# Percentage

A percentage is a way of expressing a number as a fraction of 100: a percent is a ratio whose denominator is 100, written with the symbol %<sup>[1](https://openstax.org/books/prealgebra/pages/6-1-understand-percent)</sup>. The word comes from the Latin phrase *per centum*, "by the hundred"<sup>[2](https://openstax.org/books/contemporary-mathematics/pages/6-1-understanding-percent)</sup>. Because every percentage is measured against the same total of 100, percentages provide a common scale on which proportions drawn from different totals can be compared directly<sup>[3](https://math.libretexts.org/Courses/Cosumnes_River_College/Corequisite_Codex/02%3A_Arithmetic/2.13%3A_Percents)</sup>. They are among the numerical formats most commonly encountered in adult life, describing prices, taxes, interest rates, voting preferences, and fractions of populations<sup>[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC5784045/)</sup>.

| Key fact | Detail |
|---|---|
| Definition | A percent is a ratio whose denominator is 100; 100% equals one whole<sup>[1](https://openstax.org/books/prealgebra/pages/6-1-understand-percent)</sup> |
| Core formula | part = percent × total (amount = percent × base), with the percent expressed in decimal form<sup>[2](https://openstax.org/books/contemporary-mathematics/pages/6-1-understanding-percent)</sup> |
| Conversions | 6% = 6/100 = 0.06; converting a decimal to a percent moves the decimal point two places right<sup>[1](https://openstax.org/books/prealgebra/pages/6-1-understand-percent)</sup> |
| Percentage point | The unit for the difference between two percentages, abbreviated pp; a rate rising from 10% to 12% rises by 2 percentage points<sup>[5](https://ec.europa.eu/eurostat/statistics-explained/SEPDF/cache/42722.pdf?v=2882788016648030)</sup> |
| Basis point | One percentage point equals 100 basis points; a move from 4.00% to 4.25% is 25 basis points<sup>[6](https://useconomicdata.com/learn/percentage-points/)</sup> |
| Symbol origin | The % symbol evolved from a symbol introduced in an anonymous Italian manuscript of about 1425<sup>[7](https://mathshistory.st-andrews.ac.uk//Miller/mathsym/fractions/)</sup> |
| Common error | In a study of 1,629 undergraduates, roughly half the responses to two-step percentage change questions were erroneous, even with a $100 base and 50% changes<sup>[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC5784045/)</sup> |

## Definition and notation

The word "percent" is really two words, "per cent," meaning per one hundred<sup>[1](https://openstax.org/books/prealgebra/pages/6-1-understand-percent)</sup>. Thus 25% means the ratio 25/100, 3% means 3/100, and 100% means 100/100, which equals 1, so 100% represents the whole<sup>[1](https://openstax.org/books/prealgebra/pages/6-1-understand-percent)</sup>. The [Australian Bureau of Statistics](https://www.edgepedia.org/australian-bureau-of-statistics) defines a percentage as a value expressed in relation to a whole population as a fraction of one hundred, calculated by dividing the number of times a value is observed by the total number of observations and multiplying by 100<sup>[8](https://www.abs.gov.au/statistics/understanding-statistics/statistical-terms-and-concepts/describing-frequencies)</sup>.

The symbol % itself has a traceable history. According to the historian of notation Florian Cajori, whose *A History of Mathematical Notations* traces the sign through manuscripts, an unknown Italian writer of about 1425 used a symbol which, by natural stages, developed into the modern %<sup>[9](https://hlevkin.com/hlevkin/90MathPhysBioBooks/mathHistory/Cajori_Florian_A_History_of_Mathematical_Notations_2_Vols.pdf)</sup>. The New Zealand Maths curriculum resource describes the symbol as derived from the vinculum of a fraction, /, and the two zeros from 100<sup>[10](https://meaningfulmaths.nt.edu.au/mmws/nz/resource/beginning-percentages.html)</sup>.

## Conversions

Percentages, fractions, and decimals are interchangeable representations of the same quantity. To convert a percent to a decimal, write it as a ratio with denominator 100 and divide: 6% becomes 0.06, 78% becomes 0.78, 135% becomes 1.35, and 12.5% becomes 0.125<sup>[1](https://openstax.org/books/prealgebra/pages/6-1-understand-percent)</sup>. The pattern works in reverse: converting a decimal to a percent moves the decimal point two places to the right and adds the percent sign, so 0.05 becomes 5% and 1.05 becomes 105%<sup>[1](https://openstax.org/books/prealgebra/pages/6-1-understand-percent)</sup>. To convert a fraction to a percent, first change the fraction to a decimal, then convert that decimal to a percent; 3/4 becomes 0.75 and then 75%<sup>[1](https://openstax.org/books/prealgebra/pages/6-1-understand-percent)</sup>.

