6 percentage calculators in one: basic percent, percent change, percent difference, reverse percentage, ratio to percent, and add/subtract a percentage. All include formulas and step-by-step solutions.
✓Standard Mathematical Percentage Formulas — Khan Academy & NIST — April 2026
Solves all three basic percentage questions: P% of Y = ? · X is what % of Y = ? · X is P% of what whole = ?
Leave blank to calculate thisEnter a percentage.
Leave blank to calculate thisEnter a whole number.
Leave blank to calculate thisEnter a part.
Select what you want to calculate
Measures how much a value changed relative to its starting value. % Change = [(New − Old) / |Old|] × 100
Starting value (before the change)Enter the original value.
Ending value (after the change)Enter the new value.
Compares two values without a reference direction (uses their average as denominator). % Difference = |V1 − V2| / ((V1 + V2) / 2) × 100
Enter Value 1.
Enter Value 2.
Finds the original value before a percentage increase or decrease was applied. Increase: Original = Final / (1 + P/100) · Decrease: Original = Final / (1 − P/100)
Enter the final value.
Enter the percentage.
Was the % an increase or decrease?
Converts a ratio A:B or fraction A/B into a percentage. A as % of B: (A/B) × 100 · A as share of total (A+B): A/(A+B) × 100
Enter A.
Enter B (not zero).
How to interpret the ratio
Adds or subtracts a percentage from a number (discounts, tax, tips, markups). Add P%: result = N × (1 + P/100) · Subtract P%: result = N × (1 − P/100)
Enter a number.
Enter a percentage.
Add for tax/markup, subtract for discount
Result
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📋 Step-by-Step Solution
⚠️ Disclaimer: All calculations use standard mathematical percentage formulas. Results are mathematically exact given the inputs provided.
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Sources & Methodology
✅All percentage formulas verified against Khan Academy Mathematics Curriculum and NIST Digital Library of Mathematical Functions. Percent difference formula uses the standard symmetric definition (average of two values as denominator). Percent change uses the standard asymmetric definition (original value as denominator).
Khan Academy — Percentages, Ratios & Proportions
Standard definitions and formulas for basic percentage, percent change, percent increase/decrease used in K-12 and collegiate mathematics education worldwide.
Reference for mathematical notation, ratio definitions, and dimensional analysis underlying percentage calculations. Used for percent difference and ratio conversion methodology.
Basic %: Part = (P/100)×Whole | Pct = (Part/Whole)×100 | Whole = (Part/P)×100. Percent Change: [(New−Old)/|Old|]×100. Percent Difference: [|V1−V2|/((V1+V2)/2)]×100. Reverse %: increase=Final/(1+P/100), decrease=Final/(1−P/100). Ratio to %: (A/B)×100 or A/(A+B)×100. Add/Sub: N×(1±P/100). All results rounded to 6 significant figures.
Last reviewed: April 2026
Complete Percentage Guide — All Formulas, Definitions & Real Examples
A percentage is a way of expressing a number as a fraction of 100. The word comes from the Latin per centum — “for each hundred.” Percentages are the universal language for comparing values at different scales, expressing changes over time, and communicating proportions clearly. Understanding the six core percentage calculations below covers virtually every percentage question you will ever encounter.
All 6 Percentage Formulas — Quick Reference
1. Basic %: Part = (P/100) × Whole | P% = (Part/Whole) × 100 | Whole = Part / (P/100) 2. % Change: [(New − Old) / |Old|] × 100 3. % Difference: |V1 − V2| / ((V1+V2)/2) × 100 4. Reverse %: Increase: Original = Final / (1 + P/100)
Decrease: Original = Final / (1 − P/100) 5. Ratio to %: (A/B) × 100 or A/(A+B) × 100 6. Add/Sub %: Add: N × (1 + P/100) | Subtract: N × (1 − P/100)
1. Basic Percentage — Three Questions, One Formula
Every basic percentage problem is one of three questions: (a) What is P% of Y? (b) X is what percent of Y? (c) X is P% of what whole number? All three use the same relationship: Part = (Percentage / 100) × Whole. Rearranging solves for any unknown.
