Questions: Consider a student loan of 17,500 at a fixed APR of 6% for 20 years. a. Calculate the monthly payment. b. Determine the total amount paid over the term of the loan. c. Of the total amount paid, what percentage is paid toward the principal and what percentage is paid for interest. a. The monthly payment is . b. The total payment over the term of the loan is . c. Of the total payment over the term of the loan, % is paid toward the principal and % is paid toward interest.

Solution Steps

To solve this problem, we will use the formula for calculating the monthly payment on an amortizing loan, which is given by the formula for an annuity. The monthly payment can be calculated using the principal amount, the monthly interest rate, and the number of payments. Once we have the monthly payment, we can calculate the total amount paid over the term of the loan by multiplying the monthly payment by the number of payments. Finally, we can determine the percentage of the total payment that goes toward the principal and the interest by comparing the total payment to the original loan amount.

The monthly payment \( M \) for a loan can be calculated using the formula:

\[ M = \frac{P \cdot r}{1 - (1 + r)^{-n}} \]

where:

• \( P = 17500 \) (principal),
• \( r = \frac{0.06}{12} = 0.005 \) (monthly interest rate),
• \( n = 20 \times 12 = 240 \) (total number of payments).

Substituting the values, we find:

\[ M = \frac{17500 \cdot 0.005}{1 - (1 + 0.005)^{-240}} \approx 125.3754 \]

The total amount paid over the term of the loan \( T \) is given by:

\[ T = M \cdot n \]

Substituting the values:

\[ T = 125.3754 \cdot 240 \approx 30090.1045 \]

The percentage of the total payment that goes toward the principal \( P_{\%} \) and interest \( I_{\%} \) can be calculated as follows:

\[ P_{\%} = \frac{P}{T} \cdot 100 \quad \text{and} \quad I_{\%} = 100 - P_{\%} \]

Substituting the values:

\[ P_{\%} = \frac{17500}{30090.1045} \cdot 100 \approx 58.1587 \] \[ I_{\%} = 100 - 58.1587 \approx 41.8413 \]

Final Answer