If $y$ is proportional to $x$, and $y$ is also proportional to $z$, then how are we able to arrive at the equation: $y = kxz$ ?

My understanding so far of proportionality, which comes from this Wikipedia article, is that two variables are directly proportional if their ratio yields a constant.

Thus, if $y$ is proportional to $x$, then $\frac yx = k$, where $k$ is a constant.

So, if $\frac yx = k*z$, then how are $y$ and $x$ still proportional if $z$ is not a constant?


Two variables $\alpha$ and $\beta$ are directly proportional if their ratio is constant, relative to $\alpha$ and $\beta$. That last phrase is missing from the Wiki article, because it is not relevant to the two-variable models discussed there. But it is crucial here.

With that correction, direct proportionality tells us that $$\frac{y}{x}=\lambda_1(z)$$ where $\lambda_1(z)$ is independent of $x$ and $y$, but might depend on $z$. Similarly, $$\frac{y}{z}=\lambda_2(x)$$ where $\lambda_2(x)$ is independent of $y$ and $z$, but might depend on $x$. Dividing the two equations, $$\frac{z}{x}=\frac{\lambda_1(z)}{\lambda_2(x)}$$ or, rearranging, $$\frac{\lambda_1(z)}{z}=\frac{\lambda_2(x)}{x}$$

Each side of the latter equation is constant with respect to the other side, so let their common value be $k$. Then $\lambda_1(z)=kz$, whence the claim.

  • $\begingroup$ This makes sense. Then is the definition from themathpage.com - By whatever ratio one quantity changes, the other changes in the same ratio - a more accurate definition of proportionality, especially when dealing with joint proportions? $\endgroup$ – johnnyodonnell Feb 13 at 1:00
  • $\begingroup$ @johnnyodonnell No, that definition suffers the same problem. Let $x$, $y$, and $z$ be three mutually-related variables, with $x$ directly proportional to $y$. Holding $y$ constant means $y$ changes with ratio $1$, so $x$ does too, so $x$ is constant. But changing $z$ might induce a change in $x$. $\endgroup$ – Jacob Manaker Feb 14 at 19:12
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    $\begingroup$ That definition, too, can be corrected in a way analogous to on Wiki: "Assuming all else is held constant, if one quantity changes according to some ratio, the other quantity will also change according to that ratio." $\endgroup$ – Jacob Manaker Feb 14 at 19:16
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    $\begingroup$ In short, all these definitions describe how two variables scale with respect to each other. In order to understand how two variables scale, you need to (temporarily) hold all other variables constant. $\endgroup$ – Jacob Manaker Feb 14 at 19:18

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