Question

1. peter and john just took out a 30-year amortized loan with monthly payments at 7.1% interest for $423,000 to buy a new home.

i. what will be their monthly payment for this loan? round your answer to the nearest cent. your answer must be accurate before rounding to the nearest cent. show all work.

Explanation:

Step1: Identify loan formula variables

We use the amortized loan payment formula: \( M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \), where:

• \( P = 423000 \) (principal),
• \( r = \frac{0.071}{12} \) (monthly interest rate),
• \( n = 30 \times 12 = 360 \) (total monthly payments).

Step2: Calculate monthly interest rate

\( r = \frac{0.071}{12} \approx 0.00591667 \).

Step3: Calculate \( (1 + r)^n \)

\( (1 + 0.00591667)^{360} \approx (1.00591667)^{360} \approx 8.14447 \).

Step4: Plug into payment formula

\( M = 423000 \times \frac{0.00591667 \times 8.14447}{8.14447 - 1} \)
First, numerator: \( 0.00591667 \times 8.14447 \approx 0.04819 \)
Denominator: \( 8.14447 - 1 = 7.14447 \)
Then, \( M = 423000 \times \frac{0.04819}{7.14447} \approx 423000 \times 0.006745 \approx 2853.14 \).

Answer:

The monthly payment is approximately \(\$2,853.14\).