A sailboat costs $ 29,135. You pay 5% down and amortize the rest with equal monthly payments over a 8​-year period. If you must pay 8.4% compounded​ monthly, what is your monthly​ payment? How much interest will you​ pay?

To determine the monthly payment and total interest paid on the sailboat loan, we need to follow these steps:

1. **Calculate the down payment.**
2. **Determine the loan amount.**
3. **Calculate the monthly payment using the loan amortization formula.**
4. **Calculate the total amount paid over the life of the loan.**
5. **Determine the total interest paid.**

### Step 1: Calculate the Down Payment
The down payment is 5% of the total cost of the sailboat.

\[ \text{Down Payment} = 0.05 \times 29135 \]

\[ \text{Down Payment} = 1456.75 \]

### Step 2: Determine the Loan Amount
The loan amount is the cost of the sailboat minus the down payment.

\[ \text{Loan Amount} = 29135 - 1456.75 \]

\[ \text{Loan Amount} = 27678.25 \]

### Step 3: Calculate the Monthly Payment
We use the formula for the monthly payment on an amortizing loan:

\[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \]

where:
- $ M $ is the monthly payment,
- $ P $ is the principal (loan amount),
- $ r $ is the monthly interest rate,
- $ n $ is the total number of payments.

Given:
- $ P = 27678.25 $
- Annual interest rate = 8.4%, so monthly interest rate $ r = \frac{8.4\%}{12} = 0.007 $
- $ n = 8 \times 12 = 96 $ months

Plugging in these values:

\[ M = 27678.25 \frac{0.007(1+0.007)^{96}}{(1+0.007)^{96} - 1} \]

First, calculate $(1+0.007)^{96}$:

\[ (1+0.007)^{96} \approx 1.854 \]

Then, the formula becomes:

\[ M = 27678.25 \frac{0.007 \times 1.854}{1.854 - 1} \]

\[ M = 27678.25 \frac{0.012978}{0.854} \]

\[ M = 27678.25 \times 0.01519 \]

\[ M \approx 420.28 \]

So, the monthly payment is approximately $ \$420.28 $.

### Step 4: Calculate the Total Amount Paid
\[ \text{Total Amount Paid} = M \times n \]

\[ \text{Total Amount Paid} = 420.28 \times 96 \]

\[ \text{Total Amount Paid} \approx 40346.88 \]

### Step 5: Determine the Total Interest Paid
\[ \text{Total Interest Paid} = \text{Total Amount Paid} - \text{Loan Amount} \]

\[ \text{Total Interest Paid} = 40346.88 - 27678.25 \]

\[ \text{Total Interest Paid} \approx 12668.63 \]

### Summary
- **Monthly Payment:** $ \$420.28 $
- **Total Interest Paid:** $ \$12668.63 $

Would you like more details or have any questions?

Here are 5 relative questions that expand on this information:
1. How would the monthly payment change if the interest rate were 7% instead of 8.4%?
2. What would be the monthly payment for a 10-year loan term at the same interest rate?
3. How much would you save in interest if you made an additional monthly payment of \$100?
4. What is the impact of increasing the down payment to 10% on the monthly payment and total interest paid?
5. How can you calculate the remaining loan balance after 4 years of payments?

**Tip:** Regularly paying a little extra towards the principal can significantly reduce the total interest paid over the life of the loan.

A sailboat costs $29,135. You pay 5% down and amortize the rest with equal monthly payments over an 8-year period. If you must pay 8.4% compounded monthly, what is your monthly payment? How much interest will you pay?

Learn how to calculate the monthly payment and total interest for a sailboat costing $29,135, with 5% down and an 8.4% interest rate compounded monthly over 8 years. Detailed solution using loan amortization formulas, suitable for Grades 10-12.

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