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Think about this if you change just one battery in a remote it will die faster than if you changed both the dead battery will suck the life out of the other when you go to use it. Now car batteries are similar but at the same time different, what i would do is get both tested take them out and get them tested. If just one test ok i would just replace that one, if you have the money replace both. Because let’s say the old battery is only sitting at 11 volts and has a problem reviving itself back to 11 volts once you try to start it. The new battery is sitting at 12 but once you hook the batteries together it will be at 11 and have to help the weaker battery every time thus weakening the good battery to almost the same as the weaker battery