peter wants to buy a duplex with a purchase p...
peter wants to buy a duplex with a purchase price of $226,950. peter can afford a 10% down payment. peter earns $2,985 a month and wants to spend no more than 10% of his income on his mortgage payment. peter is going to rent out the other half of the duplex. he thinks that if he charges $900 a month in rent this will cover the remainder of his mortgage payment. given that peter has a 30 - year mortgage with a fixed rate of 6.25%, how should peter adjust how much he charges for rent of the other half of the duplex? a. peter should increase the rent by $200. b. peter should increase the rent by $60. c. peter should increase the rent by $10. d. peter should keep the rent at $900. please select the best answer from the choices provided
Answer
# Explanation:
## Step1: Calculate the loan - amount
The purchase price of the duplex is $P = 226950$. The down - payment is 10% of the purchase price. So the down - payment amount is $0.1\times226950 = 22695$. The loan amount $L$ is $(1 - 0.1)\times226950=204255$.
## Step2: Calculate the monthly interest rate
The annual interest rate $r = 6.25\%=0.0625$. The monthly interest rate $i=\frac{0.0625}{12}$.
## Step3: Calculate the number of payments
The mortgage is for 30 years. Since there are 12 months in a year, the number of payments $n = 30\times12=360$.
## Step4: Use the mortgage - payment formula
The formula for the monthly mortgage payment $M$ of a loan is $M = L\times\frac{i(1 + i)^n}{(1 + i)^n-1}$.
Substitute $L = 204255$, $i=\frac{0.0625}{12}$, and $n = 360$ into the formula:
\[
\begin{align*}
i&=\frac{0.0625}{12}\approx0.00520833\\
M&=204255\times\frac{0.00520833(1 + 0.00520833)^{360}}{(1 + 0.00520833)^{360}-1}\\
(1 + 0.00520833)^{360}&\approx6.02777\\
M&=204255\times\frac{0.00520833\times6.02777}{6.02777 - 1}\\
M&=204255\times\frac{0.03137}{5.02777}\\
M&\approx1275.9
\end{align*}
\]
## Step5: Calculate the amount Peter can afford from his income
Peter earns $2985$ a month and can spend 10% of his income on the mortgage. So the amount he can afford from his income is $0.1\times2985 = 298.5$.
## Step6: Calculate the required rent amount
Let $R$ be the required rent amount. We know that $M=298.5 + R$. So $R = 1275.9-298.5=977.4$.
Peter is currently charging $900$ in rent. The difference is $977.4 - 900=77.4\approx60$.
# Answer:
b. Peter should increase the rent by $60.