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If I am going to buy $c$ number of cats and they each cost \$100, and the total is $t$

My equation would look like:

$100c = t$

Here, my dependent variable is $t$ and my independent variable is $c$

Or so I thought, I can rearrange this to get:

$t/100 = c$

In this case is $t$ the dependent? If I have the price $t$ and the output is how many cats I will buy $c$. Although, this is strange, hypothetically some old cat lady could have saved up a certain amount of money and just wants to buy as many cats as possible.

So, if this is the case, what should someone answer in a test? If they are loose like how I have just proved, then how could a student differentiate?

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2 Answers 2

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It may be counter intuitive, but while dependent and independent variables are perfectly well-defined concepts, they are not mathematical concepts - they are part of the formulation of the experimental method in science.

When one performs an experiment, one typically controls some parameter (say $x$) in the experiment, and measures an outcome (say $y$) as a result. At this point, $x$ and $y$ are discrete variables, summarizing the results of the experiment: when I held $x$ to a certain value, I measured $y$ as another value, and so on. Mathematically, we have an empirically measured function $f\colon \{x_1,\ldots,x_n\}\to \{y_1,\ldots,y_n\}$ where the $x_i$ are the finite set of values I set the parameter to, and the $y_i$ are the finite set of results I measured.

Now we typically try to fit the results of an experiment to a mathematical model, in which case we are approximating the measured function $f$ above by an abstract mathematical function, usually defined for infinitely many values. As you pointed out, once we pass to the mathematical abstraction, the roles of $x$ and $y$ can be interchanged.

In summary, the concept of dependent and independent variables refer to the setup of an experiment or measurement, and are not directly reflected in the mathematical model of the functional relationship.

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  • $\begingroup$ Alright thanks for the great answer! So on a test one should pay attention to the wording of the situation? $\endgroup$
    – BeastCoder2
    Jun 3, 2019 at 1:21
  • $\begingroup$ Yes, I would pay keen attention to the setup of the problem and ignore the actual math if I was asked to distinguish a dependent variable from an independent variable. $\endgroup$
    – pre-kidney
    Jun 4, 2019 at 7:51
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In situations like this, context is usually key.

As a rule of thumb; if I see $y=mx+c$ ie one variable on one side and many variables on the other. The solo variable is usually the dependent variable. In the above equation $y$ is dependent on $x$.

Regarding this particular example, we could ask ourselves:

  • if I buy many cats will the total price I need to pay change? Yes, since the total price is a function of the number of cats multiplied by 100.

  • Given that I buy many cats, will the price for a given cat change? Hmm, not really. Unless there is some sort of discount offer, you buying many cats will not affect the price of a single cat.

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  • $\begingroup$ But C is the number of cats, you stated that the price would be changing when I have the number of cats $c$ changing, did you make a mistake or have I? Could you please fix it or inform me of how to fix my question? Anyway, thanks for answering! :) $\endgroup$
    – BeastCoder2
    Jun 3, 2019 at 1:19
  • $\begingroup$ My mistake, by price I meant the total price, not the price for a single cat. So the total price would change the more cats you buy. But the price of a single cat would stay the same, even if you bought 100 cats. $\endgroup$
    – user679128
    Jun 3, 2019 at 1:21
  • $\begingroup$ So is your answer wrong? Look at +pre-kidney*s answer $\endgroup$
    – BeastCoder2
    Jun 3, 2019 at 1:23
  • $\begingroup$ The answer still seems to hold. With regards to the above answer, it also seems to be correct; we seem to be saying the same thing. Ie it’s about the context. You can ask the two questions I asked in my answer to get a better feel for what the independent and dependent variables. Note: that in a physics experiment, we usually have things that we try to control so that only a few things are dependent on others in an experiment. Looking at it purely from a mathematical point of view, I would use the number of variables to try and figure it out. $\endgroup$
    – user679128
    Jun 3, 2019 at 1:28

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