Definitions

# Percentage

[per-sen-tij]

In mathematics, a percentage is a way of expressing a number as a fraction of 100 (per cent meaning "per hundred"). It is often denoted using the percent sign, "%". For example, 45% (read as "forty-five percent") is equal to 45 / 100, or 0.45.

Percentages are used to express how large one quantity is relative to another quantity. The first quantity usually represents a part of, or a change in, the second quantity, which should be greater than zero. For example, an increase of \$ 0.15 on a price of \$ 2.50 is an increase by a fraction of 0.15 / 2.50 = 0.06. Expressed as a percentage, this is therefore a 6% increase.

Although percentages are usually used to express numbers between zero and one, any dimensionless proportionality can be expressed as a percentage. For instance, 111% is 1.11 and −0.35% is −0.0035.

## Proportions

Percentages are correctly used to express fractions of the total. For example, 25% means 25 / 100, or one quarter, of some total.

Percentages larger than 100%, such as 101% and 110%, may be used as a literary paradox to express motivation and exceeding of expectations. For example, "We expect you to give 110% [of your ability]"; however, there are cases when percentages over 100 can be meant literally (such as "a family must earn at least 125% over the poverty line to sponsor a spouse visa").

## Calculations

The fundamental concept to remember when performing calculations with percentages is that the percent symbol can be treated as being equivalent to the pure number constant $1/100=0.01$. For example, 35% of 300 can be written as .

To find the percentage of a single unit in a whole of N units, divide 100% by N. For instance, if you have 1250 apples, and you want to find out what percentage of these 1250 apples a single apple represents, provides the answer of 0.08%.

To calculate a percentage of a percentage, convert both percentages to fractions of 100, or to decimals, and multiply them. For example, 50% of 40% is:

It is not correct to divide by 100 and use the percent sign at the same time. (E.g. , not , which is actually .)

### An example problem

Whenever we talk about a percentage, it is important to specify what it is relative to, i.e. what the total is that corresponds to 100%. The following problem illustrates this point.

In a certain college 60% of all students are female, and 10% of all students are computer science majors. If 5% of female students are computer science majors, what percentage of computer science majors are female?

We are asked to compute the ratio of female computer science majors to all computer science majors. We know that 60% of all students are female, and among these 5% are computer science majors, so we conclude that (60 / 100) × (5/100) = 3/100 or 3% of all students are female computer science majors. Dividing this by the 10% of all students that are computer science majors, we arrive at the answer: 3% / 10% = 30 / 100 or 30% of all computer science majors are female.

This example is closely related to the concept of conditional probability.

Here are other examples:

1. What is 200% of 30?
2. :Answer: 200% × 30 = (200 / 100) × 30 = 60.
3. :
4. What is 13% of 98?
5. :Answer: 13% × 98 = (13 / 100) × 98 = 12.74.
6. :
7. 60% of all university students are male. There are 2400 male students. How many students are in the university?
8. :Answer: 2400 = 60% × X, therefore X = (2400 / (60 / 100)) = 4000.
9. :
10. There are 300 cats in the village, and 75 of them are black. What is the percentage of black cats in that village?
11. :Answer: 75 = X% × 300 = (X / 100) × 300, so X = (75 / 300) × 100 = 25, and therefore X% = 25%.
12. The number of students at the university increased to 4620, compared to last year's 4125, an absolute increase of 495 students. What is the percentual increase?
13. :Answer: 495 = X% × 4125 = (X / 100) × 4125, so X = (495 / 4125) × 100 = 12, and therefore X% = 12%.

## Percent increase and decrease

Due to inconsistent usage, it is not always clear from the context what a percentage is relative to. When speaking of a "10% rise" or a "10% fall" in a quantity, the usual interpretation is that this is relative to the initial value of that quantity. For example, if an item is initially priced at \$200 and the price rises 10% (an increase of \$20), the new price will be \$220. Note that this final price is 110% of the initial price (100% + 10% = 110%).

