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 x/yz:y/xzx/yz : 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=1:2:3.yz : 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, x/yz=1x/yz = 1 and y/xz=1/4y/xz = 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:

xyzyxz=xyyx=x2y2=x2z2y2z2=(xzyz)2=22=4. \begin{aligned} \frac{\frac{x}{yz}}{\frac{y}{xz}} &=& \frac{\frac{x}{y}}{\frac{y}{x}} = \frac{x^2}{y^2} \\ &=&\frac{x^2z^2}{y^2z^2} = (\frac{xz}{yz})^2 \\ &=& 2^2 = 4. \end{aligned}