Use PMT = (P(r/n))/[1 - (1 + r/n) ^ (- n * l)] to determine the regular payment amount, rounded to the neares dollar. Your credit card has a balance of $1500 and an annual interest rate of 18%. You decide to pay off the balance over 2 years. If there are no further purchases charged to the card, how much must you pay each month (round to nearest dollar)? Showing your work for this question is required.

1 answer

To calculate the monthly payment (PMT) using the formula given, let's identify each of the variables in the formula:

  • \( P \) = principal balance (the amount owed)
  • \( r \) = annual interest rate as a decimal
  • \( n \) = number of payments per year (monthly payments means \( n = 12 \))
  • \( l \) = number of years to pay off the debt

Given:

  • \( P = 1500 \)
  • Annual interest rate = 18%, so \( r = 0.18 \)
  • Number of years \( l = 2 \)
  • Therefore, \( n = 12 \)

Step 1: Calculate the monthly interest rate

The monthly interest rate is given by:

\[ \text{monthly interest rate} = \frac{r}{n} = \frac{0.18}{12} = 0.015 \]

Step 2: Calculate the total number of payments

The total number of payments over 2 years is:

\[ \text{total payments} = n \times l = 12 \times 2 = 24 \]

Step 3: Plug values into the PMT formula

Now substituting these values into the PMT formula:

\[ PMT = \frac{P \times \left( \frac{r}{n} \right)}{1 - \left(1 + \frac{r}{n}\right)^{-n \cdot l}} \]

Substituting the values:

\[ PMT = \frac{1500 \times 0.015}{1 - (1 + 0.015)^{-24}} \]

Step 4: Calculate the denominator

First, calculate \( 1 + 0.015 \) and then raise it to the power of \(-24\):

\[ 1 + 0.015 = 1.015 \]

Now calculate \( (1.015)^{-24} \):

\[ (1.015)^{-24} \approx 0.6530 \]

Now, compute the complete denominator:

\[ 1 - (1.015)^{-24} \approx 1 - 0.6530 \approx 0.3470 \]

Step 5: Calculate PMT

Now substitute back into the equation:

\[ PMT = \frac{1500 \times 0.015}{0.3470} \approx \frac{22.5}{0.3470} \approx 64.8 \]

Step 6: Round to the nearest dollar

Finally, rounding to the nearest dollar gives:

\[ \text{Monthly Payment } PMT \approx 65 \]

Conclusion

You must pay approximately $65 each month to pay off a credit card balance of $1500 at an 18% annual interest rate over 2 years.