Math for Grownups (5 page)

That’s when Larry reminds him about the state sales tax, so Bubba whips out his calculator watch once more. Remember how we converted percents to decimals in
Chapter 1
? In this case, the 6.85% sales tax converts to 0.0685, making it easy to multiply. (At least, it’s easy as long as you’ve got a watch calculator handy. Otherwise, you may have to break out the pencil and paper.)

$44,325 • 0.0685
=
$3,036.26

$44,325
+
$3,036.26
=
$47,361.26

But wait—there’s more. Bubba can’t drive his new baby off the lot without a title and license. He’s going to have to pay a title and licensing fee, which is probably between 1% and 1.5% of the cost of the truck (without the sales tax). To be on the safe side, Larry recommends finding 1.5% of $44,325.

0.015 • $44,325
=
$664.875

Thus Bubba can expect to pay $664.88 for the title and licensing fee. All he has to do now is add that amount to the total he found earlier:

$47,361.26
+
$664.88
=
$48,026.14

Now that’s a monster price for a monster truck.

The destination fee may not be negotiable, but other fees are. And some of these are downright phony.

• With the
documentation fee
, the dealer is trying to recoup the cost of drawing up the title, registration, and license.

•
Market adjustments
are fees added to hard-to-get vehicles.

• Don’t be fooled by the
transportation fee.
It’s not the same as a destination charge, but it is sometimes added if the dealer had to have a specific vehicle delivered for you, from another dealership.

•
Dealer prep
is the cost of preparing the vehicle for delivery.

•
Dealer additions
are becoming rare, but they may include fabric protector and undercoating.

• Finance fees are difficult to spot. To avoid them, arrange your financing before you walk onto the lot.

• Experts advise questioning all
processing and miscellaneous fees
. Buzzwords to look for include
surcharge, assessment, commission, honorarium, recompense, toll,
and
interest
.

After adding on the cost of the options and the destination fee, Bubba found the total cost of the truck by using a two-step process: He found the sales tax and then added it to the price of the truck.

Is there an easier way? Sure!

If you can add 6.85 to 100 in your head, you can find the total price in one fell swoop. That’s because the total cost of Bubba’s dream truck is 106.85% of $44,325.

1.0685 • $44,325
=
$47,361.26

Why does this work? The answer comes from elementary arithmetic, my friend. First, let’s compare the two methods. Method 1 is to multiply the cost of the truck by the sales tax rate and then add the answer to the cost of the truck.

($44,325 • 0.0685)
+
$44,325

$3,036.26
+
$44,325

$47,361.26

Now see what happens when you add 1 to the sales tax rate and then multiply by the cost of the truck.

(1
+
0.0685) • $44,325

1.0685 • $44,325

$47,361.26

This works because of something called the distributive property. When you’re multiplying a number by the sum of two numbers, you can
distribute
the first number to both of those numbers and then add.

But hold on. It’s easier to explain with variables.

a
(
b
+
c
) =
ab
+
ac

If we use the distributive property on the second method, we get the first method. Take a look:

(1
+
0.0685) • $44,325

$44,325(1
+
0.0685)
→
switch the multiplication so that it looks like the distributive property

($44,325 • 1)
+
($44,325 • 0.0685)
→
distribute $44,325

$44,325
+
($44,325 • 0.0685) → and this is what the first method looks like

It’s good to have choices.

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. So 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.
(So, 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.

Based on price alone, the new car doesn’t seem so minty fresh.

If you’re financing your car, you’ll need to come up with a down payment, which is an upfront amount you pay before you can walk—or rather drive—off the lot with your new (or used) vehicle.

If you ask your grandfather how much to put down, he’ll probably tell you at least 20% of the cost of the car. That was the typical down payment 40 years ago, and as it turns out, it’s still good advice.

Melissa has her eye on a sweet little convertible with a price tag of $42,300. Her dad has offered to pay the 20% down payment. How large is the check he needs to write?

To find 20% of a number, you need to change the percent to a decimal and then multiply:

0.20 • $42,300
=
$8,460

Melissa can do this calculation easily by using a calculator. But could she also do it in her head? Yep, if she thinks of 20% as 10% times 2.

20%
=
10% • 2

10% of $42,300 is $4,230

$4,230 • 2
=
$8,460

Either way, she needs a whopping $8,460 for her down payment.

But suppose Melissa has to cough up her own down payment, and she doesn’t have as much cash stashed as her dad. How can she figure out how much car she can afford?

Let’s say that Melissa has brown-bagged lunch and skipped dinners out for 2 years, just so that someday she could afford to buy a new car. At this point, she’s put away $4,200. If she wants to use this for a 20% down payment on a new set of wheels, what’s the most expensive car she can afford?

Here’s another way of putting this question: 20% of what is $4,200?

If your brain is in math mode, you may automatically see an equation in which you need to find
x
. But if your brain is still focusing on whether that cute hatchback comes in robin’s-egg blue, a little review is probably in order.

Luckily, there are some easy clues in the question itself:

• 20% is the same as 0.20

• when dealing with percents,
of
usually means multiplication

•
what
is the variable (the thing you need to find, or
x
)

•
is
means “equals”

When you recognize these clues, you can set up an algebraic equation (an equation to help you find
x
).

Wait! Don’t close this book! It’s not as hard as you think, so keep reading!

Why put down at least 20%? There’s a one-word answer to that question: depreciation.

Depreciation is the value that the car loses over time. You can’t buy a car for $40,000 and sell it for the same amount a year later. That’s because as soon as a car is purchased, it depreciates in value. (Real estate is different. Buildings and land often appreciate—or gain value—over time.)

On average, a car’s depreciation in the first year is 20%. If you’ve paid that much already with a down payment (and assuming you keep up-to-date on your payments), you will never be “upside-down” on your loan. In other words, you won’t owe more on the car than it’s worth.

That’s a pretty big deal, if you suddenly hit hard times and need to sell your car. If you’re upside-down, you will owe the bank the difference between the value of the car and the amount you’ve borrowed. That’s the kind of math that no one wants to do.

20% of what is $4,200?

0.20 •
x
= 4,200

0.20
x
= 4,200 (Remember, 0.20
x
is the same as 0.20 times
x
.)

Don’t panic. Solving for
x
in this equation is really simple, if you remember three rules:

1. The object of the game is to get
x
all by itself on one side of the equals sign.

2. Whatever you do to one side of the equation, you must do to the other side to keep them even.