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)
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Convert the annual interest rate to a monthly interest rate: \[ r = \frac{13.5%}{12} = \frac{0.135}{12} = 0.01125 \]
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Plug the values into the formula: \[ n = \frac{-\log(1 - \frac{0.01125 \times 4378.34}{460})}{\log(1 + 0.01125)} \]
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Calculate \( \frac{0.01125 \times 4378.34}{460} \): \[ \frac{0.01125 \times 4378.34}{460} \approx 0.1097 \]
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Calculate \( 1 - 0.1097 \): \[ 1 - 0.1097 = 0.8903 \]
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Calculate \( \log(0.8903) \): \[ \log(0.8903) \approx -0.0510 \]
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Calculate \( \log(1 + 0.01125) \): \[ \log(1.01125) \approx 0.0049 \]
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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.