And another one
	
		
	
	
		
		
			My neighbours have a 120 volt mig. It will blow a 15 amp breaker in no time so to use it properly you need to install a 20 amp breaker with correspondingly larger wire. Correct me if I'm wrong but a wire that is sized for 20 amps at 120 volts will handle 40 amps on a 240 volt circuit. 40 amps is enough to run a welder so why bother with the 120 volts?
		
		
	 
Thats a good point for those who want to weld at full power (on 120v) they need a 20A outlet.  With 15A outlet you can weld at lower settings though.  I welded for several years on a 15A circuit and popped the breaker a few times.
If neighbor put a 20A breaker in that slot and ran a 12ga wire out to a 20A rated outlet, they would have a 20A circuit which would operate the machine at the max settings.  15Amp circuits are plentiful but getting a 20A 120v circuit can be almost as difficult (for some) or expensive as getting 220v (if your panel has the slots or capacity).  But that's for welding at the machines full power setting, which is only required for heavier projects.
With regard to your next question you have a mixup somewhere.  40A@240v is 4 times the power of 20A@120v.
20A x 120volts is 2400watts over one 12ga wire to neutral/ground.
20A x 240volts is 4800watts over two 12ga wires.
40A x 240volts is 9600 watts, pretty sure you need 8ga for 40A.
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I see what you're getting at though.  Lets say you had a 20A (12-2 wire) going to the garage.  Today, it has one breaker, thus 120v on the black wire, and the white is used for neutral + ground.
You're thinking why not add another breaker in the panel (on the other leg) to the white wire, and get 20A@240v = 4800 Watts, 240v at the garage outlet.  So all you are changing is adding a breaker at the panel, and changing the outlet in the garage to 240v type, over the same 12ga wire.
That can work, 240v air conditioners use that configuration.  Looks like to get full power out of a 211 (for example) they recommend 25A but I bet a 211 would do a LOT on 20A.
======= specs for a Miller 211 ==========
with MVP Millermatic211Auto-SetTM
InputPower
230 V, 25 A, 60 Hz, Single-Phase
120 V, 20 A, 60 Hz, Single-Phase
=========================