Question

for a student loan with a principal of $15,000, an annual interest rate of 6%, and a repayment term of 15 years, what is the monthly payment according to $m = \frac{p \cdot \frac{r}{12} \cdot (1 + \frac{r}{12})^{12n}}{(1 + \frac{r}{12})^{12n} - 1}$? $150.00 $126.00 $126.57 $175.00

Explanation:

Step1: Identify given values

Principal \( P = 15000 \), annual interest rate \( r = 6\% = 0.06 \), number of years \( t = 15 \), number of months per year \( n = 12 \).

Step2: Calculate monthly interest rate

Monthly rate \( \frac{r}{n} = \frac{0.06}{12} = 0.005 \).

Step3: Calculate total number of payments

Total payments \( nt = 12\times15 = 180 \).

Step4: Substitute into the formula

Formula: \( M = \frac{P \cdot \frac{r}{n} \cdot (1 + \frac{r}{n})^{nt}}{(1 + \frac{r}{n})^{nt} - 1} \)
Substitute values:
\( M = \frac{15000 \cdot 0.005 \cdot (1 + 0.005)^{180}}{(1 + 0.005)^{180} - 1} \)

First, calculate \( (1 + 0.005)^{180} \). Using a calculator, \( (1.005)^{180} \approx 2.45409 \).

Then, numerator: \( 15000 \cdot 0.005 \cdot 2.45409 = 75 \cdot 2.45409 \approx 184.05675 \)

Denominator: \( 2.45409 - 1 = 1.45409 \)

Finally, \( M = \frac{184.05675}{1.45409} \approx 126.57 \)

Answer:

\$126.57