What are the appropriate ranges of gear ratios for 1.75 mm and 3 mm filament extruders? Most 3&nbsp;mm (mostly are actually 2.85&nbsp;mm) filament extruders have some kind of gear reduction. Many 1.75&nbsp;mm extruders are direct-drive / ungeared but some do use gears. What kind of reduction ratios are suitable or optimal?
$ \frac{(2.85\ mm)^2}{(1.75\ mm)^2}=2.65$ which is the reduction you would want assuming a 1.75 is direct drive.

In this answer I am comparing the amount of filament flowing though the extruder by using $ \pi \times r^2$.  Dividing the 2 values gives the ratio of the 2 speeds.  A gear ratio is them used to change the speed.