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What tires are you running? I’m just trying to understand, because mine cost $38 ea, and it is not a place I’d be trying to save a buck.
Short answer: Yes. It is actually very bad to have different size tires on any driven axle. Even with an open differential. On the front, this would just cause handling issues. On the rear, it causes both handling and mechanical issues.
 

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I realize this. Do the math though. That is about a quarter inch difference in diameter.

I know that this isn’t the actual tire size, but for instance, a new 23” (72.22” circumference) tire vs a used 22.75” (71.435” circumference) tire is a difference in circumference of .785. Which is actually pretty significant. Also, the used tire should have been measured at the highest wear point, because that is the real effective diameter. The difference is likely much more.
 

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I realize this. Do the math though. That is about a quarter inch difference in diameter.

I know that this isn’t the actual tire size, but for instance, a new 23” (72.22” circumference) tire vs a used 22.75” (71.435” circumference) tire is a difference in circumference of .785. Which is actually pretty significant. Also, the used tire should have been measured at the highest wear point, because that is the real effective diameter. The difference is likely much more.
Agree with you. Almost half the tread is gone. I'd replace both sides at that point. Tires seem to be under appreciated. I see all these SUV ads showing drivers blissfully blasting through snow and ice on all season tires, realizing that in real life, that would be pretty stupid. Same goes for running on old, dry rotted or petrified tires. Tires have a finite life, regardless of how many miles they have. A flat at speed is no fun. Consider the death of Paul Walker:

 

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8mm - 5mm = 3mm radius (6mm diameter)

6mm is .236”, and this is all measured at the center of the tire, not the inside edge that is most likely nearly bald.

(Your math still works out the same
11/32 - 7/32 = 4/32 = 1/8R = 1/4D)
 
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