Question

Say you take out a loan with a principal of $44,500. The interest rate is 13.11%, compounded monthly. If you make consistent monthly payments and pay off the loan over the course of six and a half years, how much interest will you have paid in total? Round dollar amounts to the nearest cent.
a.
$21,849.92
b.
$3,018.03
c.
$20,003.60
d.
$24,321.18


Please select the best answer from the choices provided

A
B
C
D

Answer 1

To calculate the total interest paid on the loan, we can follow these steps:

1. **Identify the variables:**
- Principal (P) = $44,500
- Interest rate (r) = 13.11% per year = 0.1311/12 per month
- Number of payments (n) = 6.5 years = 6.5 * 12 months = 78 months

2. **Calculate the monthly interest rate:**
\[
r_{monthly} = \frac{0.1311}{12} = 0.010925 \text{ (approximately)}
\]

3. **Use the formula for the monthly payment (M) for an amortizing loan:**
\[
M = P \cdot \frac{r_{monthly}(1 + r_{monthly})^n}{(1 + r_{monthly})^n - 1}
\]

4. **Substituting the values:**
\[
M = 44500 \cdot \frac{0.010925(1 + 0.010925)^{78}}{(1 + 0.010925)^{78} - 1}
\]

5. **Calculating \( (1 + r_{monthly})^{n} \):**
\[
(1 + 0.010925)^{78} \approx 2.3570
\]

6. **Calculating monthly payment:**
\[
M = 44500 \cdot \frac{0.010925 \cdot 2.3570}{2.3570 - 1}
\]
\[
= 44500 \cdot \frac{0.025749625}{1.3570} \approx 44500 \cdot 0.018979 \approx 844.73
\]

7. **Total payment over the life of the loan:**
\[
\text{Total payment} = M \cdot n = 844.73 \cdot 78 \approx 65905.84
\]

8. **Total interest paid:**
\[
\text{Total interest} = \text{Total payment} - \text{Principal} = 65905.84 - 44500 = 21405.84
\]

Upon rounding to the nearest cent:

The total interest paid is approximately $21,405.84.

Looking at the options provided:
- A. $21,849.92
- B. $3,018.03
- C. $20,003.60
- D. $24,321.18

The closest option matching our calculation is **A: $21,849.92**.