Percentage

A percentage is a way of expressing numbers as fractions of 100 and is often denoted using the percent sign, "%". For example, "45.1%" (read as "forty five point one percent") is equal to 0.451.

Percentages are used to express how large one quantity is in terms of another quantity. The first quantity is then usually a part of or a change in the second quantity. 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 an increase by 6%.

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 often used to express fractions of the total. By definition, all fractions of the total are less than 100%. For example, 25% means "one quarter". In this meaning, percentages larger than 100%, such as 101% and 110%, are used as a literary paradox to express motivation and exceeding of expectations. For example, "We expect you to give 110% [of your ability]".

Changes
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, a 10% increase on an item initially priced at $200 is $20, giving a new price of $220; to many people, any other usage is incorrect.

In the case of interest rates, however, it is a common practice to use the percent change differently: suppose that an initial interest rate is given as a percentage like 10%. Suppose the interest rate rises to 15%. This could be described as a 50% increase, measuring the increase relative to the initial value of the interest rate. However, many people say in practice "The interest rate has risen by 5%". To counter this confusion, the unit "percentage points" is sometimes used when referring to differences of percentages. So, in the previous example, "The interest rate has increased by 5 percentage points" would be an unambiguous expression that the rate is now 15%.

With changes, percentage can be of any positive value. For example, a 100% growth is synonymous with doubling; a growth of 100% starting from 200 units is 200 units, increasing the total to 400.

Cancellations
A common error when using percentages is to imagine that a percentage increase is cancelled out when followed by the same percentage decrease. A 50% increase from 100 is 100 + 50, or 150. A 50% reduction from 150 is 150 – 75, or 75. The end result is smaller than the 100 we started out with. This phenomenon is due to the change in the "initial" value after the first calculation. In this example, the first initial value is 100, but the second is 150.

In general, the net effect is:
 * (1 + x) (1 – x) = 1 – x2,

that is a net decrease proportional to the square of the percentage change.

To use a specific example, stock brokers came to understand that even if a stock has sunk 99%, it can nevertheless still sink another 99%. Also, if a stock rises by a large percentage, the trader still loses all of the stock's value if the stock subsequently drops 100%, meaning it has a zero value.

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 $$35%*300=35*0.01*300=105$$.

To find the percentage of a single unit in the whole, use 100/whole. For instance, if you have 1300 apples, and you want to find out what percentage of the 1300 apples a single apple represents, 100/1300 would provide 0.0769%

To calculate a percentage of a percentage, for instance if you wish to find how much 50% of 40% is, one method is to multiply the 50 by .01, 50*.01 is .50. Then, 40*.50 is 20, or 20%. 50% of a 40% of the whole is 20% of the whole.

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 females 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 .6 × .05 = .03 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% = .3 or 30% of all computer science majors are female.

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.

Related units

 * Percentage point
 * Per mille (‰) 1 part in 1,000
 * Basis point (‱) 1 part in 10,000
 * Parts per million (ppm)
 * Parts per billion (ppb)
 * Parts per trillion (ppt)
 * Baker percentage
 * Concentration