Mathematics · Applied Mathematics
Percentage Calculator
Calculate percentages, percentage change, and percentage of a total. Covers all common percentage operations.
Calculator
Formula
The basic percentage formula expresses a part as a fraction of a whole, scaled to 100. Percentage change (Δ%) computes the relative change between an initial value V₁ and a final value V₂, expressed as a percentage of the initial value. A positive result indicates an increase; a negative result indicates a decrease.
Source: Elementary arithmetic. Reference: NIST Handbook of Mathematical Functions.
How it works
The word 'percent' means 'per hundred' (Latin: per centum). Any percentage can be understood as a ratio scaled to a denominator of 100. To find what percentage A is of B, divide A by B and multiply by 100. To find X% of a number, multiply that number by X/100.
Percentage change is directional: it measures how much a value has changed relative to its starting point. A 50% increase followed by a 50% decrease does not return to the original value — it returns to 75% of it. This asymmetry is important in financial contexts.
Worked example
A product originally costs $80 and is now priced at $60. The percentage discount is:
Δ% = (60 − 80) ÷ |80| × 100 = −20 ÷ 80 × 100 = −25%
The product has been discounted by 25%.
Frequently asked questions
How do I find what percentage one number is of another?
Divide the first number by the second and multiply by 100. For example, to find what percentage 45 is of 180: (45 ÷ 180) × 100 = 25%. So 45 is 25% of 180.
What is the difference between percentage and percentage points?
Percentage points measure the arithmetic difference between two percentages. If an interest rate increases from 3% to 5%, it has risen by 2 percentage points but by 66.7% in relative terms (2 ÷ 3 × 100). Confusing these two is a common source of misinterpretation in financial and political reporting.
How do I calculate a percentage increase?
Subtract the original value from the new value, divide by the original value, and multiply by 100. Formula: ((New − Original) ÷ |Original|) × 100. A positive result is an increase, a negative result is a decrease.
How do I find the original value before a percentage increase?
Divide the final value by (1 + percentage/100). For example, if a price of $120 includes a 20% increase, the original price was $120 ÷ 1.20 = $100. This is called 'working backwards' or finding the 'pre-increase' value.
Why does a 50% loss require a 100% gain to recover?
Percentage changes are calculated relative to the current value, not the original. If $100 falls 50% to $50, you need a 100% gain on $50 to return to $100. This asymmetry means losses are mathematically harder to recover from than the equivalent gain — a fundamental principle in risk management.
Last updated: 2025-01-15 · Formula verified against primary sources.