Basic Math & Pre-Algebra For Dummies, 2nd Edition (2014)
Part III. Parts of the Whole: Fractions, Decimals, and Percents
Chapter 12. Playing with 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 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.
Making Sense of Percents
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.
Dealing with Percents Greater than 100%
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.
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. 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.
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. It's simple. 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 9 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:
1. Convert the fraction to a decimal.
For example, suppose you want to convert the fraction to a percent. To convert to a decimal, you can divide the numerator by the denominator, as shown in Chapter 11:
2. 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”).
o 0.8 = 80%
Now suppose you want to convert the fraction to a percent. Follow these steps:
1. Convert to a decimal by dividing the numerator by the denominator:
2. Convert 0.625 to a percent by moving the decimal point two places to the right and adding a percent sign (%):
o 0.625 = 62.5%
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. Other problems, 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 all of them.
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 5 is 5.
· 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 20 is 10.
· 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:
· 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 I can show you an easier way: Remember that 20% is 2 times 10%, so to find 20% of a number, move the decimal point one place to the left and double the result:
· Finding 10% of a number: Finding 10% of any number is the same as finding of that number. To do this, just move the decimal point one place to the left:
· 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 “Dealing with Percents Greater than 100%” section for details on what having more than 100% really means.)
Turning the problem around
Here's a trick that makes certain tough-looking percent problems so easy that you can do them in your head. Simply move the percent sign from one number to the other and flip the order of the numbers.
Suppose someone wants you to figure out the following:
· 88% of 50
Finding 88% of anything isn't an activity anybody looks forward to. But an easy way of solving the problem is to switch it around:
· 88% of 50 = 50% of 88
This move is perfectly valid, and it makes the problem a lot easier. It works because the word of really means multiplication, and you can multiply either backward or forward and get the same answer. As I discuss in the preceding section, “Figuring out simple percent problems,” 50% of 88 is simply half of 88:
· 88% of 50 = 50% of 88 = 44
As another example, suppose you want to find
· 7% of 200
Again, finding 7% is tricky, but finding 200% is simple, so switch the problem around:
· 7% of 200 = 200% of 7
In the preceding section, I tell you that, to find 200% of any number, you just multiply that number by 2:
Deciphering more-difficult percent problems
You can solve a lot of percent problems, using the tricks I show you earlier in this chapter. For more difficult problems, you may want to switch to a calculator. If you don't have a calculator at hand, solve percent problems by turning them into decimal multiplication, as follows:
1. Change the word of to a multiplication sign and the percent to a decimal (as I show you earlier in this chapter).
Suppose you want to find 35% of 80. Here's how you start:
2. Solve the problem using decimal multiplication (see Chapter 11).
Here's what the example looks like:
So 35% of 80 is 28.
Putting All the Percent Problems Together
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, which is why it gets top billing.
But percents crop up 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 13.) 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 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 ?
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 12-1.
Table 12-1 The Three Main Types of Percent Problems
What to Find
The ending number
50% of 2 is what?
What percent of 2 is 1?
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.
Solving percent problems with equations
Here's how to solve any percent problem:
1. Change the word of to a multiplication sign and the percent to a decimal (as I show you earlier in this chapter).
This step is the same as for more straightforward percent problems. For example, consider this problem:
o 60% of what is 75?
Begin by changing as follows:
2. Turn the word is to an equals sign and the word what into the letter n.
Here's what this step looks like:
This equation looks more normal, as follows:
3. Find the value of n.
Technically, the last step involves a little bit of algebra, but I know you can handle it. (For a complete explanation of algebra, see Part V of this book.) In the equation, n is being multiplied by 0.6. You want to “undo” this operation by dividing by 0.6 on both sides of the equation:
Almost magically, the left side of the equation becomes a lot easier to work with because multiplication and division by the same number cancel each other out:
Remember that n is the answer to the problem. If your teacher lets you use a calculator, this last step is easy; if not, you can calculate it using some decimal division, as I show you in Chapter 11:
Either way, the answer is 125 — so 60% of 125 is 75.
As another example, suppose you're faced with this percent problem:
What percent of 250 is 375?
To begin, change the of into a multiplication sign and the percent into a decimal.
Notice here that, because I don't know the percent, I change the word percent to × 0.01. Next, change is to an equals sign and what to the letter n:
Consolidate the equation and then multiply:
Now divide both sides by 2.5:
Therefore, the answer is 150 — so 150% of 250 is 375.
Here’s one more problem: 49 is what percent of 140? Begin, as always, by translating the problem into words:
Simplify the equation:
49 = n × 1.4
Now divide both sides by 1.4:
49 ÷ 1.4 = n × 1.4 ÷ 1.4
Again, multiplication and division by the same number allows you to cancel on the left side of the equation and complete the problem:
49 ÷ 1.4 = n
35 = n
Therefore, the answer is 35, so 49 is 35% of 140.