Question

alexa is buying a car and needs to take out a loan for $23,000. the bank is offering an annual interest rate of 6.3%, compounded monthly, for a 5 - year loan. using the formula below, determine her monthly payment, to the nearest dollar.

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

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

Explanation:

Step1: Calculate the monthly interest rate

The annual interest rate is 6.3% or 0.063 in decimal form. So, $r=\frac{0.063}{12}= 0.00525$.

Step2: Calculate the number of payments

The loan is for 5 years, and since payments are monthly, $n = 5\times12=60$.

Step3: Identify the principal amount

The amount borrowed $P = 23000$.

Step4: Substitute values into the formula

$M=\frac{23000\times0.00525\times(1 + 0.00525)^{60}}{(1 + 0.00525)^{60}-1}$.
First, calculate $(1 + 0.00525)^{60}$. Let $x=(1 + 0.00525)^{60}$. Using the formula for compound - interest $a(1 + r)^n$, we have $x\approx1.376894$.
Then, $23000\times0.00525 = 120.75$.
The numerator is $120.75\times1.376894\approx166.26$.
The denominator is $1.376894 - 1=0.376894$.
$M=\frac{166.26}{0.376894}\approx441$.

Answer:

441