Percentage

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A percentage is a way of expressing a proportion, a ratio or a fraction as a whole number, by using 100 as the denominator. A number such as "45%" ("45 percent" or "45 per cent") is shorthand for the fraction 45/100 or 0.45.

As an illustration,

"45 percent of human beings..."

is equivalent to both of the following:

"45 out of every 100 people..."

"0.45 of the human population..."

One way to think about percentages is to realize that "one percent", represented by the symbol %, is simply the number 1/100, or 0.01.

A percentage may be a number larger than 100; for example, 200% of a number refers to twice the number. In fact, this would be a 100% increase, while a 200% increase would give a number three times the original value. Thus one can see the relationship between percent increase and times increase.

1 Confusion from the use of percentages
- 1.1 Changes
- 1.2 Cancellations
2 An example problem
3 Word and symbol
4 Related units
5 External links

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Confusion from the use of percentages

Many confusions arise from the use of percentages, due to inconsistent usage or misunderstanding of basic arithmetic.

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Changes

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, a 10% increase on an item initially priced at $200 is $20, giving a new price of $220; to many people, any other usage is incorrect.

In the case of interest rates, however, it is a common practice to use the percent change differently: suppose that an initial interest rate is given as a percentage like 10%. Suppose the interest rate rises to 15%. This could be described as a 50% increase, measuring the increase relative to the initial value of the interest rate. However, many people say in practice "The interest rate has risen by 5%". To counter this confusion, the unit "percentage points" is sometimes used when referring to differences of percentages. So, in the previous example, "The interest rate has increased by 5 percentage points" would be an unambiguous expression that the rate is now 15%. Often also, the term "basis points" is used, one basis point being one one hundredth of a percentage point. Thus, the interest rate above increased by 500 basis points.

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Cancellations

A common error when using percentages is to imagine that a percentage increase is cancelled out when followed by the same percentage decrease. A 50% increase from 100 is 100 + 50, or 150. A 50% reduction from 150 is 150 – 75, or 75. The end result is smaller than the 100 we started out with. This phenomenon is due to the change in the "inital" value after the first calculation. In this example, the first intial value is 100, but the second is 150.

In general, the net effect is:

(1 + x)(1 – x) = 1 – x²,

that is a net decrease proportional to the square of the percentage change.

To use a specific example, stock brokers came to understand that even if a stock has sunk 99%, it can nevertheless still sink another 99%. Also, if a stock rises by a large percentage, the trader still loses all of the stock's value if the stock subsequently drops 100%, meaning it has a zero value.

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An example problem

Whenever we talk about a percentage, it is important to specify what it is relative to, i.e. what the total is 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. Among the females, only 5% 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 .6 × .05 = .03 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% = .3 or 30% of all computer science majors are female.

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Word and symbol

Image:Percent 1425.png

Image:Percent 1650.PNG

Image:Percent 18e.PNG

In British English, percent is usually written as two words (per cent). In American English, percent is the most common variant. In the early part of the twentieth century, there was a dotted abbreviation form per cent., which came from the original Latin per centum. The concept of considering values as parts of a hundred is originally Greek.

The symbol for percent "%", evolved from a symbol similar except for a horizontal line instead of diagonal (c. 1650), which in turn evolved from an abbreviation of "P cento" (c. 1425). Traditionally, the symbol follows the number to which it applies.

In Unicode, there is also an "ARABIC PERCENT SIGN" ("٪"U+066A), which has the circles replaced by square dots set on edge.

In computing, the percent character is also used for the mod operation in programming languages that derive their syntax from C. The ASCII code for the percent character is 37, or 0x25 in hexadecimal. In the textual representation of URIs, a % immediately followed by a 2-digit hexadecimal number denotes a UTF-8 octet specifying (part of) a character that might otherwise not be allowed in URIs. Names for the percent sign include percent sign; mod; grapes in ITU-T, and the humorous double-oh-seven in INTERCAL.