Problem $1$:

A six month bank loan of $$1,000$ required payments of $$175\mathrm{.}53$ per month. After three payments were made, the borrower paid off the loan for $$524\mathrm{.}03,$ a sum determined by the lender. Interest was quoted at $18$ percent nominal, compounded monthly.

a$.$ Was the monthly payment correct?

b$.$ Did the bank overcharge the borrower on the payoff and if so$,$ by how many dollars?