Diane Wallace bought a living$-$room suite on credit, signing an installment contract with a finance company that requires monthly payments of $$45\mathrm{.}47$ for three years. The first payment is made on the date of signing and interest is $18\%$ compounded monthly.

$($a$)$ What was the cash price?

$($b$)$ How much will Diane pay in total?

$($c$)$ How much of what she pays will be interest?

$($d$)$ Based on the cash price calculated in part $($a$),$ if the interest rate is changed to $15\mathrm{.}3\%$ compounded monthly, what is the new monthly payment?