Why does proportionality yield multiplicitiveness and not addition?

Question

I have a simple question:

Suppose that we say, quantity y is directly proportional to quantities a and b. This is then written as:

$$ y \propto a b $$

Why however, is it not written as:

$$ y \propto (a + b) $$

Meaning, how do we "know", that this is multiplicative, and not additive? Why is the former used all the time?

Thank you.

Answer 1

Two values are directly proportional if their ratio is constant. If you write $y \propto (a+ b)$ then you are saying that

$$ \frac{y}{a + b} = k $$

where $k$ is a constant. In this situation $y$ is not proportional to $a$ or to $b$ because $y/a$ and $y/b$ are not constant. On the other hand, if you say that $y \propto ab$ then you are saying

$$ \frac{y}{ab} = k $$

In this situation you see that (keeping $b$ fixed) $y/a$ is constant and (keeping $a$ fixed) $y/b$ is constant. So in this situation $y$ is proportional to $a$ and to $b$ as you wanted.

Answer 2

Assuming that $a$ and $b$ can vary independently of each other:

Suppose that for some particular $a,b$ we have $y/ab =k.$

If $a$ changes to $a'=a(1+d)$ while $ b$ does not change, the value of $y$ changes in such a way that $y/a$ does not change, so $y$ changes from $kab$ to $k(1+d)ab=ka'b.$

Then if $b$ changes to $b'=b(1+e)$ while $a'$ does not change, the value of $y'$ changes in such a way that $y'/b$ does not change, so $y'$ changes from $ka'b$ to $ka'(1+e)b=ka'b'.$

No matter how $a$ and $b$ change, the value $y/ab$ remains constantly equal to $k$.