The Percent Equation

Wow! When my family moved at the end of June, I didn’t mean to take an entire month and a half off! My interest just got centered on other things for a while, but I’m back now. We’re settled in our new place and summer camp for my daughter is over. School will be starting soon, and maybe we can get back to normal around here. (Heck, maybe we can find out what “normal” means, in the first place!)

In previous posts, we talked about how to convert a percent to a decimal or fraction and how to calculate an amount based on a percent. Today, I want to focus on how to use the percent equation.

There are only three players in the percent equation. If you read the previous post, you have already used them, but we didn’t really give them names because we were always using the equation the same way & didn’t need names. However, I want you to be able to calculate any piece of the equation without memorizing three different forms. In order to do that, we will have to give names to the three parts. Here goes:

Naming the Parts of the Equation

Take, for example, the following statement:

30 is 15% of 200.

In this example, 200 is called the “Base.” To me, that name makes sense because it is what we are “taking a percent of”, if you will.

Now, here is where many students get confused because the names that most textbooks give the other two numbers are contrary to our intuition, and many students throw up their hands in confusion & quit. The text books want to call the 30 in our example the “Percentage” and the 15% in our example, the “Rate.”

Although I can understand why the word “percentage” is used for the 30, I think it is confusing. People always want to associate the word “percentage” with the number that has the percent sign. They don’t know what to call the 30 or why how it is different from the 200.

A Little Confusion

Many middle grade teachers will try to get around this problem by calling them “the IS number” and “the OF number.” That’s OK for getting students to pass the test, but in my experience, this terminology doesn’t do anything to help students remember the actual function of either number or how the two numbers are related. I’ve encountered very few students who knew what to do with “the IS number” and “the OF number” a year later. No, we need better names, names that will help us remember how the two numbers are related.

What Worked for Me

I remember the first time that I really understood percents. I was actually a college freshman, sitting in chemistry class. A lot of us were missing questions involving percents, and I must admit that even though I could do the calculation if I wrote it out, I was never sure of my answer & was easily confused about the “IS number” and the “OF number.”

The professor said, “Don’t you see? A percent is just a partial divided by a whole.” EUREKA! That did it for me. A percent is a partial divided by a whole! (Let that sink in really well. Compare it to what you already know about percents.) Now, maybe those aren’t the best names in the world because they do cause a few problems when you talk about percents that are larger than 100%, but for starters, I think they will work fine. We just need to start out by talking about percents less than 100.

Names The Work

Back to our example: 30 is 15% of 200.

Obviously, 15% is the “percent.” Which number should be the base? 200 is the base because it is the number we are comparing to. 30 is the number that is being compared, so it is the “partial.” 30 is a part of 200, and the percent tells us how big a part it is.

Now, let’s check whether this makes sense. We should be able to replace the percent, the partial, the whole, and the words “is” and “divided by” with numbers or symbols.

A percent is a partial divided by a whole.
15% = 30 / 200

Remember that the words “percent” and “hundredth” are interchangeable, so now our equation reads:

.15 = 30 / 200

Is it true? You can check it quickly by hand or with a calculator. If you do, you’ll see that it works fine.

Now that we have a formula (a percent is a partial divided by a whole) for percents, we will find that we can always find a missing piece of information if we know the other two numbers.

Let’s try a few examples:

What percent of 140 is 28?

Remember “A percent is a partial divided by a whole.” In this case we want to compare 28 to 140, so 28 is the “partial” and 140 is the “base” or “whole.” We don’t know the percent, so we will use the letter x to hold its place until we figure it out.

A percent is a partial divided by a whole.
x = 28 / 140

x = 28/140 = .2 = 20%

What percent of 28 is 140?

Well, in this case, we want to compare 140 to 28, so even though it is a small number, 28 is the base (because it is being compared to) and 140 is the “partial” even though it is bigger than the base.

A percent is a partial divided by a whole.
x = 140 / 28

x = 140 / 28 = 5 = 500%

Now this brings up a question. If 100% of something is all of that something, then why should we even talk about 500%? The answer is not hard to understand. Percents are just a tool we use to help us compare numbers. Sometimes we need to compare a large number to a smaller number. For example, if you were comparing 2007 gas prices to 1992 gas prices, you would find that prices now are higher than (that is, they are more than 100% of) prices back then.

Here’s an example with words:

Last year, only 15 people attended the senior class homecoming dance. This year, 18 people are expected to attend. What is this year’s attendance as a percent of last year’s attendance?

Solution: Last year’s attendance was 15. That’s the number we are comparing to, so 15 is the “whole” or “base” number. 18 is this year’s attendance, so 18 is the “partial” number. We are looking for the percent.

A percent is a partial divided by a whole.
x = 18 / 15

x = 18 / 15 = 1.2 = 120%

Another example, this time with a different unknown:

What is 22% of 95?

We are looking for a piece (22%) of 95. So 95 is the “whole” and 22 is the percent. We are looking for the “partial.”

A percent is a partial divided by a whole.
22% = x / 95

Now just as you would break an egg before you put it in the frying pan, we need to “break the shell” on our percent before we cook it up in our equation. That means we substitute the percent symbol for the word “hundredths”, which causes us to move the decimal back two places.

22% = x / 95 so .22 = x/95 so .22 x 95 = x

x = 20.9

Another example:

18 is 50% of what?

We know 18 is part of (50% of) something, so 18 must be the partial number. We actually do not know what we are comparing to in this case. This time we are looking for that comparing number, so we are looking for the “whole.”

A percent is a partial divided by a whole.
50% = 18 / x

Break the shell on that percent. Now we have:

.50 = 18 / x

Now we solve for x.

.5 = 18 / x so (.5)(x) = 18 so x = 18/.5 so x = 36

If percents have been difficult for you, I recommend practicing these examples with paper and pencil. Don’t just read through them — it will help a lot if you will let your brain really dig in by working all the way through. It makes a huge difference to do it yourself. Also, practice with the percents you see while you are shopping or watching the news. The more you use them, the easier they get. Remember that percents are just a tool we use to let us compare two numbers. If you can remember how to place the pieces in the percent equation, you can solve any percent problem.

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