- Identify the principal \(P = \$10,000\), interest rate \(r = 0.05\), and the number of months \(n = 36\).
- Determine the monthly interest rate \(i = \frac{0.05}{12}\).
- Use the formula \(M = \frac{P \cdot i \cdot (1 + i)^n}{(1 + i)^n - 1}\) to compute the monthly payment.
- The result of the calculation is approximately \(\$299.71\), which corresponds to option (b).
- The final answer is \(\boxed{\$299.71}\).
Explanation
- Understanding the Problem and Data Analysis
We are given a student loan of \(P = \$10,000\), an annual interest rate of \(r = 5\% = 0.05\), and a repayment period of \(3\) years, which is equivalent to \(n = 3 \times 12 = 36\) monthly installments. Our goal is to find the monthly payment \(M\) using the given formula:
\[M = \frac{P \cdot i \cdot (1 + i)^n}{(1 + i)^n - 1}\]
where \(i = \frac{r}{12}\) is the monthly interest rate.
- Calculating the Monthly Repayment Amount
First, we calculate the monthly interest rate \(i\):
\[i = \frac{r}{12} = \frac{0.05}{12} \approx 0.00416667\]
Next, we substitute the values \(P = 10000\), \(i = \frac{0.05}{12}\), and \(n = 36\) into the formula:
\[M = \frac{10000 \cdot \left(\frac{0.05}{12}\right) \cdot \left(1 + \frac{0.05}{12}\right)^{36}}{\left(1 + \frac{0.05}{12}\right)^{36} - 1}\]
Using the calculation, the monthly payment \(M\) evaluates to approximately:
\[M \approx \frac{10000 \cdot 0.00416667 \cdot (1.00416667)^{36}}{(1.00416667)^{36} - 1} \approx \frac{41.6667 \cdot 1.16147}{1.16147 - 1} \approx \frac{48.3946}{0.16147} \approx 299.71\]
Thus, the monthly repayment amount is approximately \(\$299.71\).
- Conclusion
Comparing our result of \(\$299.71\) with the provided options:
A\(\$319.21\)
B\(\$299.71\)
C\(\$200.00\)
D\(\$150.00\)
The calculated monthly repayment amount matches option (b). Therefore, the correct answer is \(\boxed{\$299.71}\).
Examples
When taking out a loan, understanding how your monthly payments are calculated is crucial for financial planning. By using the amortization formula, you can determine how interest affects your total payment and how long it will take to pay off the debt, helping you budget effectively for future expenses. This is essential for managing personal finances, such as planning student loan repayments or estimating mortgage costs.