gear reduction in the differential

[Brandon505]

what is the purpose of having gear reduction in the differential when the transmission provides the different gear ratios needed? Is it just because the differential would be to bulky if they used the same size gears?

[MikeCStig]

Without gear reduction in the differential your car would be geared for unmanageable speeds. Say you have a 3.00 final drive with a 5 speed transmission (to make the math easier). Here's a fairly common set of top speeds in each gear for a somewhat quick, sporty car, but not absurdly fast:
1st: 30mph
2nd: 58mph
3rd: 90mph
4th: 112mph
5th: 150mph
Of course many cars will fall short of the top speed they're geared for due to aerodynamic restrictions, but that's a whole other story.

Without the gear reduction provided by a 3.00 final drive the top speeds in each gear would be 3 times what is listed above. While the idea of going up to 90mph in first gear sounds amazing, it really can't be done, not even by the world's craziest supercars.
With a manual transmission, the clutch would have to be in a partially engaged position for far too long resulting in too much friction which would generate too much heat and the clutch would burn out and fail way too soon.
In an automatic, it would put a hell of a load on the torque converter and the fluid inside would boil, which would melt the inside of the torque converter and possibly, under a heavy load, cause the converter to explode.

Currently the fastest car in the world (in theory) is the Koenigsegg Agera R with a theoretical top speed of 273mph. It has a 7 speed manual gearbox, and yes... like any other car uses gear reduction in the differential.
Before that was the Bugatti Veyron SS with a PROVEN top speed of 267.8 mph provided by a 16 cylinder engine with 4 turbochargers developing 1200 horsepower. It has a 7 speed dual-clutch automated manual gearbox (it's complicated) and also uses gear reduction because even a monster engine like that needs multiple gear ratios and gear reduction to perform as well as it does.

As you can tell this is quite a complex subject, and what I've listed is really just the basics, I don't know much beyond that but I hope I've helped to answer your question.