you want to buy a $500,000 home. you plan to ...
you want to buy a $500,000 home. you plan to pay 10% as a down payment and take out a 30 - year loan for the rest. do not round numbers while working on the problem. only round the final answer. since we are working with money, round to the cent. 1. how much will the loan be? 1. what will your monthly payments be if the interest rate is 5%? 1. what will your monthly payments be if the interest rate is 6%?
Answer
# Explanation:
## Step1: Calculate the loan amount
The home price is $500,000 and the down - payment is 10% of $500,000. The down - payment amount is $500,000\times0.1 = 50,000$. The loan amount $L$ is $500,000−50,000=450,000$.
## Step2: Calculate the monthly payment formula for a fixed - rate loan
The formula for the monthly payment $M$ of a fixed - rate loan is $M = P\frac{r(1 + r)^n}{(1 + r)^n-1}$, where $P$ is the principal loan amount, $r$ is the monthly interest rate, and $n$ is the total number of payments. For a 30 - year loan, $n = 30\times12=360$ months.
### Case 1: Interest rate is 5%
The annual interest rate $i = 0.05$, so the monthly interest rate $r=\frac{0.05}{12}$. The principal $P = 450,000$.
$M_1=450000\times\frac{\frac{0.05}{12}(1+\frac{0.05}{12})^{360}}{(1 + \frac{0.05}{12})^{360}-1}$
$M_1\approx2416.70$
### Case 2: Interest rate is 6%
The annual interest rate $i = 0.06$, so the monthly interest rate $r=\frac{0.06}{12}=0.005$. The principal $P = 450,000$.
$M_2=450000\times\frac{0.005(1 + 0.005)^{360}}{(1+0.005)^{360}-1}$
$M_2\approx2697.94$
# Answer:
1. $450,000.00
2. $2416.70
3. $2697.94