I am getting confused about a super basic issue.
I have two quantities: $6\pm4$ and $4\pm3$ .
So let's say $x = 6$, $\Delta x = 4$, $y = 4$, $\Delta y = 3$
Now I want to calculate the uncertainty of $z=\frac{x}{y}=1.5$.
If I just take the range of values given by the uncertainties I would get $z=\frac{x}{y}$ somewhere between $\frac{2}{7}\approx0.3$ and $\frac{10}{1}=10$.
Now if I propagate the errors according to $\frac{\Delta z}{z}=\sqrt{(\frac{\Delta x}{x})^2+(\frac{\Delta y}{y})^2}$ then I get $\frac{\Delta z}{z}\approx1.004$ which is an uncertainty of > 100%. I.e. a final result of $z\pm\Delta z\approx1.5\pm1.5$.
This confuses me because firstly, I only predict values up to a maximum of $\approx3$ whereas I predicted values up to $10$ when using the range of possible values. Secondly and more importantly, this uncertainty implies that the ratio of $z=\frac{x}{y}$ can be zero which is odd. Obviously, if I increase the uncertainties of $x$ and $y$ slightly, I can also generate negative nubers of the ratio.
Is there something I am doing wrong? Or a way to deal with this issue? Having zero or negative numbers just doesn't make physical sense at all in the context that I am using these numbers.
Thanks a lot