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Find the monthly payment.

Find the total amount to repay the loan.


Find the total amount of interest that will be paid.

To solve this problem, we'll calculate the monthly payment, the total amount to repay the loan, and the total amount of interest that will be paid using the loan information provided.

### Given Information:
- **Principal (P):** $7,500
- **Annual Interest Rate (r):** 32.49% per year
- **Loan Term (n):** 3 years

### Steps to Solve

1. **Convert the Annual Interest Rate to a Monthly Rate:**
   \[
   r_{\text{monthly}} = \frac{32.49\%}{12} = \frac{0.3249}{12} \approx 0.027075 \text{ (monthly rate)}
   \]

2. **Convert the Loan Term from Years to Months:**
   \[
   n_{\text{months}} = 3 \times 12 = 36 \text{ months}
   \]

3. **Calculate the Monthly Payment Using the Formula for an Amortizing Loan:**

The formula for the monthly payment $M$ on a loan is given by:
\[
M = P \times \frac{r_{\text{monthly}}(1 + r_{\text{monthly}})^{n_{\text{months}}}}{(1 + r_{\text{monthly}})^{n_{\text{months}}} - 1}
\]

Plugging in the values:
\[
M = 7500 \times \frac{0.027075 \times (1 + 0.027075)^{36}}{(1 + 0.027075)^{36} - 1}
\]

Let's calculate this step by step:

### Step-by-Step Calculation

First, compute $(1 + r_{\text{monthly}})^{n_{\text{months}}}$:

\[
(1 + 0.027075)^{36} \approx 2.812
\]

Now, compute the numerator:

\[
0.027075 \times 2.812 \approx 0.07618
\]

And the denominator:

\[
2.812 - 1 = 1.812
\]

Thus, the monthly payment $M$ is:

\[
M = 7500 \times \frac{0.07618}{1.812} \approx 7500 \times 0.04203 \approx 315.22
\]

**Monthly Payment $M$ ≈ $315.22**

4. **Calculate the Total Amount to Repay the Loan:**

\[
\text{Total Repayment} = M \times n_{\text{months}} = 315.22 \times 36 \approx 11,347.92
\]

5. **Calculate the Total Interest Paid:**

\[
\text{Total Interest} = \text{Total Repayment} - P = 11,347.92 - 7,500 \approx 3,847.92
\]

### Summary of Results
- **Monthly Payment:** $315.22
- **Total Amount to Repay the Loan:** $11,347.92
- **Total Interest Paid:** $3,847.92

### Would you like more details or have any questions?

Here are some related questions that might help you further:

1. How would the monthly payment change if the interest rate was lower, say 20% annually?
2. What would be the impact on the total interest paid if the loan term was extended to 5 years instead of 3 years?
3. How does the calculation differ if the interest is compounded quarterly instead of monthly?
4. Can you explain how an amortization schedule would look like for this loan?
5. What is the effect of making extra payments towards the principal on the total interest paid over the life of the loan?

**Tip:** When calculating loan payments, always check if the interest rate is annual and needs to be converted to a monthly rate for correct calculations.

Learn how to calculate the monthly payment, total repayment, and total interest paid on a loan with a detailed example. This problem covers key concepts such as interest rates, amortization formulas, and compound interest calculations. Suitable for advanced high school students.

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