1. **Problem Statement:** Jamie made a down payment of 1200 and monthly payments of 270 for 4 years and 8 months. The interest rate is 1.00% compounded quarterly. We need to find: a. The cost of the car at purchase. b. The total interest paid over the term. 2. **Given Data:** - Down payment, $D = 1200$ - Monthly payment, $P = 270$ - Term: 4 years 8 months = $4 \times 12 + 8 = 56$ months - Interest rate per quarter, $i_q = 1.00\% = 0.01$ 3. **Convert interest rate to monthly rate:** Since interest is compounded quarterly, the quarterly rate is 0.01. Monthly rate $i_m$ is found by: $$ i_m = (1 + i_q)^{1/3} - 1 = (1 + 0.01)^{1/3} - 1$$ Calculate: $$ i_m = 1.01^{1/3} - 1 \approx 1.003322 - 1 = 0.003322$$ So monthly interest rate $i_m \approx 0.003322$ (or 0.3322%). 4. **Calculate the present value of the monthly payments:** The monthly payments form an ordinary annuity. The present value $PV$ of these payments is: $$ PV = P \times \frac{1 - (1 + i_m)^{-n}}{i_m} $$ where $n = 56$ months. Calculate: $$ PV = 270 \times \frac{1 - (1 + 0.003322)^{-56}}{0.003322} $$ First calculate $(1 + 0.003322)^{-56}$: $$ (1.003322)^{-56} = \frac{1}{(1.003322)^{56}} \approx \frac{1}{1.204} = 0.8307 $$ Then: $$ PV = 270 \times \frac{1 - 0.8307}{0.003322} = 270 \times \frac{0.1693}{0.003322} = 270 \times 50.97 = 13761.9 $$ 5. **Calculate the total cost of the car:** The total cost is the down payment plus the present value of the monthly payments: $$ \text{Cost} = D + PV = 1200 + 13761.9 = 14961.9 $$ Rounded to nearest cent: 14961.90 6. **Calculate total amount paid:** Total monthly payments: $$ 270 \times 56 = 15120 $$ Total amount paid including down payment: $$ 1200 + 15120 = 16320 $$ 7. **Calculate total interest paid:** Interest paid is total amount paid minus the cost of the car: $$ \text{Interest} = 16320 - 14961.9 = 1358.1 $$ Rounded to nearest cent: 1358.10 **Final answers:** a. Cost of the car = 14961.90 b. Total interest paid = 1358.10