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.
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