How loose are the terms "independent" and "dependent" variables?

Question

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?

Answer 1

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.

Answer 2

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.