Adrian Clemons Adrian, a single man who wants to buy a house in five years, read an artide that recommended a down payment of 20 percent. With a large income and little debt, Adrian can afford to save a substantial amount of money every month. He is asking you for advice to help him reach his goal. on it is now five years later, and Adrian has saved enough money for a 20 percent down payment on a house. He will have to borrow $135,000 in a 30-year loan with an annual interest rate of 6 percent compounded monthly What will his monthly mortgage payment be? Select one:
A. $809.39
B. $779,98
C. $1.199.55
D. $397.50
The correct answer and explanation is :
To determine Adrian’s monthly mortgage payment, we will use the formula for the monthly payment of a fixed-rate mortgage loan. The formula is:
[
M = P \times \frac{r(1 + r)^n}{(1 + r)^n – 1}
]
Where:
- (M) is the monthly mortgage payment
- (P) is the principal loan amount (in this case, $135,000)
- (r) is the monthly interest rate (annual interest rate divided by 12)
- (n) is the total number of payments (loan term in years multiplied by 12)
Step-by-step Calculation:
- Principal Loan Amount (P):
Adrian needs to borrow $135,000 after making a 20% down payment. - Annual Interest Rate:
The annual interest rate is 6%. To convert this to a monthly interest rate, we divide by 12: [
r = \frac{6\%}{12} = \frac{0.06}{12} = 0.005
] - Total Number of Payments (n):
The loan term is 30 years, so the total number of monthly payments is: [
n = 30 \times 12 = 360 \text{ months}
] - Substitute into the formula: [
M = 135,000 \times \frac{0.005(1 + 0.005)^{360}}{(1 + 0.005)^{360} – 1}
] First, calculate the factors in the formula: [
(1 + 0.005)^{360} = 6.022575
] Now, substitute and simplify: [
M = 135,000 \times \frac{0.005 \times 6.022575}{6.022575 – 1}
] [
M = 135,000 \times \frac{0.030112875}{5.022575}
] [
M = 135,000 \times 0.005998
] [
M \approx 809.39
]
So, Adrian’s monthly mortgage payment is approximately $809.39.
Final Answer:
The correct answer is A. $809.39.
Explanation:
The mortgage payment is calculated using a standard mortgage formula for fixed-rate loans. This involves converting the interest rate into a monthly rate, determining the total number of payments (for a 30-year loan, it’s 360 months), and applying these to the formula. After calculating, we find that Adrian’s monthly payment is approximately $809.39. This is the amount he will need to pay each month to cover both principal and interest for the 30-year loan.