Question 1192524: I don't know if I subtract out the down payment or what to do with it. I've used so many different formulas, I've confused myself. Here's the problem: This is exactly how it is presented:
For the car loan described, give the following information.
A car dealer will sell you a used car for $6,998 with $798 down and payments of $166.51 per month for 48 months.
(a) amount to be paid:
$
(b) amount of interest:
$
(c) interest rate (Round your answer to two decimal places.)
%
(d) APR (rounded to the nearest tenth of a percent)
%
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Part (a)
You pay the car dealer $798 up front in the down payment.
Then you pay an additional $166.51 per month for 48 months (aka 4 years).
Meaning you pay an additional 166.51*48 = 7,992.48 additional dollars on top of the original $798.
In total, you'll pay 798+7,992.48 = 8,790.48 dollars.
Answer: $8,790.48
Delete the dollar sign and/or comma if needed.
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Part (b)
interest = [loan amount paid back] - [original loan amount given to customer]
interest = [ (number of months)*(monthly payment) ] - [ (car value) - (down payment) ]
interest = (48)*(166.51) - [ (6,998) - (798) ]
interest = (7,992.48) - (6,200)
interest = 1,792.48
Notes: - The concept of "total interest" is sometimes referred to as "total finance charge".
- The down payment is NOT part of the loan. The loan is purely to pay off what the down payment does not cover. As far as the bank is concerned, the car's value is $6200 because this is the amount loaned.
- The total interest is found by subtracting the amount paid back and the amount financed.
Answer: $1,792.48
Delete the dollar sign and/or comma if needed.
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Part (c)
i = 1792.48 = interest result of the previous part (b)
P = 6200 = loan amount
t = 4 = number of years
So,
i = P*r*t
1792.48 = 6200*r*4
1792.48 = 24800r
r = (1792.48)/(24,800)
r = 0.072277 approximately
r = 0.0723
This converts to 7.23%
Answer: 7.23%
Delete the percent sign if needed.
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Part (d)
The monthly payment formula is
P = (L*i)/( 1-(1+i)^(-n) )
where,
P = monthly payment
L = loan amount
i = interest rate per month
n = number of months
In this case we have,
P = 166.51
L = 6200
i = unknown and what we want to solve for.
n = 48
Plugging in the known values gets us this equation
166.51 = (6200*i)/( 1-(1+i)^(-48) )
Unfortunately, we cannot solve for i by hand since it's buried under the exponent of -48, and we have another 'i' term outside of that said exponent.
We'll have to use a graphing calculator to effectively solve the equation
166.51 = (6200*x)/( 1-(1+x)^(-48) )
which turns into
166.51 - (6200*x)/( 1-(1+x)^(-48) ) = 0
Let
f(x) = 166.51 - (6200*x)/( 1-(1+x)^(-48) )
Use your graphing calculator to find the root of f(x) which is roughly
x = 0.010882
This is the approximate location where the function curve crosses the x axis.
Keep in mind that 0 < x < 1 is the interval to focus on when it concerns interest rates in decimal form.
This shows that the monthly interest rate is approximately
i = 0.010882
As a check,
P = (L*i)/( 1-(1+i)^(-n) )
P = (6200*0.010882)/( 1-(1+0.010882)^(-48) )
P = 166.510240227634
We don't land exactly on 166.51 but we're fairly close.
The reason for this is due to rounding error.
So the annual rate is
r = 12*i = 12*0.010882 = 0.130584
which converts to 13.0584%
Rounding to one decimal place leads to 13.1%
As a general rule, the APR is usually higher than the interest rate because it involves other (hidden) fees that aren't necessarily advertised on the sticker price.
It's not entirely hidden as such details are buried in the fine print.
If the car dealership is going to advertise one percentage only, then they'll likely go for the lower percentage.
More info found here:
https://www.consumerfinance.gov/ask-cfpb/what-is-the-difference-between-an-interest-rate-and-the-annual-percentage-rate-apr-in-an-auto-loan-en-733/
https://www.rategenius.com/car-loan-interest-rate-vs-apr
Admittedly, it would be nice if convention was to stick to a single percentage value only to make things more simple.
The unfortunate part of the financial world is that the math is purposefully made more complicated than it has to be, with the goal of confusing customers and therefore charging more money than advertised or initially thought.
In my opinion, the practice is shady.
Here's a nice calculator I often use for car loan math problems
https://www.cars.com/car-loan-calculator/
Type in the following:- Car price = 6998
- Down payment = 798
- Trade in value = 0
- Length = 48 months
- APR = 13.06% (from rounding 13.0584% to two decimal places)
- Sales Tax = 0% (yes its not realistic if your state has sales tax but we'll just go with it for theoretical purposes)
The calculator will spit out a monthly payment of $167 which isn't too far from the $166.51 per month figure mentioned initially.
This helps confirm we have the correct APR value.
Answer: 13.1%
Delete the percent sign if needed.
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