Question

Phoebe realizes that she has charged too much on her credit card and has racked up $5,800 in debt if she can pay $250 each month and the card charges 12 percent APR (compounded monthly), how long will it take her to pay off the debt? (Do not round intermediate calculations and round your final answer to 2 decimal places.)

Answer 1

it will take Phoebe approximately 27.12 months to pay off her debt. Rounded to two decimal places, this is approximately 27.12 months.

To calculate how long it will take Phoebe to pay off her credit card debt of $5,800 with a monthly payment of $250 and a 12 percent APR compounded monthly, follow these steps:

Step 1: Convert the annual interest rate to a monthly interest rate:

Monthly Interest Rate (r) = Annual Interest Rate / 12

r = 12% / 12 = 1% per month (expressed as a decimal, 0.01)

Step 2: Set up the formula for the monthly payment required to pay off the debt:

`\[PMT = (P \cdot r)/(1 - (1 + r)^(-n))\]`

Where:

PMT = Monthly payment ($250)

P = Principal amount (initial debt, $5,800)

r = Monthly interest rate (0.01)

n = Number of months

Step 3: Solve for n:

`\[250 = (5800 \cdot 0.01)/(1 - (1 + 0.01)^(-n))\]`

Step 4: Simplify and solve for n:

`\[(1 + 0.01)^(-n) = 1 - (5800 \cdot 0.01)/(250)\]`

`\[(1.01)^(-n) = 1 - 0.232\]`

Now, take the natural logarithm (ln) of both sides:

`\[-n \cdot ln(1.01) = ln(1 - 0.232)\]`

Step 5: Solve for n:

`\[n = -(ln(1 - 0.232))/(ln(1.01))\]`

Step 6: Calculate n using a calculator:

`\[n \approx 27.12\]`

So, it will take Phoebe approximately 27.12 months to pay off her debt. Rounded to two decimal places, this is approximately 27.12 months.