A transformer would introduce unnecessary losses, be expensive, take a lot of space and have certain performance limitations.
I test ran a 200 horsepower three phase 480 volt motor in the shop today. According to the nameplate, at full load, this motor should pull 225 amps and run at a 94% efficiency.
Running at 480 volts 60 Hz unloaded the amp draw was 70 amps and the power meter measured 2.32 kilowatts of power being consumed by the motor. That’s only about 3.1 horsepower or about 1/72 of the power the motor draws at full load. Then I reduced the voltage to 263 volts and the amp draw dropped to 30 amps and the power meter read 1.35 kilowatts or 1.81 horsepower drawn from the power line.
Of course, an unloaded motor has an efficiency of zero no matter how little power it draws. In the Tesla model S with its induction motor driven by a variable frequency inverter, the zero load power can be reduced to nearly zero by lowering the inverter output voltage to nearly zero.
With modern inverter driven three phase induction motors, the electronic inverter sort of becomes the car’s transmission.
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Some electric cars do use a transmission. There was an electric conversion kit for the MR2 which put an electric motor in the car, but used the stock 5-speed transmission. Might still be available - I won’t look because I don’t want to be tempted (again) to spend a lot of money on it. 
For the 1991-1999 generation EV west makes a kit. About $7,500 though.
I’ve been watching the show Vintage Voltage on the Motor Trend channel, pretty interesting how they do the conversions. Depending on how much power you want you can use the stock transmission but Tesla conversions seem to becoming more common. Using the whole drive unit from a Model S.
That voltage control 3 phase motor is acting like three turbine of a torque converter. It runs on slip between three rotor and the source frequency and some wasted heat is generated. Variable frequency drive is like a torque converter running in the coupled stage.
In either case, the induction motor works on slip(the difference between rotor speed and drive frequency). Attaching a transmission at the motor output reduces that slip
That motor I tested had a winding resistance that measured 0.0201 ohms from leg to leg. At 225 amps, that gives an I^2R loss of 1526 watts. The slip losses , 1780 rpm, or 20 rpm of slip, amounts to 2.25 horsepower or 1676 watts. This motor has a nameplate efficiency of 94% so the total loss at full load would be 9500 watts, 3202 of those watts from slip and copper resistance losses, where did the remainder go? Iron losses, bearing friction, and windage mostly. Iron losses come from magnetic hysteresis and eddy currents in the core.
When the motor is only loaded half ways, the slip losses are reduced, and so are the I^2R losses in the winding, but the magnetic flux density in the iron core stays the same. Reducing the terminal voltage to around 70% both reduces the I^2R and core losses which more than compensates for the slightly higher slip due to the reduced voltage.
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Transmissions are not without losses. A lot depends on design but a typical range I have seen is around 15-18% for a manual and 18-22% for an automatic. That is a lot of power lost not to mention the additional weight and cost…no free lunch. There’s a reason things are the way they are today. Many different engineering groups have already run the numbers on cost-benefit scenarios…
Wasn’t that the one that essentially had a Prius motor though? I think it was sub-100 hp if i remember right. If I’m gonna spend thousands converting the thing to electric, I expect to scare the hell out of my passengers when I’m done. 
It’s possible, have to look again.
It doesn’t have to be fast to be frightening. My Austin America in the early 70s was underpowered, but a stone-cold hoot when cornering. It was an early FWD car, and few people knew how well they cornered as long as you didn’t let up on the gas. I remember scaring one of my friends taking a 90 degree left turn at about 30. He was in the back seat alone and bounced off the left wall, then the right wall as I fish tailed through the turn. The pavement was dry and I kept my foot in it. He was terrified, and his ride was a Firebird 400 with a 4.10 rear. I was pleased…
Heck, I can scare the hell out of my passengers with a stock Toyota Yaris.
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Yep, only 71hp from the electric motor.
https://www.evwest.com/catalog/product_info.php?products_id=219
$7600 and no battery!
Not included in the kit but you have lots of options from Ev West, at various power levels and pricing
Yeah, you can scare people in anything. I wasn’t being literal. Handling is great, as I well know owning a CRX and a non-turbo MR2. But sometimes getting pushed back in your seat is nice too, and that little “less horsepower than a 1988 CRX HF” motor ain’t gonna do that.
The electric motor has ruler flat torque from near zero rpm.
Without turbo/supercharging you’d need a gas motor rated about 2X the horsepower (142 hp) to get the same practical performance.
Yeah, but since I’m sitting at 135 right now, that would be a very expensive 7hp performance gain.
As the owner of an electric bike and scooter I can say there would be a lot more difference than 7hp.
Not for everyone of course. But a second/third vehicle for short trips, low maintenance, and to top it off sounds like George Jetson’s space car.
As someone who’s driven a Prius, that’s still a slow car, and that’s helping the electric motor out with a second engine.
My generation of MR2 is only 500 or so pounds lighter than a Prius. The acceleration would be significantly underwhelming when you factor the cost of the swap. $7600 plus a couple-three grand for the battery pack would buy a boatload of power elsewhere. Heck, if you look hard enough you could probably find a wrecked Model S in a junkyard and get the motor and possibly the battery pack for that price.
But is the Model S battery pack bigger than the MR2? You could have it on a trailer and tow it I suppose…