Question

determine the monthly payment of a loan for $3,000 at 7.5% interest compounded monthly for 36 months.
a. $93.32
b. $95.40
c. $211.33
d. $253.60

Explanation:

Step1: Recall the formula for monthly loan payment

The formula for the monthly payment \( M \) of a loan is given by:
\[ M = P \frac{\frac{r}{n}(1 + \frac{r}{n})^{nt}}{(1 + \frac{r}{n})^{nt}- 1} \]
where:

• \( P \) is the principal amount (loan amount) = \$3000
• \( r \) is the annual interest rate (in decimal) = \( 7.5\%=0.075 \)
• \( n \) is the number of times interest is compounded per year (monthly, so \( n = 12 \))
• \( t \) is the number of years. Since the loan is for 36 months, \( t=\frac{36}{12} = 3 \) years.

Step2: Calculate the monthly interest rate \( \frac{r}{n} \)

\[ \frac{r}{n}=\frac{0.075}{12}=0.00625 \]

Step3: Calculate \( (1 + \frac{r}{n})^{nt} \)

First, calculate \( nt=12\times3 = 36 \)
\[ (1 + 0.00625)^{36}=(1.00625)^{36} \]
Using a calculator, \( (1.00625)^{36}\approx1.251797 \)

Step4: Calculate the numerator and denominator

Numerator: \( \frac{r}{n}(1 + \frac{r}{n})^{nt}=0.00625\times1.251797\approx0.00782373 \)
Denominator: \( (1 + \frac{r}{n})^{nt}- 1=1.251797 - 1 = 0.251797 \)

Step5: Calculate the monthly payment \( M \)

\[ M = 3000\times\frac{0.00782373}{0.251797} \]
\[ M=3000\times0.03107 \] (approximate value after division)
\[ M\approx93.21 \] (close to \$93.32 due to more precise calculation of \( (1.00625)^{36} \))

Answer:

a. \$93.32