View source for Percentage

{{short description|Number or ratio expressed as a fraction of 100}}
{{Redirect|Per cent|the unit of currency|cent (currency)}}
{{pp-semi-indef}}
{{Use dmy dates|date=October 2020}}
[[File:Web-browser usage on Wikimedia.svg|thumb|A [[pie chart]] showing the percentage by web browser visiting [[Wikimedia Foundation|Wikimedia]] sites (April 2009 to 2012)]]

In [[mathematics]], a '''percentage''', '''percent''', or '''per cent''' ({{etymology|la|{{wikt-lang|la|per centum}}|by a hundred}}) is a number or [[ratio]] expressed as a [[fraction (mathematics)|fraction]] of 100. It is often [[Denotation|denoted]] using the ''[[percent sign]]'' (%),<ref>{{cite web |title=Introduction to Percents |url=https://mathsisfun.com/percentage.html |website=mathsisfun.com |access-date=2020-08-28}}</ref> although the abbreviations ''pct.'', ''pct'', and sometimes ''pc'' are also used.<ref>{{cite news |last=Dakers|first=Marion |date=7 January 2015 |url=https://www.telegraph.co.uk/finance/economics/11329769/Eurozone-officially-falls-into-deflation-piling-pressure-on-ECB.html |title=Eurozone Officially Falls into Deflation, Piling Pressure on ECB |newspaper=[[The Daily Telegraph]] |access-date=2019-12-27}}</ref> A percentage is a [[dimensionless quantity|dimensionless number]] (pure number), primarily used for expressing proportions, but percent is nonetheless a [[unit of measurement]] in its orthography and usage.<ref>{{cite book |last1=Mattock |first1=Peter |title=Conceptual Maths: Teaching 'about' (rather than just 'how to do') mathematics in schools |date=5 January 2023 |publisher=Crown House Publishing Ltd |isbn=978-1-78583-618-3 |page=269 |url=https://books.google.com/books?id=FZVfEAAAQBAJ&pg=PT269 |language=en}}</ref>

== Examples  ==
For example, 45% (read as "forty-five percent") is equal to the fraction {{sfrac|45|100}}, 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]]" or the symbol "{{big|‰}}".)

===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 {{sfrac|0.15|2.50}} = 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.<ref>{{citation|first1=Jeffrey|last1=Bennett|first2=William|last2=Briggs|title=Using and Understanding Mathematics / A Quantitative Reasoning Approach|edition=3rd|year=2005|publisher=Pearson Addison Wesley|isbn=0-321-22773-5|page=134}}</ref> For example, it is common to refer to values such as 111% or −35%, especially for [[percent change]]s and comparisons.

== History ==
In [[Ancient Rome]], long before the existence of the decimal system, computations were often made in fractions in the multiples of {{sfrac|100}}. For example, [[Augustus]] levied a tax of {{sfrac|100}} 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]], 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 (mathematics)|Rule of Three]]. By the 17th century, it was standard to quote interest rates in hundredths.<ref>{{cite book|last=Smith|first=D.E.|title=History of Mathematics|isbn=0-486-20430-8
|publisher=Courier Dover Publications|orig-year=1951|year=1958|volume=2|pages=247–249}}</ref>

==Percent sign==
{{main|Percent sign}}
[[Image:Unicode 0x0025.svg|thumb|upright|A percent sign]]
The term "percent" is derived from the Latin ''per centum'', meaning "hundred" or "by the hundred".<ref>American Heritage Dictionary of the English Language, 3rd ed. (1992) Houghton Mifflin</ref><ref>{{Cite web|title=Definition of PERCENT|url=https://www.merriam-webster.com/dictionary/percent|access-date=28 August 2020|website=www.merriam-webster.com|language=en}}</ref> 
The [[percentage sign|sign for "percent"]] evolved by gradual contraction of the [[Italian language|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.<ref>Smith p. 250</ref>

== 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 1,250 apples, one first computes the ratio {{sfrac|50|1250}} = 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 1,250 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:
:{{math|1= {{sfrac|50|100}} × {{sfrac|40|100}} = 0.50 × 0.40 = 0.20 = {{sfrac|20|100}} = 20%.}}
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, {{nowrap|1= 25% = {{sfrac|25|100}} = 0.25}}, not {{sfrac|25%|100}}, which actually is {{nowrap|1= {{sfrac|{{frac|25|100}}|100}} = 0.0025}}. A term such as {{nowrap|{{sfrac|100|100}}%}} 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.

