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Given that $x$ is directly proportional to $y$, and $z$ and is inversely proportional to $w$, and that $x = 4$ when $(w,y,z) = (6,8,5)$, what is $x$ when $(w,y,z)=(4,10,9)$?

Part of the answer says that

Because x is inversely proportional to $w$, when all other variables are constant, $xw$ is constant. Similarly, when the other two variables are constant, each of $\frac{x}{y}$ and $\frac{x}{z}$ is constant. We can combine all these by saying $\frac{xw}{yz}$ is constant.

I don't understand what "when all other variables are constant" or "when the other two variables are constant" means.

I also need an intuitive explanation as to how / why $xw$, $\frac{x}{y}$, and $\frac{x}{z}$ are combined together.

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Mahdi Rkioui
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  • I am deleting my answer as it does not satisfy all the parameters of the question. I still believe the author's conclusion is wrong, but the entire question may be incorrect as well. I am undecided at this point. – Deepak Jun 26 '20 at 07:16
  • @Mahdi R. I have added an example from physics in a comment to my answer that may clarify what is meant by being proportional or inversely proportionaly while all other variables (quantities) are kept constant. – trancelocation Jun 26 '20 at 08:24
  • Thanks, I now understand why they say that all other variables are kept constant. If the other variables are allowed to change, then the proportionality relationship itself will be changed. I still don't understand how the three expressions are combined, though. – Mahdi Rkioui Jun 26 '20 at 19:57
  • Does this answer your question? Joint Proportionality – ryang Jun 25 '23 at 14:55

1 Answers1

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Obviously you are dealing with so called combined proportions.

Your quantity $x$ depends on $3$ other quantities $w,y,z$.

Quantity $x$ is proportional to $y$. This means, if you vary only quantity $y$, then $x$ depends proportionally on $y$. You could write it as $x= y\cdot a$, where $a$ is a constant, which still depends on $w$ and $z$.

Now, $x$ is also proportional to $z$. If you vary only quantity $z$, then using the intermediate result from above, you can write $x = y\cdot a = y\cdot z \cdot b$, where $b$ is a constant, which still depends on $w$.

Finally, $x$ is inversely proportional to $w$. Hence, if you vary only $w$, then $x$ depends inversely proportionally on $w$. Using the result of the paragraph before you can write $x = y\cdot z \cdot b = \boxed{\frac{yz}{w}\cdot c}$ with a constant, which is independent of $w,y,z$.

So, with the first set of values you calculate $c$:

$$4 = c\frac{8\cdot 5}{6}\Leftrightarrow c= \frac 35$$

Now, you can calculate $x$ for the second set of values:

$$x = \frac 35 \frac{10\cdot 9}{4} = \frac{27}{2}$$

trancelocation
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  • How do you go from the valid expression $x = ya$ to the suspect expression $ya =yzb$? This implies $a = zb$, which implies (for nonzero $y$) that at least one of $a$ or $b$ is not constant since $z$ is variable. – Deepak Jun 26 '20 at 06:21
  • @Deepak: From the first step you have $x = y\cdot a(z,w)$. If you now vary only $z$, you get $x= y\cdot z\cdot b(w)$. – trancelocation Jun 26 '20 at 07:13
  • For the one who downvoted my answer: Let's go into physics and have a look at the gravitational law $F= G\frac{m_1m_2}{r^2}$. The gravitational force is obviously proportional to the mass $m_1$ and proportional to the mass $m_2$ and inversely proportional to the square of the distance between the bodies. Of course, each proportion is considered, when only one of the quantities varies and the others stay constant. Would the downvoter please elaborate on the reason for the downvote? – trancelocation Jun 26 '20 at 07:28