Real examples: (a) 15% of $240 = 0.15 × 240 = $36 tip. (b) 45 correct out of 60 total = (45/60) × 100 = 75% score. (c) $18 is 12% of what total budget? = 18 / 0.12 = $150 total budget.
2. Percent Change — Measuring Growth and Decline
Percent change answers: by what percentage did X change from its original value? The formula always uses the original (old) value as the denominator, making it a measure relative to the starting point. A positive result is a percent increase; a negative result is a percent decrease.
Percent Change = [(New − Old) / |Old|] × 100
Examples: A stock rose from $45 to $54: (54−45)/45 ×100 = +20% increase. A salary dropped from $70,000 to $63,000: (63,000−70,000)/70,000 ×100 = −10% decrease. Monthly website traffic went from 8,000 to 12,400: +55% increase.
3. Percent Difference — Comparing Two Equal Values
Percent difference is used when neither of two values is “original” — both are simply measurements or prices to compare. Because neither value is the reference baseline, the average of both values is used as the denominator: |V1 − V2| / ((V1+V2)/2) × 100.
Example: Two suppliers quote $85 and $100 for the same part. Percent difference = |85−100| / ((85+100)/2) × 100 = 15/92.5 × 100 = 16.2% difference. Note: if you used percent change with $85 as the base, you would get (100−85)/85 × 100 = 17.6% — a different answer because the denominator is different.
4. Reverse Percentage — Finding the Original Value
Reverse percentage undoes a percentage change to find the original value before the change was applied. This is essential for recovering pre-tax prices from VAT-inclusive prices, finding original prices from sale prices, or recovering original scores from curved marks.
Scenario
Final Value
% Applied
Formula
Original
Price after 20% increase
$120
+20%
120 / 1.20
$100
Sale price after 25% off
$75
−25%
75 / 0.75
$100
Price inc. 8% VAT
£108
+8%
108 / 1.08
£100
Curved exam score
85 points
+15%
85 / 1.15
73.9
5. Ratio to Percentage
Converting a ratio to a percentage makes proportions easier to compare and communicate. There are two interpretations: A as a percentage of B means (A/B) × 100. A as a share of the combined total (A+B) means A/(A+B) × 100.
Examples: Test score 18/24 = (18/24) × 100 = 75%. Election result: 2,400 votes for Candidate A, 1,600 for Candidate B, total 4,000 = A’s share = 2,400/4,000 × 100 = 60%. Gear ratio 3:5 means third gear covers 3/(3+5) = 37.5% of full speed range.
Adding P% to a number and then subtracting P% does NOT return to the original. This counterintuitive property comes from the fact that each percentage is applied to a different base. After adding 20% to 100 (result: 120), subtracting 20% from 120 gives 96, not 100. This is why successive percentage changes must be compounded, not added.
⚠️ Common Mistake — Percentage vs Percentage Point: If an interest rate rises from 3% to 5%, that is a 2 percentage point increase, but a 66.7% percent change in the rate. Percentage points are arithmetic differences between two percentages. Percent change is the relative change. These are completely different — confusing them is the most common error in financial and statistical reporting.
Real-World Percentage Applications
Context
Formula Used
Example
Result
Restaurant tip
Add % (Mode 6)
18% on $65 bill
$11.70 tip / $76.70 total
Sales discount
Subtract % (Mode 6)
30% off $149
$44.70 off / $104.30 sale
Sales tax
Add % (Mode 6)
8.25% on $200
$16.50 tax / $216.50 total
Pre-tax price
Reverse % (Mode 4)
$216.50 at 8.25% tax
$200 pre-tax
Grade score
Basic % (Mode 1)
42 right of 56
75%
Stock gain
% Change (Mode 2)
$180 from $150
+20%
Price comparison
% Difference (Mode 3)
$95 vs $115
18.9% difference
Survey ratio
Ratio to % (Mode 5)
3:7 agree:disagree
30% agree
Frequently Asked Questions
Percentage = (Part / Whole) × 100. Example: 25 out of 80 = (25/80) × 100 = 31.25%. To find P% of a number: result = (P/100) × number. Example: 15% of 200 = 0.15 × 200 = 30. To find the whole when you know the part and percentage: whole = part / (P/100). Example: 18 is 12% of what? = 18 / 0.12 = 150.
