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Part 1

A woman borrows 

​$8000

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.

To solve this problem, we will calculate the monthly payments using the formula for the amortization of a loan and then break down the interest paid each year. Here's how we proceed:

### Step 1: Determine the Monthly Payment
The formula for calculating the monthly payment $ M $ for a loan amortized over time is:

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

where:
- $ P $ is the principal amount (initial loan) = \$8000
- $ r $ is the monthly interest rate = $\frac{6\%}{12} = \frac{0.06}{12} = 0.005$
- $ n $ is the total number of payments = $3 \times 12 = 36$ (for 3 years)

### Step 2: Calculate Monthly Payment $ M $

Let’s compute the monthly payment first.The monthly payment $ M $ is approximately \$243.38.

### Step 3: Determine Interest Paid Each Year
To find the interest paid during each year, we need to calculate how much of each monthly payment goes toward interest and how much goes toward the principal.

- Initially, the interest portion is higher, and it decreases as the loan balance reduces.
- The formula for interest paid in any month is:
  \[
  \text{Interest for month } i = \text{Remaining balance} \times r
  \]

We will calculate the total interest paid during the first, second, and third years. Let’s start by calculating the interest for the first year (12 months), then for the second and third years.The total interest paid during each year of the loan is approximately:

- **First year**: \$411.75
- **Second year**: \$257.02
- **Third year**: \$92.74

Would you like further details on the breakdown of payments, or do you have any other questions?

Here are five related questions to expand on this topic:
1. How is the total interest on a loan calculated over its entire term?
2. What factors influence the interest portion of each payment in an amortized loan?
3. How does changing the loan term affect the monthly payment and total interest paid?
4. What is the effect of different compounding periods (e.g., daily, quarterly) on interest calculations?
5. How does paying extra toward the principal affect the interest paid over time?

**Tip:** Making additional payments toward the principal can significantly reduce the total interest paid over the life of a loan.

A woman borrows $8000 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.

Find out how to calculate the interest paid each year on an $8000 loan amortized over 3 years with monthly payments at 6% interest compounded monthly. This problem demonstrates key concepts in loan amortization and interest calculations, suitable for students in Grades 11-12.

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