A woman borrows ​$7000 at 6​% compounded​ monthly, which is to be amortized over 3 years in equal monthly payments. For tax​ purposes, she needs to know the amount of interest paid during each year of the loan. Find the interest paid during the first​ year, the second​ year, and the third year of the loan. ​[Hint: Find the unpaid balance after 12 payments and after 24​ payments.]

To solve this, we'll use the amortization formula to determine the monthly payment and then calculate the interest paid during each year by finding the unpaid balance after 12 and 24 payments.

### Step 1: Calculate the monthly payment
The monthly payment on an amortized loan can be calculated using the following formula:

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

Where:
- $ M $ = monthly payment
- $ P $ = loan principal = 7000
- $ r $ = monthly interest rate = $\frac{6\%}{12} = \frac{0.06}{12} = 0.005$
- $ n $ = total number of monthly payments = 3 years $\times$ 12 months/year = 36 months

Now, let's calculate the monthly payment:

\[
M = 7000 \cdot \frac{0.005(1 + 0.005)^{36}}{(1 + 0.005)^{36} - 1}
\]

I'll compute this first.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can manually calculate the monthly payment using the formula:

\[
M = 7000 \cdot \frac{0.005(1 + 0.005)^{36}}{(1 + 0.005)^{36} - 1}
\]

Let me know if you'd like assistance with the next steps or clarification on how to proceed with this calculation!

Here are some related questions to expand on the topic:
1. How does the amortization schedule change if the interest rate increases?
2. How is the principal repaid differently during the first and last years of the loan?
3. What is the effect of compounding frequency on the loan's interest payments?
4. How does loan duration impact monthly payments and total interest?
5. What other types of loans use amortization schedules?

**Tip:** Always double-check your interest rate conversions when using compounded rates in formulas!

This problem involves calculating the interest paid over the duration of a $7000 loan at 6% interest compounded monthly, amortized over 3 years. Using the amortization formula, the solution helps to find the interest paid during the first, second, and third years. Ideal for students studying finance or advanced high school algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation