karina bought a townhouse for $199,900. she has a 30 - year mortgage with a fixed rate of 5.5%. karinas…

karina bought a townhouse for $199,900. she has a 30 - year mortgage with a fixed rate of 5.5%. karinas monthly payments are $998.08. what percent of the purchase price was karinas down payment?\na. 7%\nb. 12%\nc. 15%\nd. 18%\nplease select the best answer from the choices provided

karina bought a townhouse for $199,900. she has a 30 - year mortgage with a fixed rate of 5.5%. karinas monthly payments are $998.08. what percent of the purchase price was karinas down payment?\na. 7%\nb. 12%\nc. 15%\nd. 18%\nplease select the best answer from the choices provided

Answer

Explanation:

Step1: Calculate the loan amount

First, find the total amount paid over 30 - year mortgage. There are $30\times12 = 360$ months. The monthly payment is $998.08$, so the total amount paid is $998.08\times360=$359308.8$. Let the loan - amount be $L$. Using the mortgage formula (or simply the total payment concept), we assume the loan amount is what is being paid off over time.

Step2: Calculate the down - payment amount

The purchase price of the townhouse is $P = 199900$. Let the down - payment be $D$. We know that $P=D + L$. We need to find $D$. Since we are not using the interest rate for this calculation (we can find the loan amount from the monthly payment and number of months), the loan amount $L$ (total amount paid over 30 years) is $998.08\times360 = 359308.8$ (this is wrong, we should use the present - value of the mortgage formula, but an alternative way is as follows). The loan amount $L$ can be calculated using the present - value of an ordinary annuity formula $PV = PMT\times\frac{1-(1 + r)^{-n}}{r}$, where $PMT = 998.08$, $r=\frac{0.055}{12}$, and $n = 360$. But a simpler way is to note that the loan amount $L$ is the present value of the monthly payments. We know that the purchase price of the house is $199900$. The loan amount $L$: The monthly payment $M = 998.08$, number of months $n = 360$, and annual interest rate $i=0.055$. Using the present - value of an ordinary annuity formula $PV = M\times\frac{1-(1+\frac{i}{12})^{-n}}{\frac{i}{12}}$. $PV=998.08\times\frac{1-(1 +\frac{0.055}{12})^{-360}}{\frac{0.055}{12}}\approx998.08\times\frac{1-(1+\ 0.0045833)^{-360}}{0.0045833}\approx998.08\times\frac{1 - 0.19566}{0.0045833}\approx998.08\times\frac{0.80434}{0.0045833}\approx998.08\times175.5$. $PV\approx174299$. The down - payment $D=199900 - 174299=25601$.

Step3: Calculate the percentage of the down - payment

The percentage of the down - payment $p=\frac{D}{P}\times100=\frac{25601}{199900}\times100\approx12.8% \approx12%$.

Answer:

B. 12%