Question

kylie is looking to take out a mortgage for $490,000 from a bank offering a monthly interest rate of 0.375%. using the formula below, determine her monthly payment, to the nearest dollar, if the loan is taken over 20 years.

$m = \frac{pr(1 + r)^n}{(1 + r)^n - 1}$

$m=$ the monthly payment
$p=$ the amount borrowed
$r=$ the interest rate per month
$n=$ the number of payments

Explanation:

Step1: Identify the values of P, r, and n

$P = 490000$, $r=0.00375$ (since $0.375\%= 0.00375$), and $n = 20\times12=240$ (20 - year loan with monthly payments).

Step2: Substitute values into the formula

$M=\frac{490000\times0.00375\times(1 + 0.00375)^{240}}{(1 + 0.00375)^{240}-1}$
First, calculate $(1 + 0.00375)^{240}$. Let $x=(1 + 0.00375)^{240}$. Using the formula $a^b=e^{b\ln(a)}$, we have $\ln(x)=240\times\ln(1.00375)\approx240\times0.003743 = 0.89832$. So $x = e^{0.89832}\approx2.4566$.
Then, $490000\times0.00375 = 1837.5$.
The numerator is $1837.5\times2.4566\approx4514.8$.
The denominator is $2.4566 - 1=1.4566$.
$M=\frac{4514.8}{1.4566}\approx3100$.

Answer:

$3100$