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Consider the following question.

$x$, $y$ and $z$ are three quantities. $x$ varies inversely with $y$ when $z$ is constant. $y$ varies inversely with $z$ when $x$ is constant. When $y = 8$ and $z = 7,$ $x = 30.$ Find $x$ if $y = 16$ and $z = 21.$

According to the solution I am referring to, $x$ varies inversely with the product $y.z$ and hence the product $x.y.z$ is constant. The answer therefore is $\frac{(8 \times 7 \times 30)}{(16 \times 21)} = 5.$

The question I have is - How is $x$ inversely proportional to $z$? As per my basic understanding, since $x$ is inversely proportional to $y$, and $y$ is inversely proportional to $z$, it implies that $x$ is directly proportional to $z$. What am I overlooking here?

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As per my basic understanding, since $x$ is inversely proportional to $y$, and $y$ is inversely proportional to $z$

That's not completely accurate.

$x$ varies inversely with $y$ when $z$ is constant. Similarly, $y$ varies inversely with $z$ when $x$ is constant.

Since $x$ and $z$ aren't necessarily constant at the same time, then $x$ varying inversely with $y$ doesn't necessarily occur at the same time that $y$ varies inversely with $z$.

How is $x$ inversely proportional to $z$?

It's not. $x$ varies inversely with $z$ when $y$ is constant.

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    A couple of questions. How did you establish the inverse proportion relation between x and z, when y is constant. (Is there a thing like transitive variation?). Second, how do we read the statement - when y = 8 and z = 7, x = 30? Which quantities are to be considered constant here? – 2017-01-08