Question

You owe $4,378.34 on a credit card at 13.5% APR. You pay $460.00 / month.
How many months does it take to pay off your balance

Answer 1

To calculate how long it will take to pay off your credit card balance, we can use the formula for the amortization of a loan. The formula to determine the number of payments (months) needed to pay off a balance is as follows:

\[
n = \frac{-\log(1 - \frac{rP}{A})}{\log(1 + r)}
\]

Where:
- \( n \) is the number of payments (months)
- \( P \) is the principal (the amount owed, $4,378.34)
- \( r \) is the monthly interest rate (annual rate / 12)
- \( A \) is the monthly payment ($460.00)

1. **Convert the annual interest rate to a monthly interest rate:**
\[
r = \frac{13.5\%}{12} = \frac{0.135}{12} = 0.01125
\]

2. **Plug the values into the formula:**
\[
n = \frac{-\log(1 - \frac{0.01125 \times 4378.34}{460})}{\log(1 + 0.01125)}
\]

3. **Calculate \( \frac{0.01125 \times 4378.34}{460} \):**
\[
\frac{0.01125 \times 4378.34}{460} \approx 0.1097
\]

4. **Calculate \( 1 - 0.1097 \):**
\[
1 - 0.1097 = 0.8903
\]

5. **Calculate \( \log(0.8903) \):**
\[
\log(0.8903) \approx -0.0510
\]

6. **Calculate \( \log(1 + 0.01125) \):**
\[
\log(1.01125) \approx 0.0049
\]

7. **Plug these values back into the formula:**
\[
n = \frac{-(-0.0510)}{0.0049} \approx \frac{0.0510}{0.0049} \approx 10.4
\]

Since \( n \) needs to be a whole number, we round up because you can't make a partial payment. Therefore, it takes approximately **11 months** to pay off the balance, assuming you make consistent payments of $460 each month.