Percentage

In mathematics, a percentage, percent, or per cent (from Latin per centum 'by a hundred') is a way to show a number or ratio as a part of 100. It helps us understand parts of a whole more easily. For example, if you have 50 out of 100 candies, you can say you have 50 percent of the candies.

People often use the percent sign (%) to show percentages. But sometimes, they also use short forms like pct. or pc. Even though a percentage is just a number, it acts like a special unit we use when we talk about how much of something there is compared to the whole.

Percentages are very useful in everyday life. They help us understand things like grades in school, discounts while shopping, or how much of a pizza has been eaten. Knowing about percentages makes it easier to compare and understand different amounts and proportions.

Examples

For example, 45% (read as "forty-five percent") is the same as the fraction 45/100, or 0.45. We often use percentages to show a part of a whole.

If 50% of the students in a class are male, that means 50 out of every 100 students are male. If there are 500 students, then 250 of them are male.

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 a 6% increase.

While many percentage values are between 0 and 100, there is no rule that says they have to be. It is common to see values such as 111% or −35%, especially when talking about percent changes and comparisons.

(Similarly, one can also express a number as a fraction of 1,000, using the term "per mille" or the symbol "‰".)

History

In Ancient Rome, people often used parts of 100 to do math long before we had our current number system. For example, a leader named Augustus used a tax of one part in a hundred on sold goods.

During the Middle Ages, using parts of 100 became very common. By the late 1400s to early 1500s, math books often showed how to use these parts for things like profits, losses, and interest rates. By the 1600s, it was normal to talk about interest rates using hundredths.

Percent sign

Main article: Percent sign

The word "percent" comes from a Latin phrase meaning "by a hundred." Over time, the symbol for percent changed from an Italian phrase meaning "for a hundred" into the "%" we use today. The symbol started as words that were shortened and changed until it became the sign we know now.

Calculations

To find a percentage, you multiply the part by 100 and then divide by the whole. For example, to find how many apples are 50 out of 1,250, you first divide 50 by 1,250 to get 0.04, and then multiply by 100 to get 4%. Another way is to multiply the part by 100 first and then divide by the whole. In this case, 50 multiplied by 100 is 5,000, and dividing by 1,250 also gives 4%.

When you need to find a percentage of another percentage, change both percentages to fractions or decimals and multiply them. For example, 50% of 40% is worked out by multiplying 0.50 by 0.40, which equals 0.20 or 20%.

It is important to always say what the percentage is out of. For example, in a college where 60% of students are female and 10% are computer science majors, and 5% of female students are computer science majors, you can find what percent of computer science majors are female by multiplying 60% by 5% to get 3% of all students, and then dividing that by the 10% who are computer science majors, giving 30%.

Variants of the percentage calculation

People calculate percentages in different ways, depending on what they need. One common method uses proportions, which helps avoid memorizing formulas. In mental arithmetic, a useful step is to figure out what 100% or 1% would be.

For example: If 42 kg is 7%, how much is 100%? We know the percentage (W) and the percent value (p %), and we want to find the basic value (G).

With general formula	With own ratio equation (Proportion)	With “What is 1%?” (Rule of 3) p % 42 kg = 100 % 7 % {\displaystyle {\frac {p\,\%}{42\,{\text{kg}}}}={\frac {100\,\%}{7\ \%}}}
p % 100 % = W G {\displaystyle {\frac {p\,\%}{100\,\%}}={\frac {W}{G}}} multiple rearrangements result in: G = W p % ⋅ 100 % {\displaystyle G={\frac {W}{p\,\%}}\cdot {100\,\%}} G = 42 kg 7 % ⋅ 100 % = 600 kg {\displaystyle G={\frac {42\,{\text{ kg}}}{7\,\%}}\cdot {100\,\%}=600\,{\text{kg}}}	G 42 kg = 100 % 7 % {\displaystyle {\frac {G}{42\,{\text{kg}}}}={\frac {100\,\%}{7\,\%}}} simple conversion yields: G = 42 kg 7 % ⋅ 100 % = 600 kg {\displaystyle G={\frac {42\,{\text{kg}}}{7\,\%}}\cdot {100\,\%}=600\,{\text{ kg}}}	42 kg : 7 7 % : 7 = 6 kg 1 % = 6 kg ⋅ 100 1 % ⋅ 100 {\displaystyle {\frac {42\,{\text{kg}}:{\color {red}7}}{7\,\%:{\color {red}7}}}={\frac {6\,{\text{ kg}}}{1\,\%}}={\frac {6\,{\text{kg}}\cdot {\color {red}100}}{1\,\%\cdot {\color {red}100}}}} without changing the last counter is: G = 6 kg ⋅ 100 = 600 kg {\displaystyle G=6\,{\text{kg}}\cdot 100=600\,{\text{ kg}}}
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

