how do i install the ink cartridge in the sharp XE-A102 register?

It doesn't have a cartridge. It uses an ink roller that sits in the little carriage in the printer. It has the word "pull" or it might be "pull up" on a little tag. Pull it and it will pop out, then replace it.
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If anyone needs a Sharp XE-A102 cash register manual, go here:
http://the-checkout-tech.com/manuals/cashregisters/Sharp9.php

Registerman
June 2010