Transitive relation in proportionality

Solution 1:

From the equation

$$V=IR$$

the statement "$V$ is proportional to $I$" means that all else held equal, $V$ is a fixed multiple of $I$ (or, rephrased, the ratio between $V$ and $I$ is a constant number).

With this sense of the words "proportional to", the relation is not transitive; indeed, you have a counterexample already.

So in what sense is the relation "proportional to" transitive? Let's define it such:

"A variable $X$ is proportional to another variable $Y$ when there is a nonzero constant number $k$ such that the equation $X=kY$ always holds true."

That definition of proportionality is transitive; indeed, let $X$, $Y$, $Z$ be variables such that $X$ is proportional to $Y$ and $Y$ is proportional to $Z$. So there are nonzero constants $k_1$ and $k_2$ such that the equations $X=k_1Y$ and $Y=k_2Z$ always holds true. Substituting yields

$$X=(k_1k_2)Z$$

and therefore $X$ and $Z$ are proportional to each other.