Formula reference
Percentage formulas — complete reference
This page lists every percentage formula used across the site. Each entry gives the formula, a plain-language definition, the correct baseline to use, a worked numerical example, and notes on when the formula applies and where it does not.
Useful starting points
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Core calculator
Apply any of these formulas interactively with the main calculator.
Open the calculatorWorked examples
See each formula applied to real numbers with full step-by-step working.
Browse examplesFAQ
Common questions about when to use each formula and how to avoid mistakes.
Read the FAQGuides
Structured learning paths linking formula knowledge to the right calculator.
Browse guidesBasic percentage formulas
These three formulas cover the most fundamental percentage questions. They share the same underlying relationship between part, whole, and percentage — rearranged depending on which value you know and which you want to find.
1. Percentage of a number
Finds a portion of a total when you know the percentage rate. Used for tips, commissions, discounts applied to prices, and any scenario where a rate is applied to a whole.
Formula: amount = (percentage / 100) × value
Example: 20% of 350: (20 / 100) × 350 = 0.20 × 350 = 70. So 20% of 350 is 70.
X% of Y calculator2. Part as a percentage of a whole
Converts a part-to-whole ratio into a percentage. Used for test scores, market share, completion rates, and any situation where you want to express a fraction as a rate per hundred.
Formula: percentage = (part / whole) × 100
Example: 36 out of 120: (36 / 120) × 100 = 0.30 × 100 = 30%. So 36 is 30% of 120.
Open the ratio calculator3. Finding the whole from a part and percentage
Works backwards from a known part and percentage to find the total. Used when a proportion is known but the whole it came from is not.
Formula: whole = (part / percentage) × 100
Example: If 45 is 15% of a total: (45 / 15) × 100 = 3 × 100 = 300. The total is 300.
Open the reverse percentage calculatorPercentage change formulas
Change formulas require an original and a new value. The original value is always the denominator — this is the most common source of percentage calculation errors. Using the wrong baseline produces a correct-looking number that answers a different question.
4. Percentage increase
Measures how much a value has grown relative to its starting point. The original (starting) value is the denominator. The result is always a positive number expressing the size of the growth.
Formula: increase (%) = ((new − original) / original) × 100
Example: From 80 to 100: ((100 − 80) / 80) × 100 = (20 / 80) × 100 = 25%.
Open the percentage increase calculator5. Percentage decrease
Measures how much a value has fallen relative to its starting point. The original value remains the denominator. The result is expressed as a positive number representing the reduction.
Formula: decrease (%) = ((original − new) / original) × 100
Example: From 200 to 170: ((200 − 170) / 200) × 100 = (30 / 200) × 100 = 15%.
Open the percentage decrease calculator6. Signed percentage change
A single signed formula that produces a positive number for increases and a negative number for decreases. This is the most general form of the percentage change formula and is used in the core calculator's percentage change mode.
Formula: change (%) = ((new − original) / |original|) × 100
Example: From 100 to 80 (a decrease): ((80 − 100) / |100|) × 100 = (−20 / 100) × 100 = −20%. From 100 to 120: ((120 − 100) / 100) × 100 = 20%.
7. Percentage difference
Compares two peer values symmetrically by using their average as the reference. Unlike percentage change, the result is the same regardless of which value is entered first. Use this when neither value is the natural starting point.
Formula: difference (%) = (|value1 − value2| / ((value1 + value2) / 2)) × 100
Example: 98 vs 102: |98 − 102| / ((98 + 102) / 2) × 100 = 4 / 100 × 100 = 4%.
Open the percentage difference calculatorReverse percentage formulas
Reverse calculations find an original value from a known final value and a percentage change. The key principle is to divide by the multiplier that created the final value — never subtract the percentage directly from the final amount.
8. Reverse a percentage increase
If a value has increased by a known percentage and you want the original, divide the final value by (1 + rate / 100). Subtracting the percentage from the final value is a common mistake and gives a different (incorrect) answer.
Formula: original = final / (1 + rate / 100)
Example: Final value 132 after a 10% increase: 132 / (1 + 0.10) = 132 / 1.10 = 120.
Open the reverse percentage calculator9. Reverse a percentage decrease
If a value has decreased by a known percentage and you want the original, divide the final value by (1 − rate / 100). This is how you find the original price before a sale discount.
Formula: original = final / (1 − rate / 100)
Example: Final value 85 after a 15% decrease: 85 / (1 − 0.15) = 85 / 0.85 = 100.
Open the reverse percentage calculatorPrice, VAT, and tax formulas
These formulas apply percentage arithmetic to pricing. Note that they perform the arithmetic for the rate entered — they do not determine what rate applies in any given jurisdiction or situation.
10a. Discount — amount saved and final price
Calculates the monetary saving from a percentage discount and the resulting sale price. Two equivalent forms: calculate the saving explicitly, or multiply the original by the complement of the rate.
Formula: saving = price × (rate / 100); final = price − saving. Equivalent: final = price × (1 − rate / 100)
Example: 20% off £150: saving = £150 × 0.20 = £30; final = £150 − £30 = £120. Or: £150 × 0.80 = £120.
Open the discount calculator10b. Add VAT or tax (net to gross)
Multiplying the net amount by (1 + rate / 100) gives the gross amount including tax. The VAT or tax element is the difference between gross and net.
Formula: gross = net × (1 + rate / 100); tax amount = gross − net
Example: Net £400 with 20% VAT: £400 × 1.20 = £480 gross; VAT = £480 − £400 = £80.
Open the VAT calculator10c. Remove VAT or tax (gross to net)
Dividing the gross amount by (1 + rate / 100) recovers the net amount. Note: this is not the same as subtracting the percentage from the gross, which would give an incorrect result.
Formula: net = gross / (1 + rate / 100); tax amount = gross − net
Example: Gross £480 with 20% VAT: £480 / 1.20 = £400 net; VAT = £480 − £400 = £80.
Open the VAT calculatorImportant notes on using these formulas
Zero and zero-multiplier cases: percentage change divides by the original value, so it is undefined when the original is zero. Reverse percentage divides the final value by (1 ± rate / 100), so it is undefined only when that multiplier is zero, such as when reversing a 100% decrease.
Negative values: the signed percentage change formula uses the absolute value of the original (|original|) in the denominator. This ensures the sign of the result reflects the direction of change rather than the sign of the starting value.
Rounding: always carry full precision through intermediate calculations and round only the final result to avoid compounding rounding errors.
VAT and tax rate source: the formulas accept any rate you enter. The correct rate to use for a given transaction depends on current law, the nature of the goods or services, and the jurisdictions involved — consult an accountant or the relevant tax authority for compliance guidance.
Apply a formula now
Open the relevant calculator to apply any formula to your own numbers, or browse the worked examples to see each formula in action.
Related percentage calculators
Use a focused calculator when your question needs a different baseline or a price-specific formula.
Percentage Calculator
Find a percentage, percentage ratio, or percentage change.
Percentage Increase Calculator
Measure how much a value has grown from its starting point.
Percentage Decrease Calculator
Work out the percentage reduction from an original value.
Percentage Difference Calculator
Compare two values when neither one is the original baseline.
Reverse Percentage Calculator
Find an original amount from a final value and percentage.
Discount Calculator
Calculate the amount saved and the final sale price.