Percentage

A percentage is a way of expressing numbers, generally those between 0 and 1. The percent sign, "%", means "divided by 100." For example, "45.1%" (read as "fourty five point one percent") is equal to 0.451. Numbers larger than 1 and smaller than 0 are sometimes also expressed as percentages. For instance, 111% is 1.11 and -0.35% is -0.0035.

Confusion from the use of percentages

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

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.

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 "initial" value after the first calculation. In this example, the first initial 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.

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. If 5% of females 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.

Word and symbol

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.. While the term has been attributed to Latin per centum, this is a pseudo-Latin construction and the term was likely originally adopted from Italian per cento or 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.

Usage in computers

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 modulo 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 (see percent-encoding). Names for the percent sign include percent sign (in ITU-T), mod, grapes (in hacker jargon), and the humorous double-oh-seven (in INTERCAL).

Related units

External links

Percentage Calculator with explanations

Elementary arithmetic

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