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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a percentage (from Latin ''per centum'' "by a hundred") is a number or
ratio In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to ...

ratio
expressed as a
fraction A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths ...
of 100. It is often denoted using the
percent sign The percent sign (or per cent sign in British English British English (BrE) is the standard dialect of the English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon Engl ...
, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also used. A percentage is a
dimensionless number In dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantity, base quantities (such as length, mass, time, and electric curre ...
(pure number); it has no
unit of measurement A unit of measurement is a definite magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematic ...
.


Examples

For example, 45% (read as "forty-five percent") is equal to the fraction , the
ratio In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to ...

ratio
45:55 (or 45:100 when comparing to the total rather than the other portion), or 0.45. Percentages are often used to express a proportionate part of a total. (Similarly, one can also express a number as a fraction of 1000, using the term "
per mille Per mille (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the ...
" or the symbol "".)


Example 1

If 50% of the total number of students in the class are male, that means that 50 out of every 100 students are male. If there are 500 students, then 250 of them are male.


Example 2

An increase of $0.15 on a price of $2.50 is an increase by a fraction of = 0.06. Expressed as a percentage, this is a 6% increase. 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 111% or −35%, especially for
percent changeIn any quantitative science The exact sciences, sometimes called the exact mathematical sciences are those sciences "which admit of absolute precision in their results"; especially the mathematical sciences. Examples of the exact sciences are m ...
s and comparisons.


History

In
Ancient Rome In historiography Historiography is the study of the methods of historian ( 484– 425 BC) was a Greek historian who lived in the 5th century BC and one of the earliest historians whose work survives. A historian is a person who stud ...
, long before the existence of the decimal system, computations were often made in fractions in the multiples of . For example,
Augustus Caesar Augustus (23 September 63 BC19 August AD 14) was the first Roman emperor The Roman emperor was the ruler of the Roman Empire during the imperial period (starting in 27 BC). The emperors used a variety of different titles through ...

Augustus
levied a tax of on goods sold at auction known as ''
centesima rerum venalium''Centesima rerum venalium'' (literally hundredth of the value of everything sold) was a 1% tax on goods sold at auction. History Tax revenues went into a fund to pay military retirement benefits ('' aerarium militare''), along with those from a new ...
''. Computation with these fractions was equivalent to computing percentages. As denominations of money grew in the
Middle Ages In the history of Europe The history of Europe concerns itself with the discovery and collection, the study, organization and presentation and the interpretation of past events and affairs of the people of Europe since the beginning of ...
, 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.


Percent sign

The term "percent" is derived from the Latin ''per centum'', meaning "hundred" or "by the hundred". The sign for "percent" evolved by gradual contraction of the
Italian Italian may refer to: * Anything of, from, or related to the country and nation of Italy ** Italians, an ethnic group or simply a citizen of the Italian Republic ** Italian language, a Romance language *** Regional Italian, regional variants of the ...

Italian
term ''per cento'', meaning "for a hundred". The "per" was often abbreviated as "p."—eventually disappeared entirely. The "cento" was contracted to two circles separated by a horizontal line, from which the modern "%" symbol is derived.


Calculations

The percent value is computed by multiplying the numeric value of the ratio by 100. For example, to find 50 apples as a percentage of 1250 apples, one first computes the ratio = 0.04, and then multiplies by 100 to obtain 4%. The percent value can also be found by multiplying first instead of later, so in this example, the 50 would be multiplied by 100 to give 5,000, and this result would be divided by 1250 to give 4%. 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; it would literally imply division by 10,000. For example, , not , which actually is . A term such as would also be incorrect, since it would be read as 1 percent, even if the intent was to say 100%. 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. :''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 In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to ...

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
conditional probability
.


Percentage 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 increase = 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 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).


Compounding percentages

Percent changes applied sequentially ''do not add up'' 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), then the final price will be $198—''not'' the original price of $200. The reason for this 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 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
interest rate An interest rate is the amount of interest In finance Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investm ...
s, 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 10.05% per annum. It is clearer to say that the interest rate increased by 5
percentage point A percentage point or percent point is the unit (measurement), unit for the Difference (mathematics), arithmetic difference of two percentages. For example, moving up from 40 percent to 44 percent is an increase of 4 percentage points, but a 10-p ...
s (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".


Word and symbol

In
British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar and usage and is employed by a populatio ...
, ''percent'' is usually written as two words (''per cent''), although ''percentage'' and ''
percentile In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a mo ...

percentile
'' are written as one word. In
American English American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of variety (linguistics), varieties of the English language native to the United States. Currently, American English is the m ...
, ''percent'' is the most common variant (but ''
per mille Per mille (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the ...
'' is written as two words). 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 ''Hansard'' is the traditional name of the transcripts of Parliamentary debates in Britain and many Commonwealth of Nations, Commonwealth countries. It is named after Thomas Curson Hansard (1776–1833), a London printer and publisher, who was t ...
'' transcripts of British Parliamentary proceedings. The term has been attributed to
Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an appa ...

