The factory says a 1.5% variance in tire rotational length is acceptable, but nothing more. The difference between a new tire with 13/32 tread depth and a worn out one with about 4/32 is close to 2% variance. If the new tire is 13/32 and your old ones have more than 8/32 you're probably OK, but if the tires are worn more than that I wouldn't replace just one.

An option for you is to buy 4 new ones, then list the three used ones on Craigslist or Fleabay for sale to recoup part of your money.

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'04 E53, 4.4 Sport
'97 E39 528i
'86 911 Carrera, track car
'96 BMW R1100R