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Part 1
A woman borrows ​$7000 at 12​% compounded​ monthly, which is to be amortized over 3 years in equal monthly payments. For tax​ purposes, she needs to know the amount of interest paid during each year of the loan. Find the interest paid during the first​ year, the second​ year, and the third year of the loan. ​[Hint: Find the unpaid balance after 12 payments and after 24​ payments.]
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Part 1
The interest paid during the first year is ​$
  
729.09.
​(Round to the nearest cent as​ needed.) 
Part 2
The interest paid during the second year is ​$
  
enter your response here.

To calculate the interest paid during the second year of the loan, we need to follow these steps:

1. **Determine the loan amount and interest rate details**:
   - Loan amount = $7000
   - Interest rate = 12% compounded monthly, which translates to a monthly interest rate of:
     \[
     i = \frac{12\%}{12} = 1\% = 0.01
     \]
   - Loan term = 3 years or 36 months.

2. **Calculate the monthly payment** using the amortization formula:
   \[
   M = \frac{P \times i}{1 - (1 + i)^{-n}}
   \]
   where:
   - $ P = 7000 $ (loan amount),
   - $ i = 0.01 $ (monthly interest rate),
   - $ n = 36 $ (number of months).

Let's calculate the monthly payment first.The monthly payment for the loan is approximately **$232.50**.

Next, we'll calculate the **unpaid balance** after 12 and 24 payments to determine the interest paid during the second year (from month 13 to month 24).

The formula for the remaining balance after $ k $ payments is:

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

We'll calculate the balance after 12 payments and 24 payments to determine how much interest was paid in the second year.The unpaid balance after 12 payments is approximately **$4939.09**, and after 24 payments, it is approximately **$2616.81**.

Now, to calculate the interest paid during the second year, we need to:

1. Compute the total interest paid from months 13 to 24. 
2. For each month, the interest is calculated as $ I_k = \text{balance} \times i $, where $ i = 0.01 $ is the monthly interest rate.

I'll now compute the interest paid during the second year (months 13-24).It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, you can share any additional information or questions!

In the meantime, here are some related questions you might explore:
1. How is the amortization schedule used to calculate interest payments for each month?
2. What is the impact of making extra payments on an amortized loan?
3. How does compounding monthly interest differ from annual compounding?
4. What are the differences between amortized loans and interest-only loans?
5. How would the total interest change if the loan term were extended?

**Tip**: When calculating loan payments, always ensure that the interest rate is adjusted to match the compounding period.

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.]

Learn how to calculate the interest paid on a $7000 loan amortized over 3 years at a 12% compounded monthly interest rate. This step-by-step solution covers loan amortization, interest calculation, and compound interest formulas. Suitable for finance students and anyone learning about loan repayments.

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