2140.11 – Triple Ratio

If x,yx, y and zz are such that yz:zx:xy=1:2:3yz : zx : xy = 1 : 2 : 3, then xyz:yxz\dfrac{x}{yz} : \dfrac{y}{xz} is

  1. 3:23 : 2

  2. 1:21 : 2

  3. 1:41 : 4

  4. 2:12 : 1

  5. 4:14 : 1

Solution

Let us first find an example of x,yx, y and zz such that yz:xz:xy=1:2:3yz : xz : xy = 1 : 2 : 3

A little fooling around suggests x=6,y=3,z=2x = 6, y = 3, z = 2 and this checks. For these numbers, xyz=1\dfrac{x}{yz} = 1 and yxz=14\dfrac{y}{xz} = \dfrac{1}{4}. If, as the problem suggests, the answer for any numbers that work is the same, then (e) is correct.

Can we prove it? It's not so bad:

  xyz  yxz=  xy  yx=x2y2=x2z2y2z2=(xzyz)2=22=4\begin{aligned}\cfrac{\ \ \cfrac{x}{yz} \ \ }{\cfrac{y}{xz}} &= \cfrac{\ \ \cfrac{x}{y}\ \ }{\cfrac{y}{x}} \\[2.5em]&= \dfrac{x^2}{y^2} \\[0.5em]&=\dfrac{x^2z^2}{y^2z^2} = \left(\dfrac{xz}{yz}\right)^2 \\[1em]&= 2^2 = 4\end{aligned}