Percentage
In mathematics, a percentage, percent, or per cent is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, although the abbreviations pct., pct, and sometimes pc are also used. A percentage is a dimensionless number, primarily used for expressing proportions, but percent is nonetheless a unit of measurement in its orthography and usage.
Examples
For example, 45% is equal to the fraction, or 0.45.Percentages are often used to express a proportionate part of a total.
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 values such as 111% or −35%, especially for percent changes and comparisons.
History
In Ancient Rome, long before the existence of the decimal system, computations were often made in fractions in the multiples of. For example, Augustus 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, 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 1,250 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 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:
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. The following problem illustrates this point.
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 × = 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.
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.Example:
42 kg is 7%. How much is 100%?
Given are W and p %.
We are looking for G.
| With general formula | With own ratio equation | With “What is 1%?” |
multiple rearrangements result in: | simple conversion yields: | without changing the last counter is: |
| Advantage: • One formula for all tasks | Advantages: • Without a formula • Easy to change over if the size you are looking for – here G – is in the top left of the counter. | Advantages: • Without a formula • Simple rule of three – here as a chain of equations • Application for mental arithmetic |
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%, the new price will be $220. Note that this final price is 110% of the initial price.Some other examples of percent changes:
- An increase of 100% in a quantity means that the final amount is 200% of the initial amount. In other words, the quantity has doubled.
- An increase of 800% means the final amount is 9 times the original.
- A decrease of 60% means the final amount is 40% of the original.
- A decrease of 100% means the final amount is zero.
Compounding percentages
Percent changes applied sequentially do not add up in the usual way. For example, if the 10% increase in price considered earlier is followed by a 10% decrease in the price, 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 are measured relative to different initial values, 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. 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 rates, 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 points. The same confusion between the different concepts of percent 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% or 38.5% ?
In financial markets, it is common to refer to an increase of one percentage point as an increase of "100 basis points".
Word and symbol
In most forms of English, percent is usually written as two words, although percentage and percentile are written as one word. In American English, percent is the most common variant.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, as well as in the Hansard transcripts of British Parliamentary proceedings. The term has been attributed to Latin per centum. 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. 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 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". 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.
However, the International System of Units and the ISO 31-0 standard require a space.