you borrow $15,000 for five years at 8% interest to buy a car. what is your monthly payment?

Explanation:

Step1: Convert annual rate to monthly rate

The annual interest rate $r = 8%=0.08$, so the monthly interest rate $i=\frac{0.08}{12}$. $i=\frac{0.08}{12}\approx0.00667$

Step2: Determine number of periods

The loan is for 5 years. Since there are 12 months in a year, the number of periods $n = 5\times12=60$.

Step3: Use loan - payment formula

The formula for the monthly payment of a loan is $M = P\times\frac{i(1 + i)^n}{(1 + i)^n-1}$, where $P$ is the principal amount. Here, $P = 15000$. $M=15000\times\frac{0.00667\times(1 + 0.00667)^{60}}{(1 + 0.00667)^{60}-1}$ First, calculate $(1 + 0.00667)^{60}$. Let $x=(1 + 0.00667)^{60}$. Using the formula $a^b=e^{b\ln(a)}$, we have $\ln(x)=60\times\ln(1.00667)\approx60\times0.00665 = 0.399$, so $x = e^{0.399}\approx1.4903$. Then, $M=15000\times\frac{0.00667\times1.4903}{1.4903 - 1}$ $M=15000\times\frac{0.00667\times1.4903}{0.4903}$ $M=15000\times\frac{0.00994}{0.4903}$ $M=15000\times0.02027$ $M = 304.05$

Answer:

$304.05$