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You are dealing with Fin Rot. The best way to prevent your fish from becoming ill is to follow the basic rules to having good aquarium management. Feed the fish specified amounts, maintain proper temperature and water conditions. Be sure to make frequent water changes and supply enough oxygen. Unfortunately, illness sometimes still occurs. When fish become ill they show signs in their behavior.
Fin or Tail Rot -Fins that are badly torn or frayed for reasons other than fighting will usually indicate fin or tail rot. Fin or tail rot is a bacterial disease that usually attacks weak or minimally damaged fin edges. Once established, the bacterium consumes the fin as it works its way up toward the fish's body. This causes the fins to appear frayed or ragged or in severe cases nearly stubs. Frayed fins can also be a sign of water problems in which case the pH, ammonia, or nitrites are too high. You should check your water conditons before treating. The best prevention is to maintain accurate water conditions. Sometimes the salt level will be too high. This can cause the effects of fin-rot.
Symptoms: The fish's tail or fins are ragged, frayed or appear to be shrinking or decaying. Goldfish may have red streaks in their veiltails.
Treatment: It is best treated with an antibiotic. There will be one best suitable at your local pet store. Make sure it states that it treats Fin or Tail Rot. In some cases a secondary infection or fungus will appear. If this is the case, treat the fish for the fin rot and then proceed to the fungus problem
I hope this helps. If you need further information, please feel free to contact me.
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