# Percent

A percentage is a way of expressing a proportion or a fraction as a whole number. A number such as "45%" ("45 percent" or "45 per cent") is actually shorthand for the fraction 45/100. In British English, percent is often written as two words (per cent), and the spelling as one word is considered by some to be "incorrect". In American English, percent is much more common, and is usually considered "correct".

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..."

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.

The symbol for percent "%" is a stylised form of the two zeros. It evolved from a symbol similar except for a horizontal line instead of diagonal (c. 1650), which in turn evolved from a symbol representing "P cento" (c. 1425).

In computing, other names for the character include: ITU-T: percent sign; mod; grapes. INTERCAL: double-oh-seven.

## 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 100\$ is 10\$, giving a new price of 110\$; 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 20%. This could be described as a 100% 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 10%," meaning 10% of 100% additional to the initial 10% (giving 20% in total), though it should mean according to the usual interpretation of percentages a 10% increase on the initial 10% (giving 11%).

To counter this confusion, the expression "percentage points" is often used. So, in the previous example, "The interest rate has increased by 10 percentage points" would be an unambiguous expression that the rate is now 20%. 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 1000 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. In general, the net effect is:

(1 + x)(1 - x) = 1 - x2

i.e. a net decrease proportional to the square of the percentage change.

Owners of dot com stocks 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, you're still broke if the stock subsequently drops 100% meaning it has a zero value.