- Identify the principal \(P = \$17,000\), annual rate \(r = 0.0588\), and duration \(t = 5\) years.
- Calculate the monthly payment: \(\text{PMT} = \frac{17,000(0.0049)}{1 - (1.0049)^{-60}} \approx \$327.71\).
- Calculate total payment: \(327.71 \times 60 = \$19,662.60\).
- Calculate total interest: \(\$19,662.60 - \$17,000 = \$2,662.60\).
- Final results: Monthly Payment = \(\boxed{\$327.71}\), Total Interest = \(\boxed{\$2,662.60}\).
Explanation
- Problem Analysis and Variable Identification
To determine the monthly payment and the total interest for Loan B, we identify the following variables:
- The principal amount \(P = \$17,000\).
- The annual interest rate \(r = 5.88\% = 0.0588\).
- The loan duration \(t = 5\) years.
- The compounding and payment frequency is monthly, so there are 12 payments per year.
- The number of monthly payments \(n = 12 \times 5 = 60\).
- The monthly interest rate \(i = \frac{r}{12} = \frac{0.0588}{12} = 0.0049\).
The formula used to calculate the monthly payment (PMT) for an amortized loan is:
\[\text{PMT} = \frac{P \cdot i}{1 - (1 + i)^{-n}}\]
- Calculating the Monthly Payment
Substituting our identified values into the PMT formula:
\[\text{PMT} = \frac{17,000 \times 0.0049}{1 - (1 + 0.0049)^{-60}}\]
First, we calculate the denominator:
\[(1 + 0.0049)^{-60} = (1.0049)^{-60} \approx 0.74542\]
\[1 - 0.74542 = 0.25458\]
Next, the numerator:
\[17,000 \times 0.0049 = 83.3\]
Finally, calculating the PMT:
\[\text{PMT} = \frac{83.3}{0.25458} \approx \$327.71\]
Rounding to the nearest cent, the monthly payment is \(\$327.71\).
- Calculating Total Payment and Total Interest
The total amount paid over the life of the loan is the monthly payment multiplied by the total number of payments:
\[\text{Total Payment} = 327.71 \times 60 = \$19,662.60\]
The total interest paid is the difference between the total payment and the original principal:
\[\text{Total Interest} = 19,662.60 - 17,000 = \$2,662.60\]
Thus, the monthly payment is \(\$327.71\) and the total interest paid for Loan B is \(\$2,662.60\).
Examples
Understanding how to calculate monthly loan payments is essential for personal financial planning. Whether you are financing a new car, taking out a student loan, or planning a home purchase, knowing how interest rates and loan terms affect your monthly budget helps you make informed borrowing decisions. By calculating these figures, you can compare different loan options to see which fits best within your financial goals.