Chapter 6
Puzzling Out Percents
IN THIS CHAPTER
Understanding what percents are
Converting percents back and forth between decimals and fractions
Solving both simple and difficult percent problems
Using the percent circle to solve three types of percent problems
Like whole numbers and decimals, percents are a way to talk about parts of a whole. The word percent literally means “for 100,” but in practice, it means closer to “out of 100.” For example, suppose 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. 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.
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. Next, I show you how to convert back and forth between percents and fractions. When you understand how conversions work, I show you the three basic types of percent problems, plus a method that makes the problems simple.
Understanding Percents Greater than 100%
100% means “100 out of 100” — in other words, everything. 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.
But lots of times, percentages larger than 100% are perfectly reasonable. For example, suppose I own a hot dog wagon and I 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. That’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 7 when I show you how to do word problems involving percents.
Converting to and from Percents, Decimals, and Fractions
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. That’s why I show you how to convert to and from percents before I show you how to solve percent problems.
Percents and decimals are very 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.
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.
Going from percents to decimals
To convert a percent to a decimal, drop the percent sign (%) and move the decimal point two places to the left. That’s all there is to it. Remember that in a whole number, the decimal point comes at the end. For example,
Changing decimals into percents
To convert a decimal to a percent, move the decimal point two places to the right and add a percent sign (%):
Switching from percents to fractions
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.
To convert a percent to a fraction, use the number in the percent as your numerator (top number) and the number 100 as your denominator (bottom number):
As always with fractions, you may need to reduce to lowest terms or convert an improper fraction to a mixed number (flip to Chapter 4 for more on these topics).
In the three examples, 
can’t be reduced or converted to a mixed number. However, 
can be reduced because the numerator and denominator are both even numbers:
And 
can be converted to a mixed number because the numerator (217) is greater than the denominator (100):
Once in a while, you may start out with a percentage that’s a decimal, such as 99.9%. The rule is still the same, but now you have a decimal in the numerator (top number), which most people don’t like to see. To get rid of it, move the decimal point one place to the right in both the numerator and the denominator:
Thus, 99.9% converts to the fraction 
.
Turning fractions into percents
Converting a fraction to a percent is really a two-step process. Here’s how to convert a fraction to a percent:
-
Convert the fraction to a decimal.
For example, suppose you want to convert the fraction
to a percent. First convert 
to a decimal by dividing the numerator by the denominator, as shown in Chapter 5: 
-
Convert this decimal to a percent.
Convert 0.8 to a percent by moving the decimal point two places to the right and adding a percent sign (as I show you earlier in “Changing decimals into percents”).
Now suppose you want to convert the fraction 
to a percent. Follow these steps:
- Convert
to a decimal by dividing the numerator by the denominator:
Therefore,
. - Convert 0.625 to a percent by moving the decimal point two places to the right and adding a percent sign (%):
Solving Percent Problems
When you know the connection between percents and fractions, which I discuss earlier in “Converting to and from Percents, Decimals, and Fractions,” you can solve a lot of percent problems with a few simple tricks. Others, however, require a bit more work. In this section, I show you how to tell an easy percent problem from a tough one, and I give you the tools to solve them all.
Figuring out simple percent problems
A lot of percent problems turn out to be easy when you give them a little thought. In many cases, just remember the connection between percents and fractions, and you’re halfway home:
- Finding 100% of a number: Remember that 100% means the whole thing, so 100% of any number is simply the number itself:
- 100% of 91 is 91
- 100% of 732 is 732
- Finding 50% of a number: Remember that 50% means half, so to find 50% of a number, just divide it by 2:
- 50% of 88 is 44
- 50% of 7 is
(or 
or 3.5)
- Finding 25% of a number: Remember that 25% equals
, so to find 25% of a number, divide it by 4: - 25% of 88 is 22
- 25% of 15 is
(or 
or 3.75)
- Finding 20% of a number: Finding 20% of a number is handy if you like the service you’ve received in a restaurant, because a good tip is 20% of the check. Because 20% equals
, you can find 20% of a number by dividing it by 5. But here’s an easier way: To find 20% of a number, move the decimal point one place to the left and double the result:
- Finding 200%, 300%, and so on of a number: Working with percents that are multiples of 100 is easy. Just drop the two 0s and multiply by the number that’s left:
- (See the earlier “Understanding Percents Greater than 100%” section for details on what having more than 100% really means.)
Deciphering more-difficult percent problems
You can solve a lot of percent problems using the tricks I show you earlier in this chapter. But what about this problem?
Ouch — this time, the numbers you’re working with aren’t so friendly. When the numbers in a percent problem become a little more difficult, the tricks no longer work, so you want to know how to solve all percent problems.
Here’s how to find any percent of any number:
-
Change the word of to a multiplication sign and the percent to a decimal (as I show you earlier in this chapter).
Changing the word of to a multiplication sign is a simple example of turning words into numbers, as I discuss in Chapters 3 and 7. This change turns something unfamiliar into a form that you know how to work with.