Percentages above 100% are meaningful whenever the part exceeds the usual whole. The OpenStax text computes 156% of 720 as 1,123.2, and the New Zealand resource notes that with 12 girls and 18 boys, the number of boys is 150% of the number of girls<sup>[2](https://openstax.org/books/contemporary-mathematics/pages/6-1-understanding-percent)</sup><sup> • </sup><sup>[10](https://meaningfulmaths.nt.edu.au/mmws/nz/resource/beginning-percentages.html)</sup>.

## The three-value model

Most percentage problems involve three quantities: the base (or total), the percent, and the part (or amount). The formula relating them is \( \text{part} = \text{percent} \times \text{total} \), with the percent expressed in decimal form<sup>[2](https://openstax.org/books/contemporary-mathematics/pages/6-1-understanding-percent)</sup>. Knowing any two values gives the third: 70% of 3,500 is 0.70 × 3,500 = 2,450; if 35% of a total is 70, the total is 70 ÷ 0.35 = 200; and if 175 out of 500 is the part, the percent is 175 ÷ 500 = 0.35, or 35%<sup>[2](https://openstax.org/books/contemporary-mathematics/pages/6-1-understanding-percent)</sup>.

Older textbooks name the same quantities differently. A nineteenth-century arithmetic defines percentage as "the process of computation in which the basis of comparison is a hundred," and lists the Base (the number on which the percentage is computed), the Rate (the number of hundredths taken), the Percentage (the result), and the Amount or Difference<sup>[11](https://files.eric.ed.gov/fulltext/ED622715.pdf)</sup>. Mathematics educators classify the three problem types as find-part, find-percent, and find-whole (called "reverse percentages" in Singapore), and research shows students differ in their understanding of the three types<sup>[12](https://onlinelibrary.wiley.com/doi/full/10.1002%2Facp.4117)</sup>.

## Percentage change, points, and basis points

The percentage change between two values equals the difference divided by the original value, multiplied by 100. The U.S. [Bureau of Labor Statistics](https://www.edgepedia.org/bureau-of-labor-statistics) uses this method for the Consumer Price Index: the CPI-U rose from 277.948 in November 2021 to 278.802 in December 2021, a difference of 0.854, which divided by the earlier value and multiplied by 100 gives a 0.3% one-month change<sup>[13](https://www.bls.gov/cpi/factsheets/calculating-percent-changes.htm)</sup>. BLS also notes that an over-the-year percent change, such as December to December, is not equal to the sum of the over-the-month changes in between<sup>[13](https://www.bls.gov/cpi/factsheets/calculating-percent-changes.htm)</sup>.

**Percentage points.** When the quantities being compared are themselves percentages, the arithmetic difference is measured in percentage points, abbreviated pp<sup>[5](https://ec.europa.eu/eurostat/statistics-explained/SEPDF/cache/42722.pdf?v=2882788016648030)</sup>. If a rate rises from 4% to 5%, the absolute difference is one percentage point, while the relative increase is 25%, because 5 ÷ 4 − 1 equals 0.25<sup>[6](https://useconomicdata.com/learn/percentage-points/)</sup>. Calling that move a "1% increase" conflates the two measures. The UNECE guide for writers gives the standard correction: saying a labor-force share "rose 5% from 15% to 20%" is wrong; it rose five percentage points<sup>[14](https://ec.europa.eu/eurostat/documents/64157/4374310/32-UNECE-making-data-meaningful-Part1-EN.pdf/82cf87d2-4bc9-431c-be96-2257cf77c810)</sup>. Federal statistical agencies typically report changes in rates in percentage points and changes in levels in percent<sup>[6](https://useconomicdata.com/learn/percentage-points/)</sup>.

In finance, small rate moves are quoted in basis points: one percentage point equals 100 basis points, so a yield moving from 4.00% to 4.25% rises 0.25 percentage point, or 25 basis points<sup>[6](https://useconomicdata.com/learn/percentage-points/)</sup>. Another subtle point is that rates and levels can move in opposite directions: when inflation drops from 5% to 3%, the rate has decreased by two percentage points even though prices continue to climb during that period<sup>[6](https://useconomicdata.com/learn/percentage-points/)</sup>.

