A woman borrows ​$5000 at 3​% 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 calculate the monthly payment, then use that to find the unpaid balance after 12 and 24 months, which will help in determining the interest paid during each year of the loan.

The loan follows the formula for amortized payments:

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

Where:
- $M$ is the monthly payment,
- $P$ is the principal (\$5000),
- $r$ is the monthly interest rate (annual interest rate divided by 12),
- $n$ is the total number of payments (number of months for the loan).

### Step 1: Calculate Monthly Payment
The annual interest rate is 3%, so the monthly interest rate $r$ is:

\[
r = \frac{3\%}{12} = 0.0025
\]

The number of monthly payments $n$ for 3 years is:

\[
n = 3 \times 12 = 36
\]

Using the formula for the monthly payment:

\[
M = 5000 \times \frac{0.0025(1 + 0.0025)^{36}}{(1 + 0.0025)^{36} - 1}
\]

Now, let's calculate the value of $M$.The monthly payment $M$ is approximately \$145.41.

### Step 2: Calculate Unpaid Balance After 12 and 24 Payments

The unpaid balance after a certain number of payments can be found using the formula:

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

Where:
- $B_k$ is the balance after $k$ payments,
- $M$ is the monthly payment,
- $r$ is the monthly interest rate,
- $k$ is the number of payments made.

#### Unpaid balance after 12 payments:
We'll calculate the balance after the first 12 months.

#### Unpaid balance after 24 payments:
We'll calculate the balance after the first 24 months.

Let's calculate both.The unpaid balance after 12 months is approximately \$3383.01, and the unpaid balance after 24 months is approximately \$1716.85.

### Step 3: Calculate Interest Paid During Each Year

The interest paid each year can be found by calculating the total amount paid during the year (12 payments) and subtracting the principal portion of the payments.

#### Interest paid in the first year:
- Total payments made in the first year = 12 payments $\times$ \$145.41.
- Principal paid in the first year = Original loan amount $-$ Balance after 12 months.

#### Interest paid in the second year:
- Total payments made in the second year = 12 payments $\times$ \$145.41.
- Principal paid in the second year = Balance after 12 months $-$ Balance after 24 months.

#### Interest paid in the third year:
- Total payments made in the third year = 12 payments $\times$ \$145.41.
- Principal paid in the third year = Balance after 24 months $-$ Final balance (0).

Let's calculate the interest for each year.The interest paid during each year of the loan is approximately:

- **First year**: \$127.89
- **Second year**: \$78.70
- **Third year**: \$28.03

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How does changing the loan term affect the interest paid?
2. What would happen if the interest rate were compounded quarterly instead of monthly?
3. How does making additional payments affect the total interest paid?
4. What is the difference between simple and compound interest in amortized loans?
5. Can you calculate the total interest paid over the life of the loan?

**Tip**: The earlier you pay off a loan, the less interest you'll pay in total, since interest is typically front-loaded in amortization schedules.

This problem involves calculating the monthly payments and unpaid balances of a $5000 loan at 3% interest compounded monthly, amortized over 3 years. It covers loan amortization, compound interest, and algebraic calculations to find the interest paid in each year. Ideal for Grades 10-12 students.

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