Question

you can afford a $450.00 per month car payment. you can get a loan at 8% interest for 5 years.
a. how expensive of a car can you afford?
i can afford a car that costs blank.
b. how much total money will you pay the loan company?
the total of all my payments will be blank.
c. how much of that money is interest?
the amount of that money that is interest will be blank.
question help: video 1 video 2

Explanation:

Step1: Identify the loan type and formula

This is a fixed - rate loan, and we can use the present value of an annuity formula to find the loan amount (the price of the car we can afford). The formula for the present value of an annuity is $PV = PMT\times\frac{1-(1 + r)^{-n}}{r}$, where $PV$ is the present value (loan amount), $PMT$ is the payment per period, $r$ is the interest rate per period, and $n$ is the number of periods.

First, we need to convert the annual interest rate to a monthly rate and the number of years to the number of months. The annual interest rate $i = 8\%=0.08$, so the monthly interest rate $r=\frac{0.08}{12}$. The number of years $t = 5$, so the number of months $n=5\times12 = 60$. The monthly payment $PMT=\$450$.

Step2: Calculate the present value (loan amount for part A)

Substitute the values into the formula:
$r=\frac{0.08}{12}\approx0.006667$, $n = 60$, $PMT = 450$

$PV=450\times\frac{1-(1 + 0.006667)^{-60}}{0.006667}$

First, calculate $(1 + 0.006667)^{-60}=(1.006667)^{-60}\approx0.671210$

Then, $1-(1.006667)^{-60}=1 - 0.671210 = 0.32879$

Then, $\frac{0.32879}{0.006667}\approx49.316$

Then, $PV = 450\times49.316\approx22192.2$

Step3: Calculate the total payment for part B

The total payment is the monthly payment times the number of months. So, Total Payment $=PMT\times n=450\times60 = 27000$

Step4: Calculate the interest for part C

The interest is the total payment minus the loan amount. So, Interest $=$ Total Payment $-PV=27000 - 22192.2=4807.8$

Answer:

A. I can afford a car that costs $\$22192.2$ (approx).
B. The total of all my payments will be $\$27000$.
C. The amount of that money that is Interest will be $\$4807.8$ (approx).