Percentage

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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a percentage (from la, per centum, "by a hundred") is a number or
ratio In mathematics, a ratio shows 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 the ...
expressed as a
fraction A fraction (from la, fractus, "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 ...
of 100. It is often denoted using the
percent sign The percent sign (sometimes per cent sign in British English) is the symbol used to indicate a percentage, a number or ratio as a fraction (mathematics), fraction of 100. Related signs include the permille (per thousand) sign and the Basis p ...
, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also used. A percentage is a
dimensionless number A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no Dimension (physics), physical dimension is assigned, with a corresponding International Sys ...
(pure number); it has no
unit of measurement A unit of measurement is a definite magnitude of a quantity Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", ...
.

# Examples

For example, 45% (read as "forty-five per cent") is equal to the fraction , the
ratio In mathematics, a ratio shows 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 the ...
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 1,000, using the term "
per mille Per mille (from New Latin, Latin , "in each thousand") is an expression that means parts per thousand. Other recognised spellings include per mil, per mill, permil, permill, or permille. The associated sign is written , which looks like a ...
" 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 change In any quantitative science, the terms relative change and relative difference are used to compare two quantities while taking into account the "sizes" of the things being compared, i.e. dividing by a ''standard'' or ''reference'' or ''starting'' va ...
s and comparisons.

# History

In
Ancient Rome In modern historiography, ancient Rome refers to Roman civilisation from the founding of the city of Rome in the 8th century BC to the collapse of the Western Roman Empire in the 5th century AD. It encompasses the Roman Kingdom (753–509 ...
, long before the existence of the decimal system, computations were often made in fractions in the multiples of . For example,
Augustus Caesar Augustus (born Gaius Octavius; 23 September 63 BC – 19 August AD 14), also known as Octavian, was the first Roman emperor; he reigned from 27 BC until his death in AD 14. He is known for being the founder of the Roman Pri ...
levied a tax of on goods sold at auction known as '' centesima rerum venalium''. Computation with these fractions was equivalent to computing percentages. As denominations of money grew in the
Middle Ages In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the Post-classical, post-classical period of World history (field), global history. It began with t ...
, 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 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 shows 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 the ...
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 In probability theory, conditional probability is a measure of the probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is ...
.

# Variants of the percentage calculation

The calculation of percentages is carried out and taught in different ways depending on the prerequisites and requirements. In this way, the usual formulas can be obtained with proportions, which saves them from having to remember them. In so-called mental arithmetic, the intermediary question is usually asked what 100% or 1% is (corresponds to). 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).

# 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 due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...
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 between two percentages. For example, moving up from 40 percent to 44 percent is an increase of 4 percentage points, but a ...
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 most forms of English, ''percent'' is usually written as two words (''per cent''), although ''percentage'' and ''
percentile In statistics, a ''k''-th percentile (percentile score or centile) is a Raw score, score ''below which'' a given percentage ''k'' of scores in its frequency distribution falls (exclusive definition) or a score ''at or below which'' a given percen ...
'' are written as one word. In
American English American English, 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. English is the Languages of the United States, most widely spoken lan ...
, ''percent'' is the most common variant (but ''
per mille Per mille (from New Latin, Latin , "in each thousand") is an expression that means parts per thousand. Other recognised spellings include per mil, per mill, permil, permill, or permille. The associated sign is written , which looks like a ...
'' 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 countries. It is named after Thomas Curson Hansard (1776–1833), a London printer and publisher, who was the first official prin ...
'' transcripts of British Parliamentary proceedings. The term has been attributed to
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
''per centum''. The concept of considering values as parts of a hundred is originally
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
. 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 achhundred'', 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 American English, sometimes called United States English or U.S. English, is the set of ...
''—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 wid ...
and the ISO 31-0 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 name ...
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 player who is a sports analyst on the television program ''Inside the NBA''. O'Neal is regarded as one of the greates ...
led the
NBA The National Basketball Association (NBA) is a professional basketball In professional sports, as opposed to amateur sports, participants receive payment for their performance. Professionalism in sport has come to the fore through a ...
with a .609 field goal percentage (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 (mathematics), fraction of games or matches a team or individual has won. The statistic is commonly used in Standings, standings or rankings to compare teams or individuals. It is defined as wins di ...
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 most often refers to: * Line (geometry) In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional object ...
of a
road A road is a linear way for the conveyance of traffic that mostly has an road surface, improved surface for use by vehicles (motorized and non-motorized) and pedestrians. Unlike streets, the main function of roads is transportation. There are ro ...
or
railway Rail transport (also known as train transport) is a means of transport Transport (in British English), or transportation (in American English), is the intentional Motion, movement of humans, animals, and cargo, goods from one location t ...
, formula for which is 100 ×  which could also be expressed as the
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given Point (geometry), point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitesimal, infinitely ...
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.

# Related units

*
Percentage point A percentage point or percent point is the unit (measurement), unit for the Difference (mathematics), arithmetic difference between two percentages. For example, moving up from 40 percent to 44 percent is an increase of 4 percentage points, but a ...
difference of 1 part in 100 *
Per mille Per mille (from New Latin, Latin , "in each thousand") is an expression that means parts per thousand. Other recognised spellings include per mil, per mill, permil, permill, or permille. The associated sign is written , which looks like a ...
(‰) 1 part in 1,000 *
Basis point A basis point (often abbreviated as bp, often pronounced as "bip" or "beep") is one hundredth of 1 percentage point. The related term '' permyriad'' means one hundredth of 1 percent. Changes of interest rates are often stated in basis points. If ...
(bp) difference of 1 part in 10,000 *
Permyriad A basis point (often abbreviated as bp, often pronounced as "bip" or "beep") is one hundredth of 1 percentage point. The related term ''#Permyriad, permyriad'' means one hundredth of 1 percent. Changes of interest rates are often stated in basis ...
(‱) 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, and in nuclear reactor engineering as a unit of reactivity. Epidemiology Statistics of Crime statistics, c ...
(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 plane, horizontal. It is a special case of the slo ...
*
Centiturn A turn is a unit of plane angle measurement equal to  radians, 360 degree (angle), degrees or 400 gradians. Subdivisions of a turn include half-turns, quarter-turns, centiturns, milliturns, etc. The closely related terms ''cycl ...

# Practical applications

*
Baker percentage Baker's percentage is a notation method indicating the proportion of an ingredient relative to the flour used in a recipe when making breads, cakes, muffins, and other baked goods. It is also referred to as baker's math, and may be indicated by a ph ...
* Volume percent

*
1000 percent "1000 percent" or "1000%" in a literal sense means to multiply by 10. In American English it is used as a metaphor meaning very high emphasis, or enthusiastic support. It was used in the 1972 United States presidential election, 1972 U.S. presiden ...
*
Relative change and difference In any quantitative science, the terms relative change and relative difference are used to compare two quantities while taking into account the "sizes" of the things being compared, i.e. dividing by a ''standard'' or ''reference'' or ''starting'' ...
*
Percent difference In any quantitative science, the terms relative change and relative difference are used to compare two quantities while taking into account the "sizes" of the things being compared, i.e. dividing by a ''standard'' or ''reference'' or ''starting'' va ...
* Percentage change *
Parts-per notation In science and engineering, the parts-per notation is a set of pseudo-units to describe small values of miscellaneous dimensionless quantity, dimensionless quantities, e.g. mole fraction or mass fraction (chemistry), mass fraction. Since these f ...
*
Per-unit system In the power systems, 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-uni ...
*
Percent point function In probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a nu ...