A store offers two payment plans. Under the installment plan, you pay $20\%$ down and $20\%$ of the purchase price in each of the next $3$

years. If you pay the entire bill immediately, you can take a $10\%$ discount from the purchase price.

a$.$ Which is the better deal if the interest rate is $5\mathrm{.}1\%?$

b$.$ How will your answer change if the four payments on the installments do not start until the end of the year?