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Given a question,

Write down an equation that embodies the following relationship

$$ a \varpropto b $$

Does the 'open' end of the $\varpropto$ symbol indicate that that's the side of the equation which needs a constant of proportionality, and hence

$$ a = bC $$

would be a correct answer, but

$$ aC = b $$

would be a wrong answer, or is there something more subtle that I've missed?

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    $\begingroup$ If $a$ is proportional to $b$, then apart from the degenerate case $a=0$, $b$ is proportional to $a$. Which we use is a matter of emphasis. We tend to write $a$ is proportional to $b$ if $b$ is "known" and we want to compute $a$. $\endgroup$ Aug 2 '14 at 14:59
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    $\begingroup$ One way of capturing the special case $C=0$ is to say that $a$ and $b$ are proportional if there are fixed numbers $X, Y$, not both zero, such that $aX+bY=0$. This is like the general case of a straight line in the plane - I was taught $y=mx+c$, but this did not cover the case of the vertical line $x=C$ - the general equation is $ax+by+c=0$ with $a,b$ not both zero. $\endgroup$ Aug 2 '14 at 15:21
  • $\begingroup$ Boths are fine. $\endgroup$ Aug 2 '14 at 19:41
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Either would be a valid equation but you'd get different values of C; I.e. if you take C to be the value that satisfies:

$a=Cb$

Then we could also write this as:

$aC^{-1}=b$

So you can view it both ways!

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  • $\begingroup$ Hmm, I asked because the textbook I'm using gives $a=bC$ as the answer, and I just wanted to make sure that I'd understood why... $\endgroup$
    – ChrisW
    Aug 2 '14 at 14:28
  • $\begingroup$ Thats the standard answer but the value C is just a constant of proportionality; so if you put it on the other side its technically the inverse of the constant theyre asking for! I.e. it would be $\frac{a}{C} = b$. $\endgroup$ Aug 2 '14 at 14:33
  • $\begingroup$ (I hope i haven't overcomplicated this) $\endgroup$ Aug 2 '14 at 14:37

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