Final answer:
To determine how long it will take Joel to pay off a $3500 laptop purchase on his credit card with a 4.95% interest rate and $65 monthly payments, a financial calculator or logarithmic method is used to find the number of months, which is then converted to years.
Step-by-step explanation:
To calculate how long it will take Joel to pay off his $3500 laptop purchase on a credit card at a 4.95% interest rate compounded monthly with monthly payments of $65, we can use the formula for the payment of an annuity with compound interest:
\[ P = \frac{R\left[1 - (1+i)^{-n}\right]}{i} \]
Where:
- P is the principal amount ($3500 in this case)
- R is the monthly payment ($65)
- i is the monthly interest rate (4.95% annual rate divided by 12 months)
- n is the number of months needed to pay off the loan
First, we convert the annual interest rate to a monthly rate:
\[ i = \frac{4.95\%}{12} \]
Next, we rearrange the formula to solve for n, the number of months. However, because the formula does not solve algebraically for n, we need either an iterative approach, like using a financial calculator or spreadsheet, or we can use the logarithmic method to find the number of payments. The formula using logarithms to solve for n is:
\[ n = \frac{\log(R) - \log(R - P × i)}{\log(1+i)} \]
After calculating n, we convert the number of months to years by dividing by 12, and then we round to the nearest tenth:
\[ years = \frac{n}{12} \]
Please note, that due to the complexity of this calculation, a financial calculator or spreadsheet is often used to find the precise number of payments.