Some other examples of percent changes:

• An increase of 100% in a quantity means that the final amount is 200% of the initial amount (100% of initial + 100% of initial = 200% of initial); in other words, the quantity has doubled.
• An increase of 800% means the final amount is 9 times the original (100% + 800% = 900% = 9 times as large).
• A decrease of 60% means the final amount is 40% of the original (100% − 60% = 40%).
• A decrease of 100% means the final amount is zero (100% − 100% = 0%).

In general, a change of $x$ percent in a quantity results in a final amount that is $100+x$ percent of the original amount (equivalently, $1+0.01x$ times the original amount).

It is important to understand that percent changes, as they have been discussed here, do not add in the usual way. For example, if the 10% increase in price considered earlier (on the \$200 item, raising its price to \$220) is followed by a 10% decrease in the price (a decrease of \$22), the final price will be \$198, not the original price of \$200.

The reason for the apparent discrepancy is that the two percent changes (+10% and −10%) are measured relative to different quantities (\$200 and \$220, respectively), and thus do not "cancel out".

In general, if an increase of $x$ percent is followed by a decrease of $x$ percent, the final amount is $\left(1+0.01x\right)\left(1-0.01x\right)=1-\left(0.01x\right)^2$ times the initial amount — thus the net change is an overall decrease by $x$ percent of $x$ percent (the square of the original percent change when expressed as a decimal number).

Thus, in the above example, after an increase and decrease of $x=10$ percent, the final amount, \$198, was 10% of 10%, or 1%, less than the initial amount of \$200.

In the case of interest rates, it is a common practice to state the percent change differently. If an interest rate rises from 10% to 15%, for example, it is typical to say, "The interest rate increased by 5%" — rather than by 50%, which would be correct when measured as a percentage of the initial rate (i.e., from 0.10 to 0.15 is an increase of 50%). Such ambiguity can be avoided by using the term "percentage points". In the previous example, the interest rate "increased by 5 percentage points" from 10% to 15%. If the rate then drops by 5 percentage points, it will return to the initial rate of 10%, as expected.

## Word and symbol

In British English, percent is usually written as two words (per cent, although percentage and percentile are written as one word). In American English, percent is the most common variant (but cf. per mille written as two words). In EU context the word is always spelled out in one word percent, despite the fact that they usually prefer British spelling, which may be an indication that the form is becoming prevalent in British spelling as well. In the early part of the twentieth century, there was a dotted abbreviation form "per cent.", as opposed to "per cent". The form "per cent." is still in use as a part of the highly formal language found in certain documents like commercial loan agreements (particularly those subject to, or inspired by, common law), as well as in the Hansard transcripts of British Parliamentary proceedings. While the term has been attributed to Latin per centum, this is a pseudo-Latin construction and the term was likely originally adopted from Italian per cento or French pour cent. The concept of considering values as parts of a hundred is originally Greek. The symbol for percent (%) evolved from a symbol abbreviating the Italian per cento.

Grammar and style guides often differ as to how percentages are to be written. For instance, it is commonly suggested that the word percent (or per cent) be spelled out in all texts, as in "1 percent" and not "1%." Other guides prefer the word to be written out in humanistic texts, but the symbol to be used in scientific texts. Most guides agree that they always be written with a numeral, as in "5 percent" and not "five percent," the only exception being at the beginning of a sentence: "Ninety percent of all writers hate style guides." Decimals are also to be used instead of fractions, as in "3.5 percent of the gain" and not "3 ½ percent of the gain." It is also widely accepted to use the percent symbol (%) in tabular and graphic material. Variations of practically all of these rules may be encountered, including in this article; the only really fast rule is to be consistent. It is important to know what method of solving the problem you would use.

There is no consensus as to whether a space should be included between the number and percent sign in English. Style guides – such as the Chicago Manual of Style – commonly prescribe to write the number and percent sign without any space in between. The International System of Units and the ISO 31-0 standard, on the other hand, require a space.