How can I figure what gear ratios are in the transaxle of my buggy if I don't know the year make or model the transaxle came out of, or if it's been overhauled or modified?
if you can come up with how many mph its running at a certain rpm rate...you can calculate the r&p along with the gear itself...you will have to do this for every gear
If it's a swingaxle it's most likely got a 4.125 or 4.375 ring & pinion...if it's 4.125, 3rd should be 1.26 and if it's 4.375 3rd could be 1.26 or 1.32.
Most IRS Type I transmissions you'll come across in the US will be AH (4.125 R&P) or AT (3.875). 2nd gear will be 2.06 and 3rd will be 1.26 99.5% of the time, so for example if you put it in second with one wheel off the ground and it takes 4¼ turns of the crank to get one turn of the wheel, you have a 4.125 R&P. (2.06x4.125)/2=4.25
If it takes 4 turns it's a 3.875 (2.06x3.875)/2=3.99
If it takes 4½ turns it's a 4.375 (2.06x4.375)/2=4.51
This assumes that 1st and 2nd gear are still stock - considering the expense involved in replacing the mainshaft to get different 1st/2nd gears, that's a pretty safe assumption unless you're dealing with a trans built by/for a hardcore drag-racer.
If in doubt you can repeat in 3rd gear, if it takes just over 2½ turns it's a 4.125, if just under it's a 3.875. A 4.375 will take either 2-3/4 or almost 3 turns depending upon which 3rd it has. Since it's not uncommon to replace 3rd with an aftermarket shorter gear, keep an open mind when using 3rd to determine R&P. The most common is 1.48 but there are many others.
Once you've confirmed what the R&P ratio is you can check to see which 4th you have, you may need to rotate the crank enough to get 2 or more turns out of the wheel to be able to discern the difference between, say, a .89 and a .93 here. Late AH transmissions had a .88 4th, earlier were all .89 unless the trans started life in a bus in which case it could be .82.