Let's break down Sally Seair's sailboat financing step by step:
1. Purchase Price and Down Payment:
- The price of the sailboat, including tax, is \[tex]$5,275.
- The down payment Sally makes is \$[/tex]500.
2. Amount Financed:
- The amount Sally finances is the purchase price minus the down payment.
- [tex]\(\$5,275 - \$500 = \$4,775\)[/tex]
3. Annual Interest Rate and Loan Term:
- The true annual interest rate is 15%.
- The loan term is 36 months.
4. Monthly Interest Rate:
- The annual interest rate is 15%, so the monthly interest rate is [tex]\(\frac{15\%}{12} = 1.25\%\)[/tex].
5. Monthly Payment Calculation:
- We need to use the formula for the monthly payment of an installment loan:
[tex]\[
M = P \frac{r(1+r)^n}{(1+r)^n - 1}
\][/tex]
where:
- [tex]\( M \)[/tex] is the monthly payment,
- [tex]\( P \)[/tex] is the loan amount (\[tex]$4,775),
- \( r \) is the monthly interest rate (0.0125),
- \( n \) is the number of payments (36).
6. Calculating Monthly Payment:
- Using the given values, we find the monthly payment to be approximately \$[/tex]165.53.
7. Total of Payments:
- The total amount paid over the course of the loan can be calculated by multiplying the monthly payment by the number of payments:
[tex]\[
c = M \times n
\][/tex]
- Substituting the values, we get:
[tex]\[
c = \$165.53 \times 36 \approx \$5,958.97
\][/tex]
8. Total of Payments and Amount Financed:
- The total of payments consists of the down payment and [tex]\( c \)[/tex]:
[tex]\[
\text{Total of payments} = \$500 + c
\][/tex]
- Substituting [tex]\( c \)[/tex], we get:
[tex]\[
\text{Total of payments} = \$500 + \$5,958.97 = \$6,458.97
\][/tex]
Summarizing the solutions:
- To the nearest penny, [tex]\( c = \$5,958.97 \)[/tex].
- Total of payments = \[tex]$4,775 (amount financed) + \$[/tex]5,958.97 = \[tex]$10,733.97.
- Monthly payment = \$[/tex]165.53.
