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Modeling with multiple variables: Taco stand

Modeling the relationship between three quantities (or more) isn't that different from modeling the relationship between two quantities. Here is an example of a model that relates different quantities related to the daily profits of a taco stand. Created by Sal Khan.

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  • hopper cool style avatar for user Hopper
    At , In '300=tp-tc', couldn't the Ts cancel?
    (1 vote)
    Default Khan Academy avatar avatar for user
    • cacteye blue style avatar for user Jerry Nilsson
      If it were 𝑡𝑝∕(𝑡𝑐), then the 𝑡's would cancel, because 𝑡∕𝑡 = 1,
      and we'd be left with 1⋅𝑝∕𝑐 = 𝑝∕𝑐

      𝑡𝑝 − 𝑡𝑐 is a different matter.
      Let's say a pizza costs $4 to make and sells for $10 (i.e. 𝑐 = 4, 𝑝 = 10).
      Then 𝑡𝑝 − 𝑡𝑐 = 10𝑡 − 4𝑡 = 6𝑡 (just like 10 bananas − 4 bananas = 6 bananas).

      So, no, the 𝑡's don't cancel.
      (2 votes)

Video transcript

- [Instructor] We're told a certain taco stand sells t tacos per day for a net profit of $300. Each taco costs c dollars to make, and is sold for p dollars. Write an equation that relates t, c, and p. So pause this video and see if you can do that. All right, now let's work this together. So let's just remind ourselves what's going on here. So we have the number of tacos sold per day is t, so t = number of tacos sold per day. They tell us the net profit of $300, so we could say 300 = net profit. They tell us that each taco costs c dollars to make, so c = cost per taco. And then p is what each taco is sold for, right over there, is sold for p dollars, so p = the price per taco. So how do we figure out what the net profit is going to be? Well, we can write it this way. Net profit is going to be equal to the total amount of money that you bring in minus the total amount of money that you have to spend. So what is going to be the total amount that you bring in? Well it's gonna be the amount you sell times... Well it's gonna be the number of tacos you sell times the price per taco. So t X p, this is how much money you're bringing in, but of course you have to also subtract out your costs. Well, what's going to be your cost? It's going to be the number of tacos times the cost per taco. That's how much we're going to spend. So the number of tacos times the cost per taco. Now here I've written everything as variables. Well, I wrote net profit out, but they told us that net profit is equal to 300. So I could write it like this. 300 = t X p, I'll do it with those same colors, tp - tc. Now, it's completely possible that they didn't give us net profit as $300 per day, and instead they said that is one of the variables, and they gave one of the other variables, and that would have been okay, we could have used the same logic. Whatever they didn't give us we could've set up as a variable of whatever they did give us, maybe they gave us the price per taco, we could've put that as a given number. Now, I know what some of y'all are thinking. You didn't get exactly this. You might've thought about it the other way. You might've thought about it as the profit is going to be equal to the number of tacos times the profit per taco. And what's going to be the profit per taco? Well, that's going to be how much each taco is sold for minus how much each taco costs. And this also would have been completely credible. And if you look at these two, they're actually algebraically equivalent. If you factor out t here, you get this expression on the right.