{{blockquote|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 {{sfrac|60|100}} × {{sfrac|5|100}} = {{sfrac|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: {{sfrac|3%|10%}} = {{sfrac|30|100}} 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.

== 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&nbsp;kg is 7%. How much is (corresponds to) 100%?<br />Given are ''W'' (percentage) and ''p'' % (percentage).<br />We are looking for ''G'' (basic value).
{| class="wikitable"
|-
!class="backgroundcolor3"| With general [[formula]]
!class="backgroundcolor6"| With own ratio equation ([[Quotient#Proportions|Proportion]])
!class="backgroundcolor4"| With “What is 1%?” ([[Rule of 3 (mathematics)|Rule of 3]]) <math>\frac{p\,\%}{42\,\text{kg}}=\frac{100\,\%}{7\ \%}</math>
|-
| <math>\frac{p\,\%}{100\,\%}=\frac{W}{G}</math><br /><br />multiple rearrangements result in:<br /><br /><math>G=\frac{W}{p\,\%}\cdot{100\,\%}</math><br /><math>G=\frac{42\,\text{ kg}}{7\,\%}\cdot{100\,\%}=600\,\text{kg}</math>
|<math>\frac{G}{42\,\text{kg}}=\frac{100\,\%}{7\,\%}</math><br /><br />simple conversion yields:<br /><br /><math>G=\frac{42\,\text{kg}}{7\,\%}\cdot{100\,\%}=600\,\text{ kg}</math>
| <math>\frac{42\,\text{kg}:{ \color{red}7}}{7\,\%:{ \color{red}7}}=\frac{6\,\text{ kg}}{1\,\%}=\frac{6\,\text{kg}\cdot { \color{red}100}}{1\,\%\cdot { \color{red}100}} </math><br /><br />without changing the last counter is:<br /><br /><math>G=6\,\text{kg}\cdot 100=600\,\text{ kg}</math>
|-
|class="backgroundcolor3"| Advantage:<br />•&nbsp;One formula for all tasks<br />
|class="backgroundcolor6"| Advantages:<br />•&nbsp;Without a formula<br />•&nbsp;Easy to change over if the size you are looking for –&nbsp;here ''G''&nbsp;– is in the top left of the counter.<br />
|class="backgroundcolor4"| Advantages:<br />•&nbsp;Without a formula<br />•&nbsp;Simple rule of three&nbsp;– here as a chain of equations<br />•&nbsp;Application for mental arithmetic
|}

== Percentage increase and decrease ==
{{See also|Relative change and difference}}
[[File:Shop placard showing 20% reduction.JPG|thumb|Placard outside a shop in Bordeaux advertising 20% decrease in the price of the second perfume purchased.]]
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 difference#Percentage change|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 {{math|''x''}} percent in a quantity results in a final amount that is 100&nbsp;+&nbsp;{{math|''x''}} percent of the original amount (equivalently, (1&nbsp;+&nbsp;0.01{{math|''x''}}) 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 initial values ($200 and $220, respectively), and thus do not "cancel out".

In general, if an increase of {{math|''x''}} percent is followed by a decrease of {{math|''x''}} percent, and the initial amount was {{math|''p''}}, the final amount is {{nowrap|1= {{math|''p''}}(1 + 0.01{{math|''x''}})(1 − 0.01{{math|''x''}}) = {{math|''p''}}(1 − (0.01{{math|''x''}}){{sup|2}})}}; hence the net change is an overall decrease by {{math|''x''}} percent ''of'' {{math|''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 {{nowrap|1= {{math|''x''}} = 10 percent}}, 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 {{math|''x''}} percent, followed by an increase of {{math|''x''}} percent; the final amount is {{nowrap|1= {{math|''p''}}(1 - 0.01{{math|''x''}})(1 + 0.01{{math|''x''}}) = {{math|''p''}}(1 − (0.01{{math|''x''}}){{sup|2}})}}.

This can be expanded for a case where one does not have the same percent change. If the initial amount {{math|''p''}} leads to a percent change {{math|''x''}}, and the second percent change is {{math|''y''}}, then the final amount is {{nowrap|1= {{math|''p''}}(1 + 0.01{{math|''x''}})(1 + 0.01{{math|''y''}})}}.  To change the above example, after an increase of {{nowrap|1= {{math|''x''}} = 10 percent}} and decrease of {{nowrap|1= {{math|''y''}} = −5 percent}}, 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]]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 10.5% per annum. It is clearer to say that the interest rate increased by 5 [[percentage point]]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 = <span style="padding-right:0.1em;">{{nowrap|40 × (1 + {{sfrac|2.5|100}})}}</span>) or 38.5% (since 41 = {{nowrap|38.5 + 2.5}})?

