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2 options, one is exact and the other is approximate but will still work. If you pull the diff cover and can see in to count the pinion gear teeth and ring gear teeth take your ring gear, divide it by the pinion, and that is your ratio. If you don't want to hassle with that jack the tires off the ground, mark a spot on the tire you can watch and a spot on the driveshaft. Then put it in neutral (chock the other axle) and turn the driveshaft by hand, counting how many revolutions the driveshaft makes to turn the tires once. If you have an open carrier it's typically easier to keep one wheel still and then divide your final number by 2. So if it takes three and a half turns at the driveshaft to turn the wheel once you probably have 3.54's. If one wheel stays still then the driveshaft should make about 7 turns to spin the tire once. If you get 4 turns it's most likely 4.10's . . . etc.
Btw, some ring gears are also stamped with their ratio but not all.
Here is a very accurate way of doing it that I copy/pasted from the web
4. Open differentials: Secure the car from rolling forwards or backwards and place the car in neutral. Raise ONE SIDE of the car off the ground (left in this example) and place jackstands under the car to safely support it.
5. Place a mark on the tire at bottom-dead-center. Crawl under the car and place a mark on the driveshaft. You'll be counting how many times the driveshaft turns in this test, so place the mark at a convenient place that you can call "zero".
6. Have a friend turn the left wheel for 20 rotations. Have them count to themselves while you count driveshaft rotations. When they are completed, divide the number of driveshaft revolutions by 10. For instance, 30 turns is a 3.00 differential ratio. 37 and a third turns is a 3.73 ratio. If you know the factory-available ratios, this will put you close enough to quickly determine which ratio you have. If you don't know the factory options, then do a larger number of wheel rotations (say, 100) and divide by 50 for a little more accuracy on that last decimal place. You're done! You can put the car back on the ground now.