Given that $yz:zx:xy = 1:2:3$ and $\tfrac{x}{yz}: \tfrac{y}{zx} = 1:k$, find $k$

Question

Given that $yz:zx:xy = 1:2:3$ and $\dfrac{x}{yz}: \dfrac{y}{zx} = 1:k.$ Find $k$.

I understand that $ k = \frac{y^2}{x^2}, y = 1,$ and $x = 2$. Therefore $k = \frac{1}{4}$. This also brings me to $z = 2$ when I put $y = 1$ and $x = 2$ into the second ratio set.

However, that brings the first ratio set to $2:4:2,$ which is $1:2:1,$ not $1:2:3$.

What am I misunderstanding? Please advise.

Answer 1

I think what you are missing out is that taking $y=1$ and $x=2$.Actually its the $x:y=1:2$ and thus $x$ can actually be $2a$ of any $y=a$ were $a$ is variable.

Furthermore,taking your current assumptions into account you will get error in the answer and hence your taking $y=1$ and $x=2$ is basically the problem in your thinking.

Answer 2

Let $yz=p$ $zx=2p$ $xy=3p$

Then, with $x=2, y=1, z$ becomes $\frac{2}{3}$. I think you miscalculate something.

Answer 3

Your error is in assuming that $y = 1$ and $x = 2$. All that you know is that $$ \frac{y}{x} = \frac{1}{2}, $$ or that $x = 2y$. You can still find $k = \bigl( \frac{1}{2} \bigr)^2 = \frac{1}{4}$.