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# Percent and Percentages

## Introduction to Percentage

### Definition of Percentage

Percent is defined as parts per hundred. You may be wondering, per hundred what? As it turns out, you can define percent for almost anything that can have a quantity associated with it. You could define a percentage of the number of pencils in an office, or of the weight of a shipment in a truck, or of the amount of time you waited for the cable guy to come to your house to connect your new account.

In order to determine percentage, an amount has to be designated as representing the whole amount of something. That amount is defined as equal to 100% . For the pencils, for instance, you would look in all of the drawers in all of the desks, and count up how many total pencils were in the office. Whatever number you arrived at would be defined as 100% of the total number of pencils in the office.

So what good is it to define a term equal to the whole amount of something? If that was all there was to it, there would be no advantage. However, using the concept of percentage allows comparisons to be made among different amounts of a quantity, by expressing those amounts relative to the defined whole amount. Percentage is a relative comparison-- by itself, it does not give information on actual quantities.

The "parts per hundred" comes into play once we have established what the actual quantity is for what we are defining as the whole. The "whole" amount, defined as 100% , can be thought of as being divided into 100 equal portions, each of which constitutes 1% of the whole. Suppose when we counted the pencils in the office we found exactly 100 pencils. Each pencil would then account for 1% of the total number of pencils. If desk #2 had 14 pencils, then desk #2 would have 14 x 1% = 14% of the total pencils in the room.

It's important to realize that the quantity 1% does not have to correspond to a complete object. Suppose the room only contained 50 pencils. If we divide 50 pencils into 100 equal parts, each part is worth one-half, or 0.5 , of a pencil-- 1% of the total number of pencils in the room corresponds to 0.5 pencils. A half of a pencil is not a very useful concept, but you can think about it another way: if a half a pencil corresponds to 1% of the total, then each complete pencil corresponds to 2% of the total. You can verify that each pencil constitutes 2% of the total by multiplying the percentage for each pencil by the number of pencils: 2% x 50 = 100% , which is what we expect-- the 50 pencils make up all, or 100% , of the pencils in the room.

If desk #3 has 6 pencils out of 50 in the room, and each pencil accounts for 2% of the total pencils, then the 6 pencils on desk #3 constitute 6 x 2% = 12% of the pencils in the room.

This introduction has mixed several sub-topics together, but you're probably already getting the idea that there is an intimate relation between percentage, fractions, and decimals. The rest of the sections on this page show how to convert among these expressions.

### Converting a Percentage to a Fraction

In many problems using percentage, it is necessary, or at least convenient, to rewrite a percentage as a fraction. In the previous section we said that desk #3 contained 6 out of 50 pencils in the office, which corresponds to 12% of the pencils in the room. For some applications, it might be easier to express 12% as a fraction.

Writing a percent value as a fraction is easy. Remember that 12% means 12 parts out of 100 parts. This is the same as saying "twelve one-hundredths", which would be written as the fraction . The fraction is not in simplest form-- it can be reduced to . Thus, the 5 pencils on desk #3 constitute 12% , or , of the total number of pencils in the office.

The general approach for converting a percentage to a fraction is to remove the percent symbol and multiply the percentage value by . The resulting fraction may not be in simplest form; if not, it can then be reduced to simplest form. Example: Write 55% as a fraction.

To convert 55% to a fraction, we remove the percent symbol and multiply the value by .

55% = 55 x = = Note that the initial fraction we obtained had to be reduced to simplest form.

Percentage values can also be greater than 100% . How can that be? Using the pencil analogy again, for the office in question there can't be more than 100% of the pencils present, because the total number of pencils is what was used to define the amount of the whole. However, suppose there is another office next door, and we want to express the number of pencils in office #2 relative to the number of pencils in the first office. If office #2 contains more pencils than does the first office, then the percentage of pencils in office #2 will be greater than 100% . We might find, for instance, that office #2 contains 135% as many pencils as the first office.

Example: Write 135% as a fraction.

Even though this percentage is greater than 100% , it is converted to a fraction in exactly the same manner as described above. The result will be an improper fraction, which will need to be converted to a mixed number and perhaps also reduced to simplest form. The reason the result will be a mixed number is because if the percentage being converted is greater than 100% , the numerator of the resulting fraction will be larger than the denominator, which will always be 100 initially (before any simplification).

