Jack and Jill have found a hill they would like to buy. They can buy the hill for $$229000.$ In talking with the bank, they have been offered a mortgage for $85\%$ of the cost $($meaning they need a $15\%$ down payment$)$ at $6\mathrm{.}2\%$ compounded monthly, with regular weekly payments for $15$ years.

a$)$ How much will the down payment be$?$

b$)$ How much will the principal of the mortgage be$?$