A woman borrows ​$7000 at 12​% 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 problem, we need to first determine the monthly payment and then calculate the unpaid balance at the end of each year to find the interest paid each year. Here's the detailed approach:

### Step 1: Formula for Monthly Payment
The formula to calculate the monthly payment $ M $ for an amortized loan is:

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

Where:
- $ P = 7000 $ (the principal loan amount),
- $ r = \frac{0.12}{12} = 0.01 $ (the monthly interest rate, since the annual rate is 12%, and it's compounded monthly),
- $ n = 3 \times 12 = 36 $ (the total number of payments over 3 years).

### Step 2: Calculate the Monthly Payment
Substitute the given values into the formula to calculate $ M $.

\[
M = 7000 \times \frac{0.01(1 + 0.01)^{36}}{(1 + 0.01)^{36} - 1}
\]

Let me compute this.The monthly payment $ M $ is approximately \$232.50.

### Step 3: Unpaid Balance After 12 and 24 Payments
We now need to calculate the unpaid balance after 12 payments (end of the first year) and 24 payments (end of the second year). The formula for the unpaid balance after $ k $ payments is:

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

We'll calculate this for $ k = 12 $ (end of the first year) and $ k = 24 $ (end of the second year).

Let me compute the unpaid balances after 12 and 24 payments.The unpaid balances are approximately:
- After 12 payments (end of the first year): \$4939.09
- After 24 payments (end of the second year): \$2616.81

### Step 4: Interest Paid During Each Year

The total interest paid each year is the difference between the total payments made and the reduction in the loan balance.

- **First Year**: In the first year, the total payments are $ 12 \times 232.50 = 2790 $. The loan balance decreases from \$7000 to \$4939.09, so the reduction in principal is $ 7000 - 4939.09 = 2060.91 $. The interest paid is the difference between the total payments and the principal reduction:

\[
\text{Interest (Year 1)} = 2790 - 2060.91 = 729.09
\]

- **Second Year**: In the second year, the total payments are again \$2790. The loan balance decreases from \$4939.09 to \$2616.81, so the reduction in principal is $ 4939.09 - 2616.81 = 2322.28 $. The interest paid is:

\[
\text{Interest (Year 2)} = 2790 - 2322.28 = 467.72
\]

- **Third Year**: In the third year, the remaining balance is paid off. The total payments are \$2790, and the balance decreases from \$2616.81 to \$0, so the reduction in principal is \$2616.81. The interest paid is:

\[
\text{Interest (Year 3)} = 2790 - 2616.81 = 173.19
\]

### Summary of Interest Paid:
- **First Year**: \$729.09
- **Second Year**: \$467.72
- **Third Year**: \$173.19

Would you like any further details or clarification on any part?

Here are 5 related questions to explore:
1. How would the interest payments change with a different loan term?
2. What if the interest were compounded weekly instead of monthly?
3. How can we compute the total interest paid over the life of the loan?
4. What is the effect of making additional payments on the loan's amortization schedule?
5. How does the loan's amortization schedule change with varying interest rates?

**Tip**: Understanding amortization helps you see how much of each payment goes to interest versus principal, which can guide financial planning.

Discover how to calculate the interest paid during each year on a $7000 loan amortized over 3 years with 12% compounded monthly. This detailed step-by-step guide uses the amortization and compound interest formulas to find the interest paid in the first, second, and third years of the loan.

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