Percentage

In mathematics, a percentage is a way of expressing a number as a fraction of 100 (per centum meaning “per hundred” in Latin). It is often denoted using the percent sign, “%”, or the abbreviation “pct”. For example, 45% (read as “forty-five percent”) is equal to 45/100, or 0.45.

Percentages are used to express how large/small one quantity is, relative to another quantity. The first quantity usually represents a part of, or a change in, the second quantity, which should be greater than zero. 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.

Although percentages are usually used to express numbers between zero and one, any dimensionless proportionality can be expressed as a percentage. For instance, 111% is 1.11 and −0.35% is −0.0035. Although this is technically inaccurate as per the definition of percent, an alternative wording in terms of a change in an observed value is “an increase/decrease by a factor of...””

Contents

Calculations

The fundamental concept to remember when performing calculations with percentages is that the percent symbol can be treated as being equivalent to the pure number constant 1/100=0.01 , for example 35% of 300 can be written as (35/100) × 300 = 105.

To find the percentage that a single unit represents out of a whole of N units, divide 100% by N. For instance, if you have 1250 apples, and you want to find out what percentage of these 1250 apples a single apple represents, 100%/1250 = (100/1250)% provides the answer of 0.08%. So, if you give away one apple, you have given away 0.08% of the apples you had. Then, if instead you give away 100 apples, you have given away 100 × 0.08% = 8% of your 1250 apples.

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:

(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.)

The easy way to calculate addition in percentage (discount 10% + 5%):

y = [(x1+x2) - (x1*x2)]

For example, in a department store promotion "discount 10%+5%", the total discount is not 15%, but:

y = [(10%%2B5%) - (10%*5%)] = [15% - 0.5%] = 14.5%

Example problems

Whenever we talk 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 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 (60/100) × (5/100) = 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: 3%/10% = 30/100 or 30% of all computer science majors are female.

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

Here are other examples:

  1. What is 200% of 30?
    Answer: 200% \times 30 = \frac{200}{100} \times 30 = \frac{2}{1} \times 30 = 2 \times 30 = 60
  2. What is 13% of 98?
    Answer: 13% \times 98 = \frac{13}{100} \times 98 = \frac{13 \times 98}{100} = 12.74
  3. 60% of all university students are female. There are 2400 female students. How many students are in the university?
    Answer: 2400 = 60% \times n, therefore n = \frac{2400}{ \tfrac{60}{100} } = \frac{2400 \times 100}{60} = 4000.
  4. There are 300 cats in the village, and 75 of them are black. What is the percentage of black cats in that village?
    Answer: 75 = n% \times 300 = \frac{n}{100} \times 300, so n = \frac{75}{300} \times 100 = 25 and therefore n% = 25%.
  5. The number of students at the university increased to 4620, compared to last year's 4125, an absolute increase of 495 students. What is the percentual increase?
    Answer: 495 = n% \times 4125 = \frac{n}{100} \times 4125, so n = \frac{495}{4125} \times 100 = 12, and therefore n% = 12%.

Percentage increase and decrease

Sometimes 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:

In general, a change of x percent in a quantity results in a final amount that is 100%2Bx percent of the original amount (equivalently, 1%2B0.01x times the original amount).

It is important to understand that percent changes, as they have been discussed here, do not add in the usual way, if applied sequentially. 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), the final price will be $198, not the original price of $200. The reason for the 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".

Another common mistake is thinking that working 50% faster means taking 50% less time to complete the task. On this account, 100% faster means twice the speed, so half the time. For example, if one traveled at 50 mph, 100% faster would be 100 mph (taking 50% less time). And 50% faster speed means 33.33% less time to travel the same distance.

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%2B0.01x)(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%2B0.01x)(1%2B0.01y)). 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". In the previous example, the interest rate "increased by 5 percentage points" from 10% to 15%. If the rate then drops by 5 percentage points, it will return to the initial rate of 10%, as expected.

Change in sign

When the first number is negative and second number is positive, the percentage change from first number to second number is negative. This often occurs in financial statements that changes from a period of loss to period in profit.

Acme Company EBIT
First quarter     (100)
Second quarter     100
Change in profitability (100 - (-100))/(-100) = -200%

In expressing a number as a percentage, the base of the comparison cannot be negative. The First number in the above example is the base of the comparison when it is expressed as a positive amount becomes Loss of 100. The change from First quarter loss to Second quarter profit becomes Percentage change in loss by -200% to turn a profit of 100.

Word and symbol

Main article: Percent sign

In British English, percent is usually written as two words (per cent, although percentage and percentile are written as one word).[1] In American English, percent is the most common variant[2] (but cf. per mille written as two words).

In the early part of the twentieth century, there was a dotted abbreviation form “per cent.”, as opposed to “per cent”. The form “per cent.” is still in use as a part of 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. While the term has been attributed to Latin per centum, this is a pseudo-Latin construction and the term was likely originally adopted from the French pour cent. The concept of considering values as parts of a hundred is originally Greek. The symbol for percent (%) evolved from a symbol abbreviating the Italian per cento. In some other languages, the form 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 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 “3 ½ percent of the gain”. 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 The Chicago Manual of Style—generally state that the number and percent sign are written without any space in between.[3] However, the International System of Units and the ISO 31-0 standard require a space.[4][5]

Related units

Other uses

It is well known that many in the media and sports worlds use clichés such as '110%' despite the fact that such phrases do not make any mathematical sense.

The word "percentage" is often misused 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%.) 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 of a road or railway.

See also

References

  1. ^ Brians, Paul. "Percent/per cent". Common Errors in English Usage. Washington State University. http://www.wsu.edu/~brians/errors/percent1.html. Retrieved 22 November 2010. 
  2. ^ "Percent (per cent)". Oxford Dictionaries. http://oxforddictionaries.com/view/entry/m_en_us1276846?rskey=eX1NKq&result=1#m_en_us1276846. Retrieved 22 November 2010. 
  3. ^ "The Chicago Manual of Style". University of Chicago Press. 2003. http://www.chicagomanualofstyle.org/. Retrieved 2007-01-05. 
  4. ^ "The International System of Units". International Bureau of Weights and Measures. 2006. http://www.bipm.org/utils/common/pdf/si_brochure_8.pdf. Retrieved 2007-08-06. 
  5. ^ "ISO 31-0 — Quantities and units – Part 0: General principles". International Organization for Standardization. 1999-12-22. http://www.iso.org/iso/en/CatalogueDetailPage.CatalogueDetail?CSNUMBER=3621. Retrieved 2007-01-05. 

External links