How to Calculate Percentages
A percentage is a way of expressing a number as a fraction of 100. The word comes from the Latin per centum, meaning "by the hundred." Percentages are used everywhere — discounts, tax rates, test scores, statistics, and financial calculations.
Finding X% of a Number
To find X% of Y, convert the percentage to a decimal by dividing by 100, then multiply by the number:
Formula: Result = (X / 100) × Y
Example: 15% of 300 = (15/100) × 300 = 0.15 × 300 = 45
Finding What Percent X is of Y
To determine what percentage one number represents of another, divide the part by the whole and multiply by 100:
Formula: Percentage = (Part / Total) × 100
Example: What percent is 75 of 250? (75/250) × 100 = 30%
Calculating Percentage Change
Percentage change measures how much a value has increased or decreased relative to its original value:
Formula: % Change = ((New - Old) / |Old|) × 100
Example: A price went from $80 to $100. Change = ((100-80)/80) × 100 = 25% increase
Finding the Whole from a Part
When you know a part and what percentage it represents, you can find the whole:
Formula: Whole = Part / (Percentage / 100)
Example: 36 is 15% of what? 36 / 0.15 = 240
Percentage Difference
Percentage difference compares two values symmetrically — neither is treated as the "original." It uses the average of both values as the base:
Formula: % Difference = |A - B| / ((|A| + |B|) / 2) × 100
Example: Values 75 and 90. Difference = |75-90| / ((75+90)/2) × 100 = 15/82.5 × 100 = 18.18%
Adding a Percentage
To increase a value by a percentage:
Formula: Result = Value × (1 + Percentage / 100)
Example: 250 + 12% = 250 × 1.12 = 280
Reverse Percentage
When you know the final amount after a percentage was added, find the original:
Formula: Original = Result / (1 + Percentage / 100)
Example: A price is $336 after 12% markup. Original = 336 / 1.12 = $300
Percentage Examples
- 20% of 500 