The following detail is needed in the affine-by-affine gluing construction of the Proj of a graded rings. Every single reference that I know of takes the claim for granted.
Let $A$ be a nonnegatively graded ring, and let $f$ and $g$ be two homogeneous elements of positive degree. Then there should be isomorphisms of rings $$\big((A_f)_0\big)_{g^{\deg f} / f^{\deg g}} \cong (A_{fg})_0 \cong \big((A_g)_0\big)_{f^{\deg g} / g^{\deg f}}$$ Here the subcript $0$ means '$0$-th graded part', while every other subscript denotes localisation at an element.
It is not clear to me at all why this should be true. I do not even have a conjectural map between the various rings, or any idea why $\deg f$ and $\deg g$ should appear.
Let me summarize the discussion from the comments so that this question may be marked as answered.
In the comments, after I suggested looking at an example, you discovered the correct map (in the case where $\deg f=\deg g$) - for an arbitrary element, we can rewrite it as follows: $$\frac{a/f^n}{(g^{\deg f}/f^{\deg g})^m}\mapsto \frac{af^{m\deg g}}{f^ng^{m\deg f}},$$ and then we just multiply by appropriate powers of $\frac{f}{f}$ and $\frac{g}{g}$ to get the bottom to be of the form $(fg)^c$ for some integer $c$ (you can solve this if you want - I don't think it's strictly necessary). The extra degree stuff is perhaps a clerical headache, but it can be dealt with.