Answered

Monthly payments may be calculated using this formula: monthly payment $=P \times \frac{r(1+r)^n}{(1+r)^n-1}$.

A student takes out a series of loans totaling $\$64,000$ with an annual interest rate of $4\%$. After graduation, he plans to pay off the loans over 8 years.



What will be his approximate monthly payment?


A\$667.00
B\$693.00
C\$780.00
D\$975.00

Verified Answer :

  • Identify the principal \(P = 64,000\), monthly rate \(r = \frac{0.04}{12}\), and total payments \(n = 96\).
  • Apply the monthly payment formula: \(M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}\).
  • Perform the calculation to find the monthly payment \(\approx \$780.11\).
  • The closest answer from the given choices is \(\boxed{\$780.00}\).

Explanation

  1. Analysis of Loan Parameters

The loan principal \(P\) is \(\$64,000\). The annual interest rate is \(4\%\), which as a decimal is \(0.04\). The student plans to pay off the loan over 8 years. To find the monthly payment, we first convert the annual interest rate into a monthly interest rate \(r\) by dividing by 12, so \(r = \frac{0.04}{12} \approx 0.003333\). We then determine the total number of monthly payments \(n\) over the 8-year term, which is \(n = 8 \times 12 = 96\).

  1. Calculating the Monthly Payment

The formula to calculate the monthly payment \(M\) is:
\[M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}\]
Substituting the known values \(P = 64000\), \(r = \frac{0.04}{12}\), and \(n = 96\) into the formula, we calculate the payment:
\[M = 64000 \times \frac{\frac{0.04}{12}(1+\frac{0.04}{12})^{96}}{(1+\frac{0.04}{12})^{96} - 1}\]
Using the parameters, the calculation yields:
\[M \approx 64000 \times \frac{0.003333 \times (1.003333)^{96}}{(1.003333)^{96} - 1} \approx 64000 \times \frac{0.003333 \times 1.3754}{1.3754 - 1} \approx 64000 \times \frac{0.0045846}{0.3754} \approx 780.11\]
Thus, the approximate monthly payment is \(\$780.11\). Comparing this result to the given options, the closest match is \(\$780.00\).

Examples

Understanding how to calculate monthly loan payments is a fundamental skill in personal finance. For instance, if you are planning to take out a student loan, a car loan, or a mortgage, this formula helps you determine how much money you need to set aside each month to manage your debt effectively. By knowing your monthly obligations beforehand, you can better plan your budget and ensure you can comfortably afford the loan repayments.

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