We need to somehow link all three of the ratios together.  We can start by connecting the last two ratios together by multiplying the last ratio by two. $z: x=1: 6=2: 12$, and since $y: z=3: 2$, we can link them together to get $y: z: x=3: 2: 12$.  Finally, since $x: w=3: 4=12: 16$, we can link this again to get: $y: z: x: w=3: 2: 12: 16$, so $w: y=(\mathbf{E}) 16: 3$. 
The ratio of $w$ to $x$ is $4: 3$, the ratio of $y$ to $z$ is $3: 2$, and the ratio of $z$ to $x$ is $1: 6$. What is the ratio of $w$ to $y$? (2020 AMC 10B Problems, Question \#3) 

  * $4: 3$ 
  * $3: 2$ 
  * $8: 3$ 
  * $4: 1$ 
  * $16: 3$ 