A sailboat costs $ 20 comma 981. You pay 5 % down and amortize the rest with equal monthly payments over a 10​-year period. If you must pay 7.2 % compounded​ monthly, what is your monthly​ payment? How much interest will you​ pay?
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Part 1
Monthly​ payments: ​$ 

To find the monthly payment and the total interest paid for the sailboat loan, we can follow these steps:

1. **Calculate the down payment:**
   The down payment is 5% of the sailboat's cost.

2. **Calculate the loan amount:**
   Subtract the down payment from the sailboat's total cost to find the principal amount that will be amortized.

3. **Calculate the monthly payment:**
   Use the formula for 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

4. **Calculate the total amount paid and the total interest:**
   The total amount paid is the monthly payment multiplied by the number of payments, and the total interest is the total amount paid minus the principal.

Let's calculate these values step-by-step.

### 1. Down Payment
The sailboat costs $ \$20,981 $ and the down payment is 5%.

\[
\text{Down Payment} = 0.05 \times 20981
\]

### 2. Principal Loan Amount
\[
\text{Principal} = \text{Total Cost} - \text{Down Payment}
\]

### 3. Monthly Payment
The interest rate is 7.2% compounded monthly, so the monthly interest rate $ r $ is:
\[
r = \frac{7.2\%}{12} = 0.006
\]
The loan is amortized over 10 years, so the number of payments $ n $ is:
\[
n = 10 \times 12 = 120
\]

### Calculation
Let's compute these values.### Results

1. **Down Payment:** $1,049.05
2. **Principal Loan Amount:** $19,931.95
3. **Monthly Payment:** $233.49 (rounded to two decimal places)
4. **Total Amount Paid:** $28,018.39
5. **Total Interest Paid:** $8,086.44

Would you like any further details or have any questions?

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**Related Questions:**

1. How would the monthly payment change if the interest rate were 5% instead of 7.2%?
2. What would be the effect on the monthly payment if the loan term were reduced to 5 years?
3. How much would the total interest paid increase if the loan were extended to 15 years?
4. What is the formula used to calculate the monthly payment for an amortizing loan?
5. How does the down payment affect the total interest paid over the loan period?
6. How would the monthly payments change if the down payment were increased to 10%?
7. Can the loan be paid off earlier? What would be the effect on the interest paid?
8. How can one calculate the balance remaining on the loan after 5 years?

**Tip:** To minimize the total interest paid, consider increasing the down payment or reducing the loan term.

Learn how to calculate monthly payments and total interest for a sailboat loan using an amortizing loan formula with compound interest. Detailed step-by-step solution for a loan of $20,981, paying 5% down and amortizing the rest over 10 years at 7.2% interest.

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