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In mathematics, a percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", or the abbreviation "pct." A percentage is a dimensionless number (pure number).

For example, 45% (read as "forty-five percent") is equal to 45/100, or 0.45. A related system which expresses a number as a fraction of 1,000 uses the terms "per mil" and "millage". Percentages are used to express how large or small one quantity is relative to another quantity. The first quantity usually represents a part of, or a change in, the second quantity. For example, an increase of $ 0.15 on a price of $ 2.50 is an increase by a fraction of 0.15/2.50 = 0.06. Expressed as a percentage, this is therefore a 6% increase. While percentage values are often between 0 and 100 there is no restriction and one may, for example, refer to 111% or −35%.


In Ancient Rome, long before the existence of the decimal system, computations were often made in fractions which were multiples of 1/100. For example Augustus levied a tax of 1/100 on goods sold at auction known as centesima rerum venalium. Computation with these fractions was similar to computing percentages. As denominations of money grew in the Middle Ages, computations with a denominator of 100 become more standard and 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.[1]

Percent sign[edit]

Main article: percent sign





(50/100) × (40/100) = 0.50 × 0.40 = 0.20 = 20/100 = 20%.

It is not correct to divide by 100 and use the percent sign at the same time. (E.g. 25% = 25/100 = 0.25, not 25% / 100, which actually is (25/100) / 100 = 0.0025. A term such as (100/100)% would also be incorrect, this would be read as (1) percent even if the intent was to say 100%.)




This example is closely related to the concept of conditional probability.

Percentage increase and decrease[edit]


Some other examples of percent changes:



Compounding percentages[edit]


In general, if an increase of x percent is followed by a decrease of x percent, and the initial amount was p, the final amount is p((1+0.01x)(1-0.01x))=p(1-(0.01x)^2); thus the net change is an overall decrease by x percent of 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 x=10 percent, the final amount, $198, was 10% of 10%, or 1%, less than the initial amount of $200.

This can be expanded for a case where you do not have the same percent change. If the initial percent change is x and the second percent change is y, and the initial amount was p, then the final amount is p((1+0.01x)(1+0.01y)). To change the above example, after an increase of x=10 and decrease of y=-5 percent, the final amount, $209, is 4.5% more than the initial amount of $200.

In the case of interest rates, it is a common practice to state the percent change differently. If an interest rate rises from 10% to 15%, for example, it is typical to say, "The interest rate increased by 5%" — rather than by 50%, which would be correct when measured as a percentage of the initial rate (i.e., from 0.10 to 0.15 is an increase of 50%). Such ambiguity can be avoided by using the term "percentage points" (pp). In the previous example, the interest rate "increased by 5 pp" from 10% to 15%. If the rate then drops by 5 percentage points, it will return to the initial rate of 10%, as expected.

Word and symbol[edit]

Main article: Percent sign

In British English, percent is sometimes written as two words (per cent, although percentage and percentile are written as one word).[5] In American English, percent is the most common variant[6] (but cf. per mille written as two words).



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.[8] However, the International System of Units and the ISO 31-0 standard require a space.[9][10]

Related units[edit]

Other uses[edit]


As "percent" it is used to describe the steepness of the slope of a road or railway, formula for which is  100 \frac{\text{rise}}{\text{run}} which could also be expressed as the tangent of the angle of inclination times 100. The is the ratio of distances a vehicle would advance vertically and horizontally, respectively, when going up- or downhill, expressed in percent.


Practical applications[edit]

See also[edit]


  1. ^ Smith, D.E. (1951∨1958). History of Mathematics 2. Courier Dover Publications. pp. 247–249. ISBN 0-486-20430-8.  Check date values in: |date= (help)
  3. ^ Smith p. 250
  4. ^ "Percentage Calculator". 
  6. ^ "Percent (per cent)". Oxford Dictionarie. Retrieved 22 November 2010. 