How to calculate savings over the life of a car loan?

Question

I'm working through the maths in this, only the relevant parts of which I quote:

...On a \$25,000 car loan through the manufacturer for four years, your monthly payment would be about [1.] \$520 at 0% interest or [2.] \$541 with a 1.9% interest rate.

$1. = 25000/48 \text{ months } = \$520$.

I think that I need this formula : $\text{Monthly Payment} = PV \times \dfrac{R}{[1 - (1 + R)^{-n}]} \qquad (♦) $
where PV = Present Value (beginning value or amount of loan),
R = Periodic Interest Rate
n = # of interest periods for overall time period

$2. $ $PV = 25,000, \quad R = 1.9\%/12 = 0.019/12, \quad n = 48 \text{ months}$.

If you opted for the manufacturer rebate and a five-year loan term through a bank or credit union, you'd spend more on interest but your monthly payment would be substantially lower than the four-year manufacturer loan. For example, with a \$2,500 manufacturer's rebate, you'd lower your financed amount to \$22,500. [3.] At a $5\%$ interest rate, your monthly payment would be \$424, [4.] while at a $4\%$ interest rate your monthly payment would be about \$414.

$3.$ Just use the formula $(♦)$. $PV = 22,500, \quad R = 5\%/12, \quad n = 60 \text{ months}$.
$4.$ Same as $3$, but $R = 4\%/12$.

...On a \$25,000 car with a choice of a \$2,500 rebate or 1.9% financing over five years, it's a better deal to take the manufacturer rebate and get financing elsewhere. At a 4 percent interest rate with the \$2,500 rebate, you'd save $\color{darkred}{ [5.] \; \$1,364 }$ in total payments. With a 5 percent loan rate with the \$2,500 rebate, you'd still save $\color{darkred}{ [6.] \; \$750 }$ over the life of the loan.

$I.$ Would someone please explain how to calculate the red $\color{darkred}{5, 6}$?
$II.$ Based only on the given info, how would you know that to calculate $R$, the given interest rates must be divided by 12?

Answer 1

I: It looks like they get 5 and 6 by taking the cash rebate from the price of the car and calculating the interest on that lowered value (\$22,500 instead of $25,000). Not sure why you couldn't just put it towards the price of the car at the dealership and get the 1.9%, but oh well.

In other words, calculate 1.9% for 5 years on \$25,000, compare to 4% for 5 years on \$22,500 and 5% for 5 years on $22,500 and you should get those values in 5 and 6.

II: You divide the interest by 12 since it is a monthly payment and there are 12 months in the year. The only assumptions may be that it is an annual interest rate (which is pretty standard). As an example the payments were quarterly you would instead divide by 4. So you divide the interest rate by the number of payments in a year (would have been nice if they gave you that information).

Hope this helps.