Question

1. melissa wants to check the accuracy of the finance charge on her promissory note. she has a $6,000, 4-year loan at an apr of 3.11%. round to the nearest cent.

a. what is her monthly payment?
b. what is the total of all her monthly payments?
c. what is the finance charge?

Explanation:

Step1: Identify the loan details

Loan amount \( P = \$6000 \), APR \( r = 3.11\% = 0.0311 \), loan term \( t = 4 \) years. Since it's a monthly payment, the number of payments \( n = t \times 12 = 4 \times 12 = 48 \). The monthly interest rate \( i = \frac{r}{12} = \frac{0.0311}{12} \).

Step2: Calculate the monthly payment (a)

The formula for the monthly payment of a loan is \( M = P \times \frac{i(1 + i)^n}{(1 + i)^n - 1} \).

First, calculate \( i = \frac{0.0311}{12} \approx 0.0025917 \).

Then, \( (1 + i)^n = (1 + 0.0025917)^{48} \). Let's calculate \( (1.0025917)^{48} \approx e^{48 \times 0.0025917} \) (using the approximation \( e^x \) for small \( x \), but more accurately, using a calculator: \( 1.0025917^{48} \approx 1.1317 \)).

Now, \( \frac{i(1 + i)^n}{(1 + i)^n - 1} = \frac{0.0025917 \times 1.1317}{1.1317 - 1} = \frac{0.002933}{0.1317} \approx 0.02227 \).

Then, \( M = 6000 \times 0.02227 \approx \$133.62 \).

Step3: Calculate total monthly payments (b)

Total payments \( = M \times n = 133.62 \times 48 \approx \$6413.76 \).

Step4: Calculate finance charge (c)

Finance charge \( = \) Total payments \( - \) Principal \( = 6413.76 - 6000 = \$413.76 \).

Answer:

a. The monthly payment is approximately \(\$133.62\).
b. The total of all monthly payments is approximately \(\$6413.76\).
c. The finance charge is approximately \(\$413.76\).