Percentage formulas
Every percentage formula on one page, with the thing that actually causes mistakes made explicit: which one answers which question.
All of these come from a single idea. A percentage is a fraction whose denominator is fixed at 100, so (Part ÷ Whole) × 100 is the root formula and everything below is that equation rearranged to solve for a different unknown.
Percentage of a number
You know the percentage and the whole, and want the part.
(Percentage ÷ 100) × Number
(20 ÷ 100) × 500 = 100
Use for tips, tax amounts, commissions and shares of a total.
What percent one number is of another
You know the part and the whole, and want the percentage.
(Part ÷ Whole) × 100
(25 ÷ 200) × 100 = 12.5%
Use for exam scores, completion rates and market share.
Finding the whole
You know the part and the percentage, and want the whole.
Part ÷ (Percentage ÷ 100)
50 ÷ 0.2 = 250
Use for pre-discount prices, pre-tax amounts and totals implied by a deposit.
Percentage increase
How much a value grew, measured against where it started.
((New − Original) ÷ Original) × 100
((150 − 100) ÷ 100) × 100 = 50%
The original value is the reference point, which is why it goes underneath.
Percentage decrease
How far a value fell. Identical to increase, written so the answer comes out positive.
((Original − New) ÷ Original) × 100
((200 − 150) ÷ 200) × 100 = 25%
Quoting “a 25% decrease” is conventional; “a −25% change” says the same thing.
Percentage change
Movement in either direction. A positive result is a rise, a negative one a fall.
((New − Original) ÷ |Original|) × 100
((125 − 100) ÷ 100) × 100 = 25%
Directional: swapping the two values changes the answer.
Percentage difference
The gap between two peer values, measured against their average.
(|A − B| ÷ ((A + B) ÷ 2)) × 100
(|100 − 120| ÷ 110) × 100 = 18.18%
Symmetric: swapping the two values leaves the answer unchanged.
Discount and final price
The money off, and the price you actually pay.
Price × ((100 − Discount) ÷ 100)
100 × 0.8 = 80
Multiplying by what is left is one step; finding the discount and subtracting is two.
The four that get confused
Increase, decrease, change and difference are routinely used interchangeably. They are not interchangeable, and on the same pair of numbers they give different answers.
| Formula | Reference point | Order matters? | 100 → 120 |
|---|---|---|---|
| Increase | The original value | Yes | 20% increase |
| Decrease | The original value | Yes | Not applicable — the value rose |
| Change | The original value | Yes | +20% |
| Difference | The average of both | No | 18.18% |
Percentage points are a fifth thing
When the values themselves are percentages, the difference between them is measured in percentage points, not percent. A rate moving from 4% to 5% has risen by 1 percentage point and by 25%. Both statements are true, and choosing the flattering one is a well-worn way of making a small change sound large.
Frequently asked questions
What is the basic percentage formula?
How do I convert a percentage to a decimal?
How do I convert a decimal or fraction to a percentage?
Which formula should I use for percentage difference?
Put these formulas to work
- Percentage CalculatorAnswer the three core percentage questions — a percentage of a number, a share as a percentage, and the missing total.
- Percentage Change CalculatorOne tool for movement in either direction — it tells you whether the value rose or fell.
- Percentage Difference CalculatorCompare two values where neither came first — measured against their average.
- Discount CalculatorSee the money off and the price you actually pay before you get to the till.