Question

You have decided to buy a new car. The total cost, including options, fees and taxes, is $30,760. The dealer offers three financing options as long as you make a down payment of at least 15% and you decide to put down exactly 15%Option 1a 3-year loan with an interest rate of 7%Option 2a 5-year loan with an interest rate of 5%Your goal is to pay as little total interest on the loan as possible, making regular monthly payments on one of these two financing options. For each option, show the monthly payment amount and the total interest that will have been paid over the entire term of the loan.

Answer 1

Total cost of the car is $30,760

Down payment is gotten below:

$\begin{gathered} 15\text{\% of 30,760} \\ =\frac{15}{100}\times30,760=4614 \end{gathered}$

Remaining balance to pay is;

$\begin{gathered} 30,760-4614 \\ =26,146 \end{gathered}$

To calculate the second option, we use the loan formula written below:

$\begin{gathered} P=\frac{\text{PMT}}{i}\lbrack1+\frac{1}{(1+i)^n}\rbrack \\ P=\frac{26416}{\frac{5}{12}}\lbrack1+\frac{1}{(1+\frac{5}{12})^{60}}\rbrack \\ P=29,910.15 \\ \text{Total payment is \ 29,910.15 after 5 years} \\ \text{Monthly payment = \ 498.50} \\ \text{Total interest = 29,910.15-26416 = \ 34}94.15 \end{gathered}$

To calculate the first option, we still use the same formula as seen below:

$\begin{gathered} P=\frac{\text{PMT}}{i}\lbrack1+\frac{1}{(1+i)^n}\rbrack \\ P=\frac{26416}{\frac{7}{12}}\lbrack1+\frac{1}{(1+\frac{7}{12})^{36}}\rbrack \\ P=29,363.38 \\ \text{Total payment after 3 years is \ 29,363.38} \\ \text{Monthly payment = \ 815.65} \\ \text{Total interest }=\text{ 29,363.38-26416 = \ }2947.38 \end{gathered}$

For Option 1:

Monthly payment amount is $815.65

Total interest is $2,947.38

For Option 2:

Monthly payment amount is $498.50

Total interest is $3494.15