Your argument doesn't look wrong as such (though I haven't checked it in details), but it does make it rather difficult to see the forest for trees since you start writing about an unknown angle $x$ from the beginning. That means that you never get to note the fundamental underlying fact:
27 ordinary minutes of arc equals 50 "centesimal minutes"
which can be seen by noting that both are exactly 1/200 of a right angle (and, as Euclid says, all right angles are equal).
Once you've seen that, consider an arbitrary angle $\theta$. There are some number $y$ (rational or irrational) of 200ths of a right angle in $\theta$, so $\theta$ is simultaneously $27y$ minutes of arc and $50y$ centesimal minutes. The ratio between these is $\frac{27y}{50y} = \frac{27}{50}$.
If this feels too much like "cheating" because it assumes that the ratio is already known (which it is! it's given to you by the problem statement!), then you could also start by measuring in right angles instead, and get the ratio $\frac{5400}{10000}$ which simplifies algebraically to $\frac{27}{50}$.
