change the new battery three month ago . the car can not start after run short distance. we recharge the battery and it only can run about 30-40 KM .the problem show the battery has not been charged and battery light is red . I went to the car shop and they advice to replace the alternator, I am bit confuse because this 2002 R50 run about 78000 KM. is it the belt or other issue create the battery not charge problem? could you please advice if it need to change the alternator and what I should check it

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Sounds like a faulty alternator. I would check charging system voltage and current to be sure.

- Nick