## Successive changes and asymmetry

Sequential percentage changes multiply rather than add. When a price of $100 rises by 50% and then falls by 50%, the final price is $75, not $100, because the second change operates on a different base<sup>[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC5784045/)</sup>. The same asymmetry appears in markets: after Black Monday's 22.6% one-day drop in the [Dow Jones Industrial Average](https://www.edgepedia.org/dow-jones-industrial-average) on October 19, 1987, a subsequent 29.2% increase was required to reverse it<sup>[15](https://dlab.sauder.ubc.ca/sjdm/journal/16/16222/jdm16222.html)</sup>. Parker's 1997 analysis expresses the general relationship as a function, with the relative decrease needed to offset a relative increase x given by \( f(x) = \frac{100x}{100 + x} \), and connects successive percentage changes to the geometric mean<sup>[16](https://onlinelibrary.wiley.com/doi/10.1111/j.1949-8594.1997.tb17385.x)</sup>.

## History

The concept has two historical streams. A [University of California](https://www.edgepedia.org/university-of-california) dissertation building on Parker and Leinhardt's review traces commercial percentage thinking to ancient practices: the earliest records of taxation and interest rates are from India around 300 B.C.E., calculated per months per hundreds, and China's *Arithmetic in Nine Sections* (200–100 B.C.E.) contained the "Rule of Three" for proportional calculation<sup>[17](https://escholarship.org/content/qt63c4h4g8/qt63c4h4g8_noSplash_17b3419c7ca6424c03a27603e1ffe005.pdf?t=r0yr3h)</sup>. By the end of the Middle Ages, 100 was a standard base in Italian ledgers around 1200 C.E., and during the Renaissance the standardization of 100 as a base amount made the concept multiplicative<sup>[17](https://escholarship.org/content/qt63c4h4g8/qt63c4h4g8_noSplash_17b3419c7ca6424c03a27603e1ffe005.pdf?t=r0yr3h)</sup>. In the 17th century the written terms compressed toward the modern symbol, a progression summarized as per cento → per 100 → p cento → per c° → % <sup>[17](https://escholarship.org/content/qt63c4h4g8/qt63c4h4g8_noSplash_17b3419c7ca6424c03a27603e1ffe005.pdf?t=r0yr3h)</sup>. The use of percent as a general tool of normalization has been linked to the birth of statistics in the late 18th and early 19th centuries, when large datasets needed a common basis for comparison<sup>[17](https://escholarship.org/content/qt63c4h4g8/qt63c4h4g8_noSplash_17b3419c7ca6424c03a27603e1ffe005.pdf?t=r0yr3h)</sup>.

## Learning difficulties

Despite daily exposure, percentages are hard to learn. The review by Melanie Parker and Gaea Leinhardt, a landmark study of percent in mathematics education research, attributes the difficulty to the concept's dual meaning (fraction or ratio) and sense (function or statistic), and to a concise language that has lost its explicit referents and uses misleading additive terminology for multiplicative meanings<sup>[18](https://journals.sagepub.com/doi/10.3102/00346543065004421)</sup>.

Empirical studies confirm the problem persists into adulthood. In the study of 1,629 undergraduates described above, roughly half the responses to two-step percentage change questions were erroneous, and the most frequent wrong answer was that the final amount equaled the initial amount, reflecting an additive rather than multiplicative interpretation<sup>[4](https://pmc.ncbi.nlm.nih.gov/articles/PMC5784045/)</sup>. A comparative survey of lower secondary students in Japan and Finland found that accuracy on base problems, "percent of percent" problems, and "percent based on percent" problems remained below 50% even among ninth graders<sup>[19](https://journals.sagepub.com/doi/10.1177/27527263261453341)</sup>. Documented misconceptions include ignoring the % sign, the "numerator rule" error of replacing a percent sign with a decimal point (converting 0.9 to 9% instead of 90%), and reversing divisor and dividend in base-finding problems<sup>[19](https://journals.sagepub.com/doi/10.1177/27527263261453341)</sup>.

