Percent Introduced

The word percent simply means “out of 100.” The language of percent is used by everyone frequently as a convenient and intuitive was to avoid using the terminology of numerical fractions. What we will try to do in this section of notes is describe very precisely how you can use the terminology of percent to clearly and accurately communicate numerical information.

Familiar examples of the use of the word percent are:

• percents which express a part or fraction of a whole.

• percents which express a rate of change in some quantity. In a way, this is not really different than percents which express a fraction of a whole because the percent as a rate of change is expressing the amount of change as a fraction of the original quantity.

Doing Arithmetic With Percents

The thing to remember is that percents represent the numerator of a fraction that always has a denominator of 100. So, when we say, for example, 25%, we are really speaking of the fraction or 0.25 in decimal form.

So conversion between a percent, an actual fraction with a numerator and denominator, and a decimal fraction is very easy to do.

If you start with the percent then

• to get the fractional equivalent, just write a fraction with the percent in the numerator and a denominator of 100. For example

• to get a decimal fraction equivalent, just divide the percent by 100 (which is the same thing as moving the decimal point two places leftwards). For example

• If you start with a proper fraction or a decimal fraction, just multiply by 100. For example

• This rule also works if the starting fraction has a denominator which is not equal to 100. For example

• It may be that this conversion results in digits to the right of the decimal point. For example

• rounded to two decimal places. How many decimal places should be retained in the final percent that is calculated depends on the situation.

Finally, notice that percents can be (and often are) larger than 100. Such percents just correspond to fractions and decimal numbers which are bigger than 1. For example

The meaning of percents bigger than 100% will depend on the situation. When a percent is expressing a part or fraction of a whole, it makes no sense to have a value bigger than 100% (since then the part would be more than the total stuff that it is part of). However, if the percent is referring to a rate of increase, values bigger than 100% just mean that the quantity has more than doubled in size.

(People occasionally speak of negative percents to indicate rates of decrease. We will need to look at expressing rates of decrease very carefully, but probably the complicated notion of negative percents should be avoided to avoid confusion.)