Basic Math and Pre-Algebra For Dummies (60 page)

This time, three trailing zeros aren't enough to get my answer, so I attach a few more and continue:

At last, the division works out evenly, so again the answer is a terminating decimal. Therefore,
.

Sometimes when you try to convert a fraction to a decimal, the division
never
works out evenly. The result is a
repeating decimal
— a decimal that cycles through the same number pattern forever.

You may recognize these pesky little critters from your calculator, when an apparently simple division problem produces a long string of numbers.

For example, to change
to a decimal, begin by dividing 2 by 3. As in the last section, start by adding three trailing zeros, and see where it leads:

At this point, you still haven't found an exact answer. But you may notice that a repeating pattern has developed in the division. No matter how many trailing zeros you attach to the number 2, the same pattern continues forever. This answer, 0.666 … , is an example of a repeating decimal. You can write
as

The bar over the 6 means that, in this decimal, the number 6 repeats forever. You can represent many simple fractions as repeating decimals. In fact,
every
fraction can be represented either as a repeating decimal or as a terminating decimal — that is, as an ordinary decimal that ends.

Now suppose you want to find the decimal representation of
. Here's how this problem plays out:

This time, the pattern repeats every other number — 4, then 5, then 4 again, and then 5 again, forever. Attaching more trailing zeros to the original decimal only strings out this pattern indefinitely. So you can write

This time, the bar is over both the 4 and the 5, telling you that these two numbers alternate forever.

 Repeating decimals are an oddity, but they aren't hard to work with. In fact, as soon as you can show that a decimal division is repeating, you've found your answer. Just remember to place the bar only over the numbers that keep on repeating.

 Some decimals never end and never repeat. You can't write them as fractions, so mathematicians have agreed on some shorter ways of naming them so that writing them out doesn't take, well, forever.

Chapter 12

In This Chapter

Understanding what percents are

Converting percents back and forth between decimals and fractions

Solving both simple and difficult percent problems

Using equations to solve three different types of percent problems

Like whole numbers and decimals, percents are a way to talk about parts of a whole. The word
percent
means “out of 100.” So if you have 50% of something, you have 50 out of 100. If you have 25% of it, you have 25 out of 100. Of course, if you have 100% of anything, you have all of it.

In this chapter, I show you how to work with percents. Because percents resemble decimals, I first show you how to convert numbers back and forth between percents and decimals. No worries — this switch is easy to do. Next, I show you how to convert back and forth between percents and fractions — also not too bad. When you understand how conversions work, I show you the three basic types of percent problems, plus a method that makes the problems simple.

The word
percent
literally means “for 100,” but in practice, it means closer to “out of 100.” For example, suppose that a school has exactly 100 children — 50 girls and 50 boys. You can say that “50 out of 100” children are girls — or you can shorten it to simply “50 percent.” Even shorter than that, you can use the symbol %, which means
percent.

Saying that 50% of the students are girls is the same as saying that
of them are girls. Or if you prefer decimals, it's the same thing as saying that 0.5 of all the students are girls. This example shows you that percents, like fractions
and decimals, are just another way of talking about parts of the whole. In this case, the whole is the total number of children in the school.

You don't literally have to have 100 of something to use a percent. You probably won't ever really cut a cake into 100 pieces, but that doesn't matter. The values are the same. Whether you're talking about cake, a dollar, or a group of children, 50% is still half, 25% is still one-quarter, 75% is still three-quarters, and so on.

Any percentage smaller than 100% means less than the whole — the smaller the percentage, the less you have. You probably know this fact well from the school grading system. If you get 100%, you get a perfect score. And 90% is usually A work, 80% is a B, 70% is a C, and, well, you know the rest.

Of course, 0% means “0 out of 100” — any way you slice it, you have nothing.

100% means “100 out of 100” — in other words, everything. So when I say I have 100% confidence in you, I mean that I have complete confidence in you.

What about percentages more than 100%? Well, sometimes percentages like these don't make sense. For example, you can't spend more than 100% of your time playing basketball, no matter how much you love the sport; 100% is all the time you have, and there ain't no more.

But a lot of times, percentages larger than 100% are perfectly reasonable. For example, suppose I own a hot dog wagon and sell the following:

• 10 hot dogs in the morning
  
• 30 hot dogs in the afternoon
  

The number of hot dogs I sell in the afternoon is 300% of the number I sold in the morning. It's three times as many.

Here's another way of looking at this: I sell 20 more hot dogs in the afternoon than in the morning, so this is a
200% increase
in the afternoon — 20 is twice as many as 10.

Spend a little time thinking about this example until it makes sense. You visit some of these ideas again in Chapter
13
, when I show you how to do word problems involving percents.

To solve many percent problems, you need to change the percent to either a decimal or a fraction. Then you can apply what you know about solving decimal and fraction problems. For this reason, I show you how to convert to and from percents before I show you how to solve percent problems.

Percents and decimals are similar ways of expressing parts of a whole. This similarity makes converting percents to decimals, and vice versa, mostly a matter of moving the decimal point. It's so simple you can probably do it in your sleep (but you should probably stay awake when you first read about the concept).

Percents and fractions both express the same idea — parts of a whole — in different ways. So converting back and forth between percents and fractions isn't quite as simple as just moving the decimal point back and forth. In this section, I cover the ways to convert to and from percents, decimals, and fractions, starting with percents to decimals.

 To convert a percent to a decimal, drop the percent sign (%) and move the decimal point two places to the left. It's simple. Remember that, in a whole number, the decimal point comes at the end. For example,

 To convert a decimal to a percent, move the decimal point two places to the right and add a percent sign (%):

Converting percents to fractions is fairly straightforward. Remember that the word percent means “out of 100.” So changing percents to fractions naturally involves the number 100.