## Related units

The percentage is one member of a family of fixed-denominator ratios. Eurostat's harmonized index of consumer prices weights product categories in per mille (‰), parts per thousand, with weights summing to 1,000 rather than 100<sup>[20](https://ec.europa.eu/eurostat/web/european-statistics-learning-hub/statistics-for-beginners/inflation)</sup>. Statistical agencies also express proportions as fractions or as ratios per 100,000 people when counts are small<sup>[21](https://www.pc.gov.au/closing-the-gap-data/explain/interpret-data/)</sup>. The choice of denominator is conventional rather than mathematical; the fixed base exists so that proportions from different totals can be stated and compared on one scale<sup>[3](https://math.libretexts.org/Courses/Cosumnes_River_College/Corequisite_Codex/02%3A_Arithmetic/2.13%3A_Percents)</sup>.

## References

1. [6.1 Understand Percent, OpenStax Prealgebra](https://openstax.org/books/prealgebra/pages/6-1-understand-percent)
2. [6.1 Understanding Percent, OpenStax Contemporary Mathematics](https://openstax.org/books/contemporary-mathematics/pages/6-1-understanding-percent)
3. [2.13: Percents, Mathematics LibreTexts](https://math.libretexts.org/Courses/Cosumnes_River_College/Corequisite_Codex/02%3A_Arithmetic/2.13%3A_Percents)
4. [Danan & Gelman, "The problem with percentages," Philosophical Transactions of the Royal Society B (PMC5784045)](https://pmc.ncbi.nlm.nih.gov/articles/PMC5784045/)
5. [Glossary: Percentage point, Eurostat Statistics Explained](https://ec.europa.eu/eurostat/statistics-explained/SEPDF/cache/42722.pdf?v=2882788016648030)
6. [Percentage points versus percentage change, US Economic Data](https://useconomicdata.com/learn/percentage-points/)
7. [Earliest Uses of Symbols for Fractions, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk//Miller/mathsym/fractions/)
8. [Describing frequencies, Australian Bureau of Statistics](https://www.abs.gov.au/statistics/understanding-statistics/statistical-terms-and-concepts/describing-frequencies)
9. [Florian Cajori, A History of Mathematical Notations](https://hlevkin.com/hlevkin/90MathPhysBioBooks/mathHistory/Cajori_Florian_A_History_of_Mathematical_Notations_2_Vols.pdf)
10. [Beginning with percentages, NZ Maths](https://meaningfulmaths.nt.edu.au/mmws/nz/resource/beginning-percentages.html)
11. [The New Normal Written Arithmetic (ERIC ED622715)](https://files.eric.ed.gov/fulltext/ED622715.pdf)
12. [Ngu, "Instructional efficiency: The role of prior knowledge and cognitive load," Applied Cognitive Psychology](https://onlinelibrary.wiley.com/doi/full/10.1002%2Facp.4117)
13. [Calculating percent changes, U.S. Bureau of Labor Statistics](https://www.bls.gov/cpi/factsheets/calculating-percent-changes.htm)
14. [Making Data Meaningful Part 1: A guide to writing stories about numbers, UNECE/Eurostat](https://ec.europa.eu/eurostat/documents/64157/4374310/32-UNECE-making-data-meaningful-Part1-EN.pdf/82cf87d2-4bc9-431c-be96-2257cf77c810)
15. [Percentage changes and investor behavior, Journal of Decision Making (dlab.sauder.ubc.ca)](https://dlab.sauder.ubc.ca/sjdm/journal/16/16222/jdm16222.html)
16. [Parker, "The Ups and Downs of Percent," School Science and Mathematics (1997)](https://onlinelibrary.wiley.com/doi/10.1111/j.1949-8594.1997.tb17385.x)
17. [An Interdisciplinary Analysis of the Concept of Percent, eScholarship (University of California)](https://escholarship.org/content/qt63c4h4g8/qt63c4h4g8_noSplash_17b3419c7ca6424c03a27603e1ffe005.pdf?t=r0yr3h)
18. [Parker & Leinhardt, "Percent: A Privileged Proportion," Review of Educational Research (1995)](https://journals.sagepub.com/doi/10.3102/00346543065004421)
19. [Kumakura et al., "Understanding of percentage among lower secondary school students in Japan and Finland" (2026)](https://journals.sagepub.com/doi/10.1177/27527263261453341)
20. [Inflation, Eurostat European statistics learning hub](https://ec.europa.eu/eurostat/web/european-statistics-learning-hub/statistics-for-beginners/inflation)
21. [How to interpret the data, Closing the Gap, Australian Productivity Commission](https://www.pc.gov.au/closing-the-gap-data/explain/interpret-data/)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Elementary and formal arithmetic*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license