Q:

# Kevin has a credit card debt of $2544 with an 18.75% interest rate, compounded daily. To pay it off, he plans to make monthly payments of$200. (6) a) How many payments will it take to pay off? b) What will be his last payment?

Accepted Solution

A:
To calculate the number of payments it will take to pay off the credit card debt and the amount of the last payment, we can use the formula for the future value of an annuity. a) Number of Payments: The formula for the future value of an annuity is given by: FV = P * [(1 + r)^n - 1] / r Where: FV = Future Value (total debt) P = Payment amount ($200) r = Interest rate per period (daily compounded) n = Number of periods (number of payments) We need to solve for n, the number of payments. In this case, the future value (FV) is the total debt of$2544. $2544 =$200 * [(1 + 0.1875/365)^n - 1] / (0.1875/365) To solve for n, we can rearrange the formula and solve for n: [(1 + 0.1875/365)^n - 1] / (0.1875/365) = $2544 /$200 Simplifying the equation: [(1 + 0.0005137)^n - 1] / 0.0005137 = 12.72 Now we can solve this equation for n using logarithms: (1 + 0.0005137)^n - 1 = 12.72 * 0.0005137 (1 + 0.0005137)^n = 1 + (12.72 * 0.0005137) n = log((1 + 0.0005137)/(1 + (12.72 * 0.0005137))) / log(1 + 0.0005137) Using a calculator, we find that n β 16.29. Therefore, it will take approximately 17 payments to pay off the credit card debt. b) Last Payment: To find the amount of the last payment, we can subtract the total amount paid in the previous payments from the total debt. Amount of the last payment = Total debt - (Number of payments - 1) * Payment amount Amount of the last payment = $2544 - (17 - 1) *$200 Amount of the last payment = $2544 - 16 *$200 Amount of the last payment = $2544 -$3200 Amount of the last payment = -\$656 The negative value indicates that there is no last payment since the debt is fully paid off after the 17th payment.