vanessa bought a house for $268,500. she has a 30 - year mortgage with a fixed rate of 6.25%. vanessas…

Explanation:

Step1: Calculate the total mortgage amount

The mortgage is for 30 years. There are 30 * 12 = 360 months. The monthly payment is $1,595.85. So the total amount paid over 30 years is $1,595.85×360 = $574,506.

Step2: Calculate the down - payment

The price of the house is $268,500. Let the down - payment be $x$. Then the mortgage amount is the price of the house minus the down - payment. But we know from the monthly payments the total amount paid on the mortgage. However, we can also use the loan - amount formula concept in reverse. The total amount paid on the mortgage is the loan amount (price of house - down payment) plus the interest. But a simpler way is to note that the loan amount $L$ paid in monthly installments of $M = 1595.85$ for $n=360$ months. The loan amount $L = M\times n=1595.85\times360$. The down - payment $D$ is the price of the house $P$ minus the loan amount $L$. First, find the loan amount $L = 1595.85\times360=574506$ (this is wrong approach as we should use present - value of annuity formula, but we can also work backward). The correct way: The present - value of an ordinary annuity formula for the loan amount $A$ is $A = M\times\frac{1-(1 + r)^{-n}}{r}$, where $M$ is the monthly payment, $r=\frac{0.0625}{12}$ is the monthly interest rate and $n = 360$ months. But we can also work as follows: The total amount paid over 30 years in monthly payments is $1595.85\times360 = 574506$. The loan amount (the amount she borrowed) is what she will pay back in present - value terms. Let's assume we work with the fact that the price of the house is $P = 268500$. The loan amount $L$ paid in monthly installments of $M = 1595.85$ for $n = 360$ months. The loan amount $L$ (using the present - value of annuity formula $PV = M\times\frac{1-(1+\frac{i}{12})^{-12t}}{\frac{i}{12}}$, where $i = 0.0625$ and $t = 30$) or simply by multiplying monthly payment by number of months (a simplification). The loan amount $L=1595.85\times360 = 574506$ (this is wrong as it doesn't account for interest properly. The correct present - value of annuity calculation: $r=\frac{0.0625}{12}$, $n = 360$, $M = 1595.85$, $PV=\ 1595.85\times\frac{1-(1+\frac{0.0625}{12})^{-360}}{\frac{0.0625}{12}}\approx228225$. The down - payment $D=268500 - 228225=40275$.

Answer:

C. $40,275.00$