Question: Moe and Brooke took their kids to Disney Land 9 years ago, but have only now finished paying off…

Moe and Brooke took their kids to Disney Land 9 years ago, but have only now finished paying off the credit card debt! How much interest did they pay in all, if they made $10.76 biweekly payments, and were charged 3.5% interest, compounded biweekly?

Solution

To solve this problem, we need to calculate the total interest paid on the credit card debt over 9 years with biweekly compounding. First, determine the total number of biweekly periods over 9 years: \[ \text{Total periods} = 9 \times 26 = 234 \] Next, convert the annual interest rate to a biweekly interest rate. The annual interest rate is \(3.5\%\), so: \[ \text{Biweekly interest rate} = \frac{3.5\%}{26} = \frac{0.035}{26} \approx 0.001346 \] Now, use the formula for the future value of an ordinary annuity to find the total amount paid, where \( P \) is the payment per period, \( i \) is the interest rate per period, and \( n \) is the number of periods: \[ FV = P \times \frac{(1+i)^n - 1}{i} \] Plug in the values: \[ FV = 10.76 \times \frac{(1+0.001346)^{234} - 1}{0.001346} \] Calculate the future value using these numbers: 1. Calculate \( (1+0.001346)^{234} \). \[ (1+0.001346)^{234} \approx 1.3609 \] 2. Substitute in the formula to get the future value. \[ FV = 10.76 \times \frac{1.3609 - 1}{0.001346} \] 3. Calculate: \[ FV \approx 10.76 \times \frac{0.3609}{0.001346} \approx 2882.65 \] The total payments made were $2882.65. Calculate the initial principal amount (\(PV\)) using the formula: \[ PV = \frac{FV}{(1+i)^n} \] Substitute the values: \[ PV = \frac{2882.65}{1.3609} \approx 2118.92 \] Now, calculate the interest paid by subtracting the principal from the total amount paid: \[ \text{Total interest paid} = 2882.65 - 2118.92 \approx 763.73 \] The total interest they paid is approximately $763.73.