu Dec 17 '20 at 17:54
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    To your question: it is not clear what $x$ is, nor what your formula is meant to compute. If $x$ is the number of people ($=$ number of tickets) and you are trying to compute the fair price of a single ticket, then your formula is obviously wrong. If you had $400$ people, you would charge each of them $3.75\times 400+1=1501$, which is far too high. – lulu Dec 17 '20 at 17:57
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    your friend has the right formula. You could also write it at $3.75 + \frac {400}x$ You can spread the venue cost over the attendees. But each attendee has their own food cost. – Doug M Dec 17 '20 at 18:04

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The equation you are probably looking for is

$$T=3.75+{400\over x}$$

where $T$ represents the price of a ticket and $x$ represents the number of tickets you plan to sell (i.e., the number of people who will attend the event). There are two important points here. First, when formulating a mathematical description of a problem, it's important to explicitly say what the variables you use represent. Second, when writing down an equation for a mathematical description of a problem, it important to actually write down an equation, and not just one side of it.

Where I said "probably," it's because you might actually be looking for the equation in the equivalent form

$$T={3.75x+400\over x}$$

(which uses your friend's expression on its right hand side). An interpretation of the equation in this form is that you are computing the total cost of the event in the numerator on the right hand side and then dividing by the number of tickets to get the ticket price, whereas in the first equation you are thinking of the ticket price as comprised of two parts: the cost of the ticket buyer's food (the $3.75$) and their share of the venue (the $400/x$).

Barry Cipra
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