Latin
''per centum''. The concept of considering values as parts of a hundred is originally
Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...
. The symbol for percent (%) 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
ach Acetylcholine (ACh) is an organic chemical , CH4; is among the simplest organic compounds. In chemistry Chemistry is the scientific discipline involved with Chemical element, elements and chemical compound, compounds composed of atoms, mo ...
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 ''The Chicago Manual of Style'' (abbreviated in writing as ''CMOS'' or ''CMS'', or sometimes as ''Chicago'') is a style guide for American English published since 1906 by the University of Chicago Press. Its 17 editions have prescribed writi ...
''—generally state that the number and percent sign are written without any space in between. However, the
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms_and_initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wi ...
and the
ISO 31-0 ISO 31-0 is the introductory part of International Organization for Standardization, international standard ISO 31 on physical quantity, quantities and physical unit, units. It provides guidelines for using physical quantities, quantity and unit sy ...
standard require a space.


Other uses

The word "percentage" is often a
misnomer A misnomer is a name that is incorrectly or unsuitably applied. Misnomers often arise because something was named long before its correct nature was known, or because an earlier form of something has been replaced by a later form to which the nam ...
in the context of sports statistics, when the referenced number is expressed as a decimal proportion, not a percentage: "The
Phoenix Suns The Phoenix Suns are an American professional basketball team based in Phoenix, Arizona. They compete in the National Basketball Association (NBA), as a member of the league's Western Conference (NBA), Western Conference Pacific Division (NBA), P ...
'
Shaquille O'Neal Shaquille Rashaun O'Neal ( ; born March 6, 1972), known commonly as "Shaq" ( ), is an American former professional basketball Basketball is a team sport in which two teams, most commonly of five players each, opposing one another on a ...

Shaquille O'Neal
led the
NBA The National Basketball Association (NBA) is a professional basketball Professional sports, as opposed to amateur sports, are sport Sport pertains to any form of Competition, competitive physical activity or game that aims to use, m ...
with a .609
field goal percentage Field goal percentage in basketball Basketball is a team sport in which two teams, most commonly of five players each, opposing one another on a rectangular Basketball court, court, compete with the primary objective of #Shooting, shooting a ...
(FG%) during the 2008–09 season." (O'Neal made 60.9% of his shots, not 0.609%.) Likewise, the
winning percentage In sports, a winning percentage is the fraction A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain ...
of a team, the fraction of matches that the club has won, is also usually expressed as a decimal proportion; a team that has a .500 winning percentage has won 50% of their matches. The practice is probably related to the similar way that batting averages are quoted. As "percent" it is used to describe the steepness of the
slope In mathematics, the slope or gradient of a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', ...
of a
road A road is a wide way leading from one place to another, typically one with a specially prepared surface which vehicles and bikes can use. Roads consist of one or two roadway A carriageway (British English British English (BrE) is the ...

road
or
railway Rail transport (also known as train transport) is a means of transferring passengers and goods on wheeled vehicle A vehicle (from la, vehiculum) is a machine that transports people or cargo. Vehicles include wagons, bicycles, motor veh ...

railway
, formula for which is 100 ×  which could also be expressed as the
tangent In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
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 In chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during a ...
and
mole percent In chemistry, the mole fraction or molar fraction (''xi'') is defined as unit of the amount of substance, amount of a constituent (expressed in mole (unit), moles), ''ni'' divided by the total amount of all constituents in a mixture (also expressed ...
.


Related units

*
Percentage point A percentage point or percent point is the unit Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit ...
difference of 1 part in 100 *
Per mille Per mille (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of ...
(‰) 1 part in 1,000 *
Basis point A basis point (often abbreviated as bp, often pronounced as "bip" or "beep") is (a difference of) one hundredth of a percent or equivalently one percent of one percent or one ten thousandth. A very rarely used term, permyriad, means parts per te ...
(bp) difference of 1 part in 10,000 *
Permyriad A basis point (often abbreviated as bp, often pronounced as "bip" or "beep") is (a difference of) one hundredth of a percent or equivalently one percent of one percent or one ten thousandth. A very rarely used term, permyriad, means parts per te ...
(‱) 1 part in 10,000 *
Per cent mille A per cent mille or pcm is one one-thousandth of a percent. It can be thought of as a "milli-percent". It is commonly used in epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, ...
(pcm) 1 part in 100,000 *
Grade (slope) The grade (also called slope, incline, gradient, mainfall, pitch or rise) of a physical feature, landform or constructed line refers to the tangent of the angle of that surface to the horizontal. It is a special case of the slope In mathemati ...
* Centiturn


Practical applications

*
Baker percentageBaker's percentage is a notation method indicating the proportion of an ingredient relative to the flour used in a recipe when making bread Bread is a staple food prepared from a dough of flour and water Water is an Inorganic compou ...
*
Volume percent In chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds composed of atoms, mo ...


See also

* 1000 percent * Relative change and difference *
Percent differenceIn any quantitative science The exact sciences, sometimes called the exact mathematical sciences are those sciences "which admit of absolute precision in their results"; especially the mathematical sciences. Examples of the exact sciences are ma ...
* Percentage change *
Parts-per notation In science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific method, builds and Taxonomy (general), organizes knowledge in the form of Testability, testable explanations and predictio ...
*
Per-unit systemIn the power systems analysis field of electrical engineering, a per-unit system is the expression of system quantities as fractions of a defined base unit quantity. Calculations are simplified because quantities expressed as per-unit do not chang ...
*
Percent point function In probability Probability is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculu ...


References

{{Fractions and ratios 100 (number) Elementary arithmetic Fractions (mathematics)