*When you’re looking at your personal finances, the big expenses stand out. That means purchasing a car is a huge consideration, and deciding between new and used can make your mind turn to mush. Do dealer and automaker incentives–like free financing or cash back–make a big difference? Sometimes yes, sometimes no.*

*Today, I’m bringing you an excerpt from my book, *Math for Grownups*. Use this math, and you can make an educated vehicular purchase, speedy quick.*

Used cars are generally less expensive than new ones, unless you’re deciding between a pre-owned Hummer and a brand new Hyundai, of course.

But how do dealer and automaker incentives stack up to buying used?

Check it out!

Roxanne is trying to decide between two cars. Her local dealership has a current model priced at $25,000, including tax. But online she saw the same car—pre-owned—for $15,000. The used car is in excellent condition and certified. Plus, the warranty transfers, so price is her only real consideration.

The dealership is offering free financing. And the automaker has a $2,000 cash-back program. That means she’ll pay exactly $23,000 for the car and no interest at all.

But to finance the used car, she’ll have to get a loan. To compare the prices, she’ll need to find out how much she’ll pay *in all* for the used car. That means she needs to know what interest on a loan will cost.

In order to calculate that, she’ll need to know the principal (the amount she is borrowing and the basis of the interest calculation). That means the principal is $15,000. She’ll also need to know the interest rate. Her bank is offering a 6% interest rate on car loans, for a period of 4 years. The interest is compounded annually, so once a year, the interest rate is calculated and added to the loan amount. Thus compounding interest means that in every year for the term of the loan, except the first year, Roxanne is paying interest on the interest she paid the year before (and the year before that . . . and you get the idea).

Roxanne can use an online calculator, or she can turn to a really simple formula:

*A* = *P*(1 + *r*)^{n}

Okay, breathe. It only *looks* hard. It’s not difficult at all if you remember the order of operations—that is, what you do first, then second, and so on.

First, do anything inside the parentheses. Next, take care of exponents—those are the little numbers at the right top of another number. They tell how often to multiply the bigger number by itself. (Thus 4^{2}means 4 4, and 16^{5} means 16 16 16 16 16.) Then multiply or divide. And finally, add or subtract. In other words, Please Excuse My Dear Aunt Sally, or PEMDAS:

**P**arentheses**E**xponents**M**ultiplication**D**ivision**A**ddition**S**ubtraction

Ready to apply this formula? With PEMDAS, you can do it!

*A* is the total amount she’ll owe

*P* is the principal

*r* is the interest rate per compounding period

*n* is the number of compounding periods

Roxanne’s principal (or the amount she’s borrowing) is $15,000, so *P* = $15,000. Her interest compounds yearly, so her rate is 6%. To make it easier to multiply, she can convert that percent to a decimal: *r *= 6% = 0.06. And because the compounding period is annual, and the length of the loan is 4 years, *n* = 4.

*A* = $15,000(1 + 0.06)^{4}

First add the numbers inside the parentheses.

*A* = $15,000(1.06)^{4}

Now calculate the exponent. Remember, 1.06^{4} = 1.06 • 1.06 • 1.06 • 1.06.

*A* = $15,000(1.26)

Last step! Just multiply.

*A* = $18,900

So, Roxanne would pay $18,900 total if she finances the purchase of the used car.

That’s a heck of lot less than the $23,000 she’d pay for the new car. And she hasn’t even figured in her down payment yet.

Why does that change anything? Because after making a down payment, she would be paying interest on less principal (remember, that’s the amount she’ll be borrowing). How would a $1,500 down payment affect her decision?

For the used car, she’d finance $13,500 instead of $15,000.

*A = *$13,500(1 + 0.06)^{4}

*A* = $13,500 • 1.26

*A* = $17,010

So the total she’ll pay for the used car is $17,010.

And for the new car? She just needs to subtract her down payment from the adjusted price: $23,000 – $1,500, or $21,500.

Judged on the basis of price alone, the new car doesn’t seem so minty fresh.

*Do you have questions about using this formula? What about questions about buying cars and fitting the payments into your monthly budget? (On Friday, I’ll talk about the year-round cost of owning a car, a consideration that is critical at the buying stage. And later this month, we’ll take a closer look at compound interest.)*