135% = 135 x = = = In this case, the initial fraction obtained was , which was then reduced to simplest form ( ) , and then expressed as a mixed number ( ) . It doesn't matter whether you first reduce the fraction to simplest form and then convert it to a mixed number, or first convert to a mixed number and then reduce the fractional portion of the mixed number to simplest form-- you'll end up with the same result either way.

Percentage values are not restricted only to integer values, such as 4% or 73% . They can also be expressed as decimal values, such as 2.91%, fractional values, such as , or mixed number values, such as .

Example: Write as a fraction.

This conversion is done the same way as outlined above: we want to multiply by . Since is a mixed number, it must first be converted to an improper fraction before multiplying. You can verify for yourself as an exercise that is equal to . = = x = = Summary: = This last example and the earlier examples on this page used several concepts covered in earlier tutorials. If necessary, you can review those concepts:

### Converting a Percentage to a Decimal

Rewriting a percentage as a decimal is very easy. All that needs to be done is to remove the percent symbol and multiply the percentage value by 0.01 . You'll recognize that this is really the same thing we did to rewrite a percentage as a fraction-- to convert a percentage to a fraction we multiplied by , and expressed as a decimal is 0.01 . In both instances, then, we are multiplying the percentage value by 1 one-hundredth. Converting a percentage value to a decimal is the easier of the two processes, though, because multiplying a number by 0.01 can be accomplished simply by moving the decimal point two places to the left.

Example: Write 26% as a decimal.

All that needs to be done is to remove the percent symbol, and move the decimal point two places to the left. The original value was not written with a decimal point, but 26 is the same as 26. , and moving the decimal point two places to the left gives us .26 . For aesthetic purposes we can add a 0 in front of the decimal point, but it's not required.

Summary: 26% = 0.26

Another example: Write 4.38% as a decimal.

We need to remove the percent symbol and move the decimal point two places to the left. The original value only has one digit to the left of the decimal point, so a 0 can be inserted to the left of the 4 . Ordinarily one would not write out this step, but you can perform the step mentally in problems like this-- adding a 0 gives 04.38 , which isn't really a proper way to write a number, but it doesn't change the number's value. Now the decimal point can be moved two places to the left, to give .0438 .

Summary: 4.38% = 0.0438

### Converting a Fraction to a Percentage

To convert a fraction to a percentage, you simply multiply the fraction by 100% .

Example: Write as a percentage.

We need to multiply by 100% . Recall that when multiplying a fraction by a whole number, the whole number can be expressed as a fraction too, as the number itself over 1 . Thus, 100 can be expressed as . Now the two fractions can be multiplied normally: In this case, multiplying the fraction by resulted in an improper fraction, which was then expressed as a mixed number. Without any further specified requirements, it's fine to leave the answer in the form , although sometimes you might want to express the mixed number as a decimal. You can verify for yourself that is equal to .

Summary: Another example: Write as a percentage.

We just multiply by 100% , as shown below: Summary: ### Converting a Decimal to a Percentage

Converting a decimal to a percentage is probably the easiest conversion discussed on this page. All that needs to be done is to multiply the decimal by 100% , just as we did when converting a fraction to a percentage. However, since 100 is an even power of ten, multiplying a decimal by 100 can be accomplished simply by moving the decimal point in the original decimal. If necessary, you can review the section on Multiplying Decimals by Powers of Ten. Recall that to multiply a decimal by 100 , the decimal point is moved two places to the right. Don't forget to add the percent symbol when you're finished, as we are really multiplying by 100% , not just by 100 .

Example: Write 0.89 as a percentage.

We want to multiply 0.89 by 100% , which is done by moving the decimal point two places to the right and adding the percent symbol. That gives us 089.% . The 0 that was originally on the left of the decimal point is now superfluous and can be dropped, and since there are now no digits to the right of the decimal point, the decimal point can also be dropped.

Summary: 0.89 = 89%

Another example: Write 1.945 as a percentage.

We multiply 1.945 by 100% , which is accomplished by moving the decimal point two places to the right and adding the percent symbol, giving us 194.5% .

Summary: 1.945 = 194.5%