Percent Change = [(New Value − Old Value) / |Old Value|] × 100. Positive result = increase. Negative result = decrease. Example: price from $50 to $65: [(65−50)/50] × 100 = +30% increase. Score from 90 to 72: [(72−90)/90] × 100 = −20% decrease. Always use the original (starting) value in the denominator.
Percent change uses one specific starting value as the reference (denominator). It measures how much one value changed from another. Percent difference uses the average of both values as the denominator — it compares two values without designating either as the baseline. Use percent change when you have a clear before/after (old/new). Use percent difference when comparing two equal-standing values, like two quotes or two measurements.
Original = Sale Price / (1 − Discount%/100). Example: item on sale for $75 after 25% off: original = $75 / (1 − 0.25) = $75 / 0.75 = $100. For a price after a percentage increase: Original = Final / (1 + Increase%/100). Example: item is $126 after a 5% price increase: original = $126 / 1.05 = $120. Use Mode 4 (Reverse %) above to solve instantly.
Both use the same percent change formula: [(New − Old) / Old] × 100. Percent increase: new is bigger than old, result is positive. Percent decrease: new is smaller than old, result is negative. Example salary increase: $50,000 to $57,500 = [(57,500−50,000)/50,000] × 100 = +15%. Example price reduction: $120 to $90 = [(90−120)/120] × 100 = −25%.
(A/B) × 100 gives A as a percentage of B. A/(A+B) × 100 gives A as a share of the combined total. Example: score 18/24 = (18/24) × 100 = 75%. Example: vote count 600:400 = 600/(600+400) × 100 = 60% for first candidate. Use Mode 5 (Ratio to %) above, and select the appropriate interpretation in the dropdown.
A percentage point is the arithmetic difference between two percentages: if interest rates rise from 3% to 5%, that is 2 percentage points. Percent change in the rate is [(5−3)/3] × 100 = 66.7%. They measure completely different things. Reporting one when you mean the other is a very common error in journalism and financial reporting. Always clarify whether you mean percentage points or percent change when discussing rate or proportion movements.
Total = Price × (1 + Tax%/100). Example: $85 item with 8.5% sales tax = $85 × 1.085 = $92.23. Tax amount = $85 × 0.085 = $7.23. To recover pre-tax price from a tax-inclusive price: Pre-tax = Total / (1 + Tax%/100). Example: $92.23 at 8.5% tax = $92.23 / 1.085 = $85. Use Mode 6 to add tax, Mode 4 to reverse it.
Because each percentage is calculated on a different base. A 20% increase on 100 gives 120. A 20% decrease on 120 gives 96, not 100. The increase was 20% of 100 (= 20). The decrease was 20% of 120 (= 24). More is subtracted than was added because the base changed. A 50% gain and 50% loss: 100 × 1.5 = 150 × 0.5 = 75 — a 25% net loss, not break even.
Sale price = Original × (1 − Discount%/100). Discount amount = Original × (Discount%/100). Example: 30% off $150: sale price = $150 × 0.70 = $105. Discount = $150 × 0.30 = $45. To find the discount percentage given original and sale price: Discount% = [(Original − Sale) / Original] × 100 = [(150−105)/150] × 100 = 30%.
Percentage error = |(Measured − True) / True| × 100. It measures how far a measurement is from the true or accepted value, expressed as a percentage of the true value. Example: you measure a 10 cm rod as 10.3 cm: error = |(10.3−10)/10| × 100 = 3% error. This is a form of percent change where the “true” value is the reference denominator. Use Mode 2 with the true value as “Old” and measured value as “New.”
Grade % = (Points Earned / Total Points) × 100. Example: 42 out of 56 questions correct = (42/56) × 100 = 75%. For weighted grades: multiply each grade by its weight, sum the weighted scores, divide by total weight. Example: Test (60%): 80, Homework (40%): 90. Weighted grade = (80×0.6 + 90×0.4) / 1.0 = (48 + 36) = 84%. Use Mode 1 (Basic %) for simple grade percentage.