Suppose you want to find 35% of 80. Here’s how you start:
-
Solve the problem using decimal multiplication (see Chapter 5 for details).
Here’s what the example looks like:
So 35% of 80 is 28.
As another example, suppose you want to find 12% of 31. Again, start by changing the percent to a decimal and the word of to a multiplication sign:
Now you can solve the problem with decimal multiplication:
So 12% of 31 is 3.72.
Applying Percent Problems
In the preceding section, “Solving Percent Problems,” I give you a few ways to find any percent of any number. This type of percent problem is the most common — that’s why it gets top billing.
But percents are commonly used in a wide range of business applications such as banking, real estate, payroll, and taxes. (I show you some real-world applications when I discuss word problems in Chapter 7.) And depending on the situation, two other common types of percent problems may present themselves.
In this section, I show you these two additional types of percent problems and how they relate to the type you now know how to solve. I also give you the percent circle, a simple tool to make quick work of all three types.
Identifying the three types of percent problems
Earlier in this chapter, I show you how to solve problems that look like this:
50% of 2 is ?
And the answer, of course, is 1. (See “Solving Percent Problems” for details on how to get this answer.) Given two pieces of information — the percent and the number to start with — you can figure out what number you end up with.
Now suppose instead that I leave out the percent but give you the starting and ending numbers:
?% of 2 is 1
You can still fill in the blank without too much trouble. Similarly, suppose that I leave out the starting number but give the percent and the ending number:
50% of ? is 1
Again, you can fill in the blank.
If you get this basic idea, you’re ready to solve percent problems. When you boil them down, nearly all percent problems are like one of the three types I show in Table 6-1.
TABLE 6-1 The Three Main Types of Percent Problems
|
Problem Type |
What to Find |
Example |
Type 1 |
The ending number |
50% of 2 is what? |
Type 2 |
The percentage |
What percent of 2 is 1? |
Type 3 |
The starting number |
50% of what is 1? |
In each case, the problem gives you two of the three pieces of information, and your job is to figure out the remaining piece. In the next section, I give you a simple tool to help you solve all three of these types of percent problems.
Introducing the percent circle
The percent circle is a simple visual aid that helps you make sense of percent problems so that you can solve them easily. The trick to using a percent circle is to write information in it correctly. For example, Figure 6-1 shows how to record the information that 50% of 2 is 1.
FIGURE 6-1: In the percent circle, the ending number is at the top, the percentage is at the left, and the starting number is at the right.
Notice that as I fill in the percent circle, I change the percentage, 50%, to its decimal equivalent, 0.5 (for more on changing percents to decimals, see “Going from percents to decimals” earlier in this chapter).
Here are the two main features of the percent circle:
- When you multiply the two bottom numbers together, they equal the top number:
- If you make a fraction out of the top number and either bottom number, that fraction equals the other bottom number:
and 
These features are the heart and soul of the percent circle. They enable you to solve any of the three types of percent problems quickly and easily.
Most percent problems give you enough information to fill in two of the three sections of the percent circle. But no matter which two sections you fill in, you can find out the number in the third section.
Finding the ending number
Suppose you want to find out the answer to this problem:
What is 75% of 20?
You’re given the percent and the starting number and asked to find the ending number. To use the percent circle on this problem, fill in the information as Figure 6-2 shows.
FIGURE 6-2: Putting 75% of 20 into a percent circle.
Because 0.75 and 20 are both bottom numbers in the circle, multiply them to get the answer:
- So 75% of 20 is 15.
As you can see, this method is essentially the same one I show you earlier in this chapter in “Deciphering more-difficult percent problems,” where you translate the word of as a multiplication sign. You still use multiplication to get your answer, but with the percent circle, you’re less likely to get confused.
Discovering the percentage
In the second type of problem, I give you both the starting and ending numbers, and I ask you to find the percentage. Here’s an example:
What percent of 50 is 35?
In this case, the starting number is 50 and the ending number is 35. Set up the problem on the percent circle as Figure 6-3 shows.
FIGURE 6-3: Determining what percent of 50 is 35.
This time, 35 is above 50, so make a fraction out of these two numbers:
This fraction is your answer, and all you have to do is convert the fraction to a percent as I discuss earlier in this chapter in “Turning fractions into percents.” First, convert 
to a decimal:
Now convert 0.7 to a percent:
Tracking down the starting number
In the third type of problem, you get the percentage and the ending number, and you have to find the starting number. For example,
15% of what number is 18?
This time, the percentage is 15% and the ending number is 18, so fill in the percent circle as Figure 6-4 shows.
FIGURE 6-4: Working out the answer to “15% of what number is 18?”
Because 18 is above 0.15 in the circle, make a fraction out of these two numbers:
This fraction is your answer; you just need to convert to a decimal as I show you in Chapter 5. Divide 18 by 0.15:
In this case, the “decimal” you find is the whole number 120, so 15% of 120 is 18.