Quantitative Reasoning

Hi ya'll would someone check my reasoning behind this question/

Robots X, Y, and Z each assemble components at their respective constant rates. If rx is the ratio of robot X's constant rate to robot Z's constant rate and ry is the ratio of robot Y's constant rate to robot Z's constant rate, is robot Z's constant rate the greatest of the three?
1. rx < ry
2. ry < 1

First, I set all the info into a ratio(fraction):

rx= (rate of x)/(rate of z)
ry= (rate of y)/(rate of z)

1. rx < ry
This would be same as (rate of x)/(rate of z) < (rate of y)/(rate of z)
Only info that can be deduced from this is that (rate of x) < (rate of y)

2. ry < 1
This would be same as (rate of y)/(rate of z) < 1
Multiply each side by (rate of z)
Only info deduced from 2 is that (rate of y) < (rate of z)

1 & 2
rx < ry < 1
This would be same as (rate of x)/(rate of z) < (rate of y)/(rate of z) < 1
Multiply each ratio by (rate of z)
We now have (rate of x) < (rate of y) < (rate of z)

ANSWER IS C