Key takeaway
The percentage formula is really three: P = (part ÷ whole) × 100 for a share of a total, value = (P ÷ 100) × number for a percent of something, and change% = ((new − old) ÷ old) × 100 for growth or decline. All three come from 'out of 100' — the % sign just means ÷100.
Scope and assumptions
These are general arithmetic forms; domain-specific rules (grading weights, tax inclusion) may adjust the base. Confirm in your context. This site is informational and not official.
Short answer: The percentage formula is really three: P = (part ÷ whole) × 100 for a share of a total, value = (P ÷ 100) × number for a percent of something, and change% = ((new − old) ÷ old) × 100 for growth or decline. All three come from "out of 100". This page is the formula reference (forms and derivations); for the step-by-step how-to that picks the formula by question type, see How to Calculate Percentage.
Quick navigation
- How to calculate → how-to-calculate-percentage
- What a % is → what-is-a-percentage
- Find the base → how-to-find-base-from-a-percent
- The calculator → Percentage Calculator
1. Where the formula comes from
"Percent" means per hundred. So any percentage is just a fraction whose denominator is 100:
25% = 25 / 100 = 0.25
That single idea gives every percentage formula — you are always building a fraction and then scaling it to a base of 100.
2. Form 1 — part of a whole
P = (part ÷ whole) × 100
You answered 17 of 25 questions.
(17 ÷ 25) × 100 = 68%
The "whole" (25) is the denominator. This is the form for scores, shares, and survey results.
3. Form 2 — percent of a number
value = (P ÷ 100) × number
What is 25% of 80?
(25 ÷ 100) × 80 = 0.25 × 80 = 20
Here you start from a percent and a total and recover the part.
4. Form 3 — percent change
change% = ((new − old) ÷ old) × 100
A value goes 40 → 50.
((50 − 40) ÷ 40) × 100 = +25%
The base is the old value. A 50 → 40 move gives −20%, not the same magnitude, because the base differs.
5. Formula comparison
| Goal | Formula | Solve for |
|---|---|---|
| Share of total | (part ÷ whole) × 100 | P |
| Part from % | (P ÷ 100) × number | value |
| Change | ((new − old) ÷ old) × 100 | change% |
| Whole from % | part ÷ (P ÷ 100) | whole |
The last row is the rearranged Form 2 — see how-to-find-base-from-a-percent.
6. Common mistakes
- Using the wrong base in change — always the old value, never the new.
- Dropping ÷100 in Form 2 — 25 × 80 = 2000, not 20.
- Confusing Form 1 and Form 2 — "17 of 25" and "17% of 25" are different questions (68% vs 4.25).
- Rounding the fraction before ×100 — round only the final percentage.
- Treating % as a unit you can add directly — convert to decimals first when combining.
7. The three forms at a glance
| Goal | Formula | You solve for |
|---|---|---|
| Share of total | (part ÷ whole) × 100 | the percent |
| Part from percent | (P ÷ 100) × number | the value |
| Change | ((new − old) ÷ old) × 100 | the move |
| Whole from percent | part ÷ (P ÷ 100) | the base |
All four are the same "out of 100" idea, just rearranged. Pick by the unknown, and the formula writes itself. If you ever see a result that looks 100× too large, you used a percent where the decimal belonged.
8. Why the percent sign hides a division
The % symbol is a quiet instruction to divide by 100, and forgetting it is the root of almost every percentage error. When you see 25%, read it as 25 ÷ 100 = 0.25 before any other operation. Treat the sign as part of the number, not a label you can drop. Once that habit is automatic, the three formula forms stop feeling like separate rules and start feeling like one idea written three ways. The sign also tells the reader the denominator is fixed at 100, which is exactly why percents compare cleanly across totals of any size.
FAQ
Q: What is the main percentage formula? P = (part ÷ whole) × 100 — the share of a total, expressed out of 100.
Q: How do I write percent of a number as a formula? value = (P ÷ 100) × number. So 25% of 80 = (25 ÷ 100) × 80 = 20.
Q: What is the percent change formula? ((new − old) ÷ old) × 100. The base is always the old value.
Q: Why does % mean divide by 100? Percent means "per hundred", so 25% = 25/100 = 0.25; the sign is shorthand for ÷100.
Q: Can one formula do all three? No — the three are rearrangements for different unknowns (percent, value, or base). Pick by what you are solving for.
Sources: Khan Academy percentage basics; Maths is Fun — Introduction to Percentages.
Disclaimer: These are general arithmetic forms; domain-specific rules (grading weights, tax inclusion) may adjust the base. Confirm in your context. This site is informational and not official.
Sources and references
- Khan Academy percentage basicshttps://www.khanacademy.org/math/pre-algebra/x31947de81e3f6453:percentages
- Maths is Fun — Introduction to Percentageshttps://www.mathsisfun.com/percentage.html