your credit card statement shows the following:\nstatement\nnew balance $2,471.20\nminimum payment due…

your credit card statement shows the following:\nstatement\nnew balance $2,471.20\nminimum payment due $40.00\ndue date june 8\ninterest calculation\napr on purchases 19.24%\nyou have decided to pay $150 each month on this credit card, and will not make any new purchases.\nhow long will it take to pay off this card?\n____ months

your credit card statement shows the following:\nstatement\nnew balance $2,471.20\nminimum payment due $40.00\ndue date june 8\ninterest calculation\napr on purchases 19.24%\nyou have decided to pay $150 each month on this credit card, and will not make any new purchases.\nhow long will it take to pay off this card?\n____ months

Answer

Explanation:

Step1: Calculate monthly interest rate

First, convert the annual percentage rate (APR) to a monthly rate by dividing by 12 (number of months in a year) and converting the percentage to a decimal. [ r = \frac{0.1924}{12} \approx 0.016033 ]

Step2: Use loan payoff formula

We use the formula for the number of periods to pay off a loan: [ n = -\frac{\ln\left(1 - \frac{P \times r}{M}\right)}{\ln(1 + r)} ] Where:

  • (P = 2471.20) (initial balance),
  • (M = 150) (monthly payment),
  • (r \approx 0.016033) (monthly interest rate) First calculate the term inside the logarithm: [ 1 - \frac{2471.20 \times 0.016033}{150} \approx 1 - \frac{39.62}{150} \approx 1 - 0.2641 = 0.7359 ] Then calculate the logarithms: [ \ln(0.7359) \approx -0.306, \quad \ln(1 + 0.016033) \approx 0.01591 ] [ n \approx -\frac{-0.306}{0.01591} \approx 19.23 ] Since we cannot pay a fraction of a month, we round up to the next whole month.

Answer:

19 months (rounded up to the full month needed to pay off the balance)