What is a Percentage?
In mathematics, a percentage is a number or ratio that represents a fraction of 100. It is a highly common way to represent a dimensionless relationship between two numbers. Other similar methods include ratios, fractions, and decimals. Percentages are often denoted by the symbol "%" written after the number, but they can also be denoted by writing "percent" or "pct".
For example, 35% is mathematically equivalent to the decimal 0.35, or the fractions 35/100 and 7/20. Percentages are widely used in everyday life—from calculating sales discounts and tipping at a restaurant, to figuring out loan interest rates and analyzing statistical data.
Percentages are computed by multiplying the value of a ratio by 100. For instance, if 25 out of 50 students in a classroom are male, the ratio is 25/50 (or 0.5). Multiplying this decimal by 100 yields 50, meaning 50% of the students are male.
How to Use This Percentage Calculator
Our tool is designed to solve several different types of percentage problems instantly. To get started, select the type of problem you are trying to solve from the dropdown menu:
- What is [X]% of [Y]? — Use this to find a specific fraction of a whole number. Example: Finding a 20% tip on a $50 bill.
- [X] is what % of [Y]? — Use this to determine what percentage one number is relative to another. Example: Discovering your test score if you got 45 out of 60 questions right.
- [X] is [Y]% of what? — Use this to work backward and find the original whole number. Example: Knowing that $20 represents 10% of a total budget, and finding the full budget.
- Percentage Difference: — Calculate the relative difference between two separate numbers. This is often used in scientific or statistical comparisons.
- Percentage Change: — Determine the new value after increasing or decreasing an initial number by a certain percent.
Percentage Formulas and Methods
While percentages can be approached intuitively, they are governed by straightforward algebraic equations. The core percentage formula is an equation involving three variables:
Where P is the percentage in decimal form, V₁ is the initial or base value, and V₂ is the resulting value. When solving equations manually, you must convert the percentage into a decimal by dividing it by 100. Our calculator handles this conversion automatically behind the scenes.
Percentage Difference Formula
The percentage difference between two values is calculated by dividing the absolute value of the difference between the two numbers by the average of those two numbers. Multiplying the result by 100 yields the solution in percent form.
Percentage Change (Increase/Decrease) Formula
Percentage increase and decrease are calculated by computing the difference between two values and comparing that difference to the initial value. Mathematically, this calculates how much the initial value has changed relative to its starting point.
To increase a number by a specific percentage, you multiply the original number by (1 + the decimal form of the percent). To decrease, multiply by (1 - the decimal form of the percent).
Decrease Formula: Value × (1 - (Percentage / 100))
Frequently Asked Questions
An easy mental math trick for calculating 20% is to first find 10% of the number, which you can do simply by moving the decimal point one place to the left. Once you have 10%, just multiply that number by 2 to get 20%.
Percentage change is used when there is a clear "old" value and "new" value, measuring how much a single variable has grown or shrunk over time. Percentage difference is used when comparing two independent values (like the height of two different buildings) where neither is clearly the "original" value. It compares the difference between them to their average.
Reversing a percentage increase is not as simple as subtracting the same percentage. If a $100 item increases by 10%, it becomes $110. If you decrease $110 by 10%, it becomes $99, not $100. To correctly reverse an increase, divide the new value by (1 + the percentage as a decimal). For example, $110 / 1.10 = $100.
Yes. A percentage larger than 100% simply means the value is greater than the whole it is being compared to. For instance, if a company's profits this year are 150% of last year's profits, it means they made 1.5 times as much money.
Percentages provide a standardized denominator of 100, which makes it much easier to compare different ratios quickly. For example, comparing fractions like 3/7 and 5/12 in your head is difficult, but comparing their percentage equivalents (42.8% and 41.6%) is instantly intuitive.