'Not to care a rap' definitions:

Definition of 'Not to care a rap'

From: GCIDE

Rap \Rap\, n. [Perhaps contr. fr. raparee.] A popular name for any of the tokens that passed current for a half-penny in Ireland in the early part of the eighteenth century; any coin of trifling value. [1913 Webster]
Many counterfeits passed about under the name of raps. --Swift. [1913 Webster]
Tie it [her money] up so tight that you can't touch a rap, save with her consent. --Mrs. Alexander. [1913 Webster]
Not to care a rap, to care nothing.
Not worth a rap, worth nothing. [1913 Webster]