Placard outside a shop in Bordeaux advertising 20% decrease in the price of the second perfume purchased.

When we talk about a "10% rise" or a "10% fall" in something, we usually mean it’s compared to the starting value. For example, if something costs $200 and the price goes up by 10% (an extra $20), the new price will be $220. This new price is 110% of the original price (100% + 10% = 110%).

Here are some examples of percent changes:

An increase of 100% means the final amount is double the original amount.
An increase of 800% means the final amount is nine times the original amount.
A decrease of 60% means the final amount is 40% of the original amount.
A decrease of 100% means the final amount is zero.

A change of x percent in a quantity results in a final amount that is 100 + x percent of the original amount. It’s important to know that increasing and then decreasing by the same percentage doesn’t bring you back to the start. For example, a 25% increase (from $100 to $125) needs a 20% decrease (from $125 back to $100) to return to the original amount.

Compounding percentages

When we apply percent changes one after another, they don't simply add up. For example, if a $200 item first goes up by 10% to $220, and then goes down by 10%, the price ends up at $198, not back to $200. This happens because each change is based on a different starting amount.

If we start with an amount and it first increases by a certain percent and then decreases by the same percent, the final amount will always end up a little less than the original. This is because the two changes multiply together, not add up. For example, a 10% increase followed by a 10% decrease leaves the amount 1% lower than it started.

Percent changes can be applied in any order and will have the same result. When talking about changes in interest rates or election results, it's important to be clear whether we are talking about percentage points or percent changes to avoid confusion.

Word and symbol

Main article: Percent sign

In most types of English, the word percent is usually written as two words, per cent, but percentage and percentile are written as one word. In American English, percent is the most common way to write it, while per mille is written as two words.

The symbol for percent (%) came from a short way to write the Italian words for "per cento." Some other languages use words like procent or prosent instead. Some languages even have their own special ways to say percentages.

Different style guides have different rules about writing percentages. Some say to always write out the word "percent," like "1 percent," while others say it’s okay to use the % symbol, especially in science. Most agree that you should always use a number with it, like "5 percent," not "five percent," except at the start of a sentence: "Ten percent of all writers love style guides." Decimals are usually used instead of fractions, like "3.5 percent" instead of "3+1⁄2 percent." However, bonds and other financial titles sometimes use fractions, like "3+1⁄2% Unsecured Loan Stock 2032 Series 2."

Most style guides say to write the number and the % symbol with no space between them, but some standards, like the International System of Units and ISO 31-0, say to use a space.

Other uses

People sometimes use the word "percentage" incorrectly in sports. For example, when they say a player like Shaquille O'Neal had a .609 field goal percentage, they really mean he made 60.9% of his shots, not 0.609%. The same goes for a team's winning percentage; a .500 winning percentage means the team won 50% of its games.

We also use percentages to describe how steep a road or railway is. This is called the grade or slope. It tells us how much a vehicle goes up or down compared to how far it moves forward, shown as a percentage.

Percentages can also show what parts make up a mixture, either by mass percent or by mole percent.

Related units

Percentage point difference of 1 part in 100
Per mille (‰) 1 part in 1,000
Basis point (bp) difference of 1 part in 10,000
Permyriad (‱) 1 part in 10,000
Per cent mille (pcm) 1 part in 100,000
Centiturn

Practical applications

Percentages are used in many everyday situations. For example, in baking, baker percentage helps describe ingredient amounts relative to the flour weight. Another use is volume percent, which measures how much of one liquid is in another, like in alcoholic drinks.