A sailboat costs $27,661. You pay 15% down and amortize the rest with equal monthly payments over a 12 year period. If you must pay 6.9% compounded​ monthly, what is your monthly​ payment? How much interest will you​ pay?

To solve this problem, we'll follow two steps: 

1. **Find the monthly payment** using the loan amortization formula.
2. **Calculate the total interest** paid over the loan period.

### Step 1: Calculate the Monthly Payment
The formula for calculating the monthly payment on an amortized loan is:

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

Where:
- $M$ is the monthly payment.
- $P$ is the loan principal (amount to be financed).
- $r$ is the monthly interest rate (annual rate divided by 12).
- $n$ is the total number of payments (months).

#### Information from the problem:
- The total cost of the sailboat is **$27,661**.
- You pay **15% down**, so the down payment is $27,661 \times 0.15 = 4,149.15$.
- The loan principal is $27,661 - 4,149.15 = 23,511.85$.
- The annual interest rate is **6.9%** compounded monthly, so the monthly interest rate is $\frac{6.9}{12} = 0.575\%$, or $0.00575$ as a decimal.
- The loan period is **12 years**, which gives $n = 12 \times 12 = 144$ months.

Now, let's calculate the monthly payment using this formula.

### Step 2: Calculate the Total Interest Paid
Once we find the monthly payment, the total interest can be found by calculating the difference between the total amount paid over the life of the loan and the loan principal:

\[
\text{Total Interest} = (M \times n) - P
\]

Let me compute these values for you.The monthly payment for the sailboat will be approximately **$240.54**.

The total interest paid over the 12-year period will be approximately **$11,125.90**.

Would you like further details or clarification on how these values were calculated? Here are some related questions to consider:

1. How would the monthly payment change if the interest rate was lower?
2. What happens to the total interest paid if the loan term is shorter?
3. How would an increased down payment affect the monthly payment?
4. Can I use the same formula for other types of loans?
5. How do early repayments impact the total interest paid?

**Tip:** A larger down payment reduces both the loan amount and the total interest paid over time.

A sailboat costs $27,661. You pay 15% down and amortize the rest with equal monthly payments over a 12-year period. If you must pay 6.9% compounded monthly, what is your monthly payment? How much interest will you pay?

Calculate the monthly payment and total interest for a $27,661 sailboat loan with a 15% down payment, amortized over 12 years at 6.9% interest compounded monthly. This problem involves concepts like loan amortization and compound interest, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation