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In mathematics, a percentage () is a number or ratio expressed as a fraction of 100. It is often Denotation using the percent sign (%), although the abbreviations pct., pct, and sometimes pc are also used. A percentage is a dimensionless number (pure number), primarily used for expressing proportions, but percent is nonetheless a unit of measurement in its orthography and usage.
(Similarly, one can also express a number as a fraction of 1,000, using the term "per mille" or the symbol "".)
While many percentage values are between 0 and 100, there is no mathematical restriction and percentages may take on other values. For example, it is common to refer to values such as 111% or −35%, especially for and comparisons.
As denominations of money grew in the Middle Ages, computations with a denominator of 100 became increasingly standard, such that from the late 15th century to the early 16th century, it became common for arithmetic texts to include such computations. Many of these texts applied these methods to profit and loss, interest rates, and the Rule of Three. By the 17th century, it was standard to quote interest rates in hundredths.
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:
Whenever communicating about a percentage, it is important to specify what it is relative to (i.e., what is the total that corresponds to 100%). The following problem illustrates this point.
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 × = 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: = or 30% of all computer science majors are female.
This example is closely related to the concept of conditional probability.
Because of the commutative property of multiplication, reversing expressions does not change the result; for example, 50% of 20 is 10, and 20% of 50 is 10.
Example:
42 kg is 7%. How much is (corresponds to) 100%?
Given are W (percentage) and p % (percentage).
We are looking for G (basic value).
multiple rearrangements result in: | simple conversion yields: | without changing the last counter is: |
| Advantage: • One formula for all tasks | Advantages: • Without a formula • Easy to change over if the size you are looking for – here G – is in the top left of the counter. | Advantages: • Without a formula • Simple rule of three – here as a chain of equations • Application for mental arithmetic |
Some other examples of percent changes:
In general, a change of percent in a quantity results in a final amount that is 100 + percent of the original amount (equivalently, (1 + 0.01) times the original amount).
In general, if an increase of percent is followed by a decrease of percent, and the initial amount was , the final amount is ; hence the net change is an overall decrease by percent of 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 , the final amount, $198, was 10% of 10%, or 1%, less than the initial amount of $200. The net change is the same for a decrease of percent, followed by an increase of percent; the final amount is .
This can be expanded for a case where one does not have the same percent change. If the initial amount leads to a percent change , and the second percent change is , then the final amount is . To change the above example, after an increase of and decrease of , the final amount, $209, is 4.5% more than the initial amount of $200.
As shown above, percent changes can be applied in any order and have the same effect.
In the case of , a very common but ambiguous way to say that an interest rate rose from 10% per annum to 15% per annum, for example, is to say that the interest rate increased by 5%, which could theoretically mean that it increased from 10% per annum to 10.5% per annum. It is clearer to say that the interest rate increased by 5 (pp). The same confusion between the different concepts of percent(age) and percentage points can potentially cause a major misunderstanding when journalists report about election results, for example, expressing both new results and differences with earlier results as percentages. For example, if a party obtains 41% of the vote and this is said to be a 2.5% increase, does that mean the earlier result was 40% (since 41 = ) or 38.5% (since 41 = )?
In financial markets, it is common to refer to an increase of one percentage point (e.g. from 3% per annum to 4% per annum) as an increase of "100 basis points".
In the early 20th century, there was a dotted abbreviation form " per cent.", as opposed to " per cent". The form " per cent." is still in use in 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. The term has been attributed to Latin per centum. The percent sign (%) evolved from a symbol abbreviating the Italian per cento. In some other languages, the form procent or prosent is used instead. Some languages use both a word derived from percent and an expression in that language meaning the same thing, e.g. Romanian procent and la sută (thus, 10% can be read or sometimes written ten for each hundred, similarly with the English one out of ten). Other abbreviations are rarer, but sometimes seen.
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: "Ten percent of all writers love style guides." Decimals are also to be used instead of fractions, as in "3.5 percent of the gain" and not " percent of the gain". However the titles of bonds issued by governments and other issuers use the fractional form, e.g. "% Unsecured Loan Stock 2032 Series 2". (When interest rates are very low, the number 0 is included if the interest rate is less than 1%, e.g. "% Treasury Stock", not "% Treasury Stock".) It is also widely accepted to use the percent symbol (%) in tabular and graphic material.
In line with common English practice, style guides—such as The Chicago Manual of Style—generally state that the number and percent sign are written without any space in between.
As "percent" it is used to describe the grade or slope, the steepness of a road or Rail tracks, formula for which is 100 × which could also be expressed as the tangent of the angle of inclination times 100. This is the ratio of distances a vehicle would advance vertically and horizontally, respectively, when going up- or downhill, expressed in percent.
Percentage is also used to express composition of a mixture by mass percent and mole percent.
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