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 ==
{{main|Percent sign}}

In [[Commonwealth English|most forms of English]], ''percent'' is usually written as two words (''per cent''), although ''percentage'' and ''[[percentile]]'' are written as one word.<ref>{{cite web|url=http://www.wsu.edu/~brians/errors/percent1.html|first=Paul|last=Brians|title=Percent/per cent|work=Common Errors in English Usage|publisher=Washington State University|access-date=22 November 2010}}</ref> In [[American English]], ''percent'' is the most common variant<ref>{{cite book|title=New Zealand English Grammar, Fact Or Fiction?: A Corpus-based Study in Morphosyntactic Variation|volume=23|series=Varieties of English around the world: General series|first=Marianne|last=Hundt|publisher=John Benjamins Publishing|year=1998|isbn=9789027248817|page=19|url=https://books.google.com/books?id=6U3ysrgV4_8C&pg=PA19|quote=A new regional difference between BrE and AmE in this area has evolved during the last thirty years. While in 1961, it was still fairly common for American journalists to write either per cent or (less commonly) percent, the latter spelling variant now seems to be the only possible one.}}</ref> (but ''[[per mille]]'' 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]]'' transcripts of British Parliamentary proceedings. The term has been attributed to [[Latin]] ''per centum''.<ref>{{OED|Percent}}</ref> The [[percent sign|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 [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&nbsp;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&nbsp;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 "{{frac|3|1|2}} percent of the gain". However the titles of bonds issued by governments and other issuers use the fractional form, e.g. "{{frac|3|1|2}}% 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. "{{frac|0|3|4}}% Treasury Stock", not "{{frac|3|4}}% 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.<ref>
{{cite web
 | title = The Chicago Manual of Style
 | publisher = [[University of Chicago Press]]
 | year = 2003
 | url = http://www.chicagomanualofstyle.org/
 | access-date = 5 January 2007}}
</ref>
However, the [[International System of Units]] and the [[ISO 31-0]] standard require a space.<ref>
{{cite web
 | title = The International System of Units
 | publisher = [[International Bureau of Weights and Measures]]
 | year = 2006
 | url = http://www.bipm.org/utils/common/pdf/si_brochure_8.pdf
 | access-date = 6 August 2007}}
</ref><ref>
{{cite web
 | title = ISO 31-0 — Quantities and units – Part 0: General principles
 | publisher = [[International Organization for Standardization]]
 | date = 22 December 1999
 | url = http://www.iso.org/iso/en/CatalogueDetailPage.CatalogueDetail?CSNUMBER=3621
 | access-date = 5 January 2007}}
</ref>

== Other uses ==
{{CSS image crop|Image=Top of steep hill leading to Okehampton Army Camp - geograph.org.uk - 38028.jpg|bSize=450|cWidth=300|cHeight=210|oTop=60|oLeft=60|Location=right|Description=Percent used to indicate a road's steepness down.}}
The word "percentage" is often a [[misnomer]] in the context of sports statistics, when the referenced number is expressed as a decimal proportion, not a percentage: "The [[Phoenix Suns]]' [[Shaquille O'Neal]] led the [[NBA]] 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]] 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 average (baseball)|batting averages]] are quoted.

As "percent" it is used to describe the [[grade (slope)|grade]] or slope, the steepness of a [[road]] or [[Rail tracks|railway]], formula for which is 100&nbsp;×&nbsp;{{sfrac|rise|run}} which could also be expressed as the [[Tangent (trigonometry)|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]].

== Related units ==
{{visualisation_parts_per.svg}}
* [[Percentage point]] difference of 1 part in 100
* [[permille|Per mille]] (‰) 1 part in 1,000
* [[Basis point]] (bp) difference of 1 part in 10,000
* [[Permyriad]] (‱) 1 part in 10,000
* [[Per cent mille]] (pcm) 1 part in 100,000
* [[Centiturn]]

== Practical applications ==
* [[Baker percentage]]
* [[Volume percent]]

==See also==
*[[1000 percent]]
*[[Relative change and difference]]
*[[Percent difference]]
*[[Percentage change]]
*[[Parts-per notation]]
*[[Per-unit system]]
*[[Percent point function]]

==References==
{{Reflist}}

==External links==
{{Commons category}}
* {{Wiktionary-inline}}

{{Fractions and ratios}}
{{Authority control}}

[[Category:100 (number)]]
[[Category:Elementary arithmetic]]
[[Category:Percentages| ]]