# Change Expressed as a Percentage

For most any percentage problem you come across, you will need to use some formulation of the basic percentage equation, which relates the three basic quantities involved in a percentage. These are known as the Whole, Percentage Amount, and Part.

## Change Expressed as a Percentage

For pretty much any percentage problem that you come across you’re going to need to use some formulation of the basic percentage equation. The basic percentage equation is the equation that relates the 3 basic quantities involved in a percentage.

You have the **whole**, which is the number you’re considering as your base amount here. You have the **percentage amount**, which tells you what percentage of the whole you’re talking about, and you have the **part**, which is the number that is that percentage of the whole.

This is 1 formulation of the equation: the part is equal to the whole, times the percentage. Now if for instance, you’re looking for the whole when you’re given the part and the percentage, you would formulate the equation as a whole, equals the part, divided by the percentage.

If you were looking for the percentage, given the whole and the part, you would formulate it as percentage is equal to the part, divided by the whole. Pretty much any percentage problem you come across is going to require you to use 1 or more of these 3 formulations, and I say 1 or more because most percentage problems will require you to use either more than 1 of these, or the same 1 multiple times.

Let’s look at an example and I’ll demonstrate what I mean by that. Let’s suppose that you’re buying a new computer and its regular price is 1,500 dollars, but the company is currently having a sale where you get 20 percent off, but then after the 20 percent off you have to pay 5 percent sales tax.

You’ve taken 20 percent off and then you’ve added back in 5 percent sales tax, and what you’re wondering is what is your actual percentage off that you see? What percentage of the original price are you actually paying here (so what is real percentage off)? This is an example of a problem you might see.

There’s going to be (3 different things we have to) 3 different operations we have to perform here to get to our final answer. The first thing is we’re taking 20 percent off of 1500, and that’s going to be this top form of the equation, because we’re given a whole and a percentage.

Now if we use 20 as the percentage in this equation, we’re going to get the amount that’s taken off of 1500, instead of the amount after the 20 percent is taken off. If we wanted to skip directly to the amount after 20 percent is taken off, we need to use 80 percent as the percentage in this equation, because 100 percent minus 20 percent is 80 percent.

Let’s go ahead and do that, just to skip a little step there. The part that remains after the 20 percent is taken off is going to be 1500 (times 20 percent, sorry) times 80 percent. 1500 times 80 percent is 1200. After the 20 percent is taken off you’re left with 1200, but now you have to add in 5 percent tax.

The 1200 becomes the new whole, and the 5 percent becomes the new percent, so we’re still using this top equation (this will be the second time now). Our new equation is P equals 1200, times 5 percent. Now the easiest way to work this problem mentally is probably going to be to note that 1 percent of 1200 is 12, and then you’d multiply that by 5 to get 5 percent.

We have 12 times 5 and that’s 60 dollars, so the tax is going to be an additional 60 dollars, which added back into the 1200 is going to be 1260. Now we’re wondering what is the real percentage off, or to state it another way, (what percentage is the difference between) what percentage of 1500 is the difference between 1500 and 1260.

To get the difference we’ll subtract 1260 from 1500 and that will give us 240, so this is the actual amount less than original price that we’re paying. The question now becomes—we’re looking for a percentage, so we need to use this equation. The part is going to be to 240, and the whole is going to be 1500.

Our new equation is going to become percentage is equal to 240, divided by 1500. I’m not going to work out the division here, but if you wanted to plug it into a calculator the percentage that you would get by dividing these 2 numbers would be 16 percent.

There we used 3 different formulations of the same equation to get to our answer, we used this top 1 twice, and then we used the bottom 1 for the final step there. This is an example of what you can encounter in some of the more complicated percentage problems, multiple uses of different forms of the basic percentage equation.