Consider a loan of ​$15 comma 000 for a period of 20 years at a fixed APR of 9.5​%. Complete parts​ (a) through​ (c).  Of the total amount​ paid, what percentage is for the principal and what percentage is for​ interest?

Let's break down the loan problem step-by-step:

### Given:
- Principal loan amount (P): $15,000
- Annual percentage rate (APR): 9.5% (as a decimal, 0.095)
- Loan term (t): 20 years

The loan is likely an amortized loan, meaning the payments are the same every month, but the amount going toward principal and interest will change over time. We can calculate the total amount paid and determine the portions that go toward the principal and interest using the loan payment formula.

#### Step 1: Calculate Monthly Payment
The formula for the monthly payment $ M $ is:
\[
M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1}
\]
Where:
- $ P $ is the principal ($15,000)
- $ r $ is the monthly interest rate (APR divided by 12)
- $ n $ is the total number of payments (number of years multiplied by 12)

1. Calculate the monthly interest rate:
   \[
   r = \frac{9.5\%}{12} = \frac{0.095}{12} \approx 0.007917
   \]

2. Calculate the total number of payments:
   \[
   n = 20 \times 12 = 240 \text{ months}
   \]

3. Substitute values into the monthly payment formula:
\[
M = \frac{15,000 \times 0.007917 \times (1 + 0.007917)^{240}}{(1 + 0.007917)^{240} - 1}
\]
I will compute this next.

#### Step 2: Calculate Total Amount Paid
Once we have the monthly payment, we can calculate the total amount paid over the life of the loan:
\[
\text{Total amount paid} = M \times n
\]

#### Step 3: Calculate Interest Paid
To find the total interest paid, subtract the original loan amount from the total amount paid:
\[
\text{Total interest paid} = \text{Total amount paid} - \text{Principal}
\]

#### Step 4: Calculate Percentages
1. Percentage paid toward the principal:
\[
\text{Percentage for principal} = \frac{\text{Principal}}{\text{Total amount paid}} \times 100
\]
2. Percentage paid toward the interest:
\[
\text{Percentage for interest} = 100\% - \text{Percentage for principal}
\]

Let me calculate these now.Here are the results for the loan:

- **Monthly payment**: $139.82
- **Total amount paid** over 20 years: $33,556.72
- **Total interest paid**: $18,556.72

### Percentages:
- **Percentage paid toward principal**: 44.7%
- **Percentage paid toward interest**: 55.3%

This means that of the total amount paid over the loan period, 44.7% is for the principal and 55.3% is for interest.

Do you have any questions or need more details?

Here are 5 related questions you might consider:
1. What if the loan term was shorter, like 10 or 15 years?
2. How would the percentages change if the APR was lower, say 5%?
3. What happens if extra payments are made toward the principal each month?
4. How would an increase in the loan amount affect the total interest paid?
5. How can you minimize the amount of interest paid on a loan?

**Tip**: Making extra payments toward the principal early in the loan term can significantly reduce the amount of interest you pay over the life of the loan.

Consider a loan of $15,000 for a period of 20 years at a fixed APR of 9.5%. Of the total amount paid, what percentage is for the principal and what percentage is for interest?

Learn how to calculate monthly payments, total interest paid, and the percentage breakdown for a $15,000 loan at 9.5% APR over 20 years. Detailed step-by-step solution with formulas for loan payment, total interest, and percentage of principal versus interest.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation