The agitator on my 16 year feeble Kenmore doesn't seem to be to be working correctly . Is it time to capture a unknown washer? ...

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Answers:    Probably. I once had a repairman communicate me my washer wasn't
worth fixing when the agitator quit. My husband managed to draw from it to work, and it lasted another 6 months. If you can draw from it working, good, later start shopping for your next washer,
because if you don't have need of it now, you will soon! 16 years is great for a washer!
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Most credible it is time. Even if you get this fixed my guess would be that something else will dance wrong with it.