Find the monthly payment for a $30$ year loan of $100000$ with an APR of $4\%$ ?

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Q. Find the monthly payment for a

30

year loan of

100000

with an APR of

4\%

Convert APR to monthly rate: Convert the annual percentage rate (APR) to a monthly interest rate. $\newline$ The APR is given as $4\%$ , which needs to be converted to a monthly rate since we are calculating a monthly payment. To do this, divide the APR by $12$ (the number of months in a year). $\newline$ Monthly interest rate = APR / $12$ $\newline$ = $4\%$ / $12$ $\newline$ = $0.04$ / $12$ $\newline$ = $0.003333\ldots$
Convert rate to decimal: Convert the monthly interest rate to a decimal. $\newline$ To use the monthly interest rate in calculations, it needs to be in decimal form rather than a percentage. $\newline$ Monthly interest rate (in decimal) = $0.003333\ldots$
Determine number of payments: Determine the number of monthly payments over the life of the loan. Since the loan is for $30$ years and payments are monthly, multiply the number of years by $12$ . Number of monthly payments = $30$ years $\times 12$ months/year $= 360$ payments
Use payment formula: Use the formula for calculating the monthly payment on an installment loan. $\newline$ The formula is: $\newline$ Monthly payment = $P \times \left(\frac{r(1+r)^n}{((1+r)^n - 1)}\right)$ $\newline$ Where: $\newline$ $P$ = principal amount (the initial amount of the loan) $\newline$ $r$ = monthly interest rate (in decimal form) $\newline$ $n$ = total number of payments
Plug values into formula: Plug the values into the formula to calculate the monthly payment. $\newline$ $P = \$100,000$ $\newline$ $r = 0.003333...$ $\newline$ $n = 360$ $\newline$ Monthly payment = $\$100,000 \times (0.003333...(1+0.003333...)^{360}) / ((1+0.003333...)^{360} - 1)$
Calculate monthly payment: Calculate the monthly payment using the values provided. $\newline$ This step involves complex calculations that are typically done with a calculator or financial software. For the sake of this example, we will simplify the calculation process. $\newline$ Monthly payment $\approx$ $\$100,000 \times (0.003333... \times (1.003333...)^{360}) / ((1.003333...)^{360} - 1)$
Perform final calculation: Perform the calculation to find the monthly payment. Using a calculator or financial software to handle the exponentiation and division, we find: Monthly payment $\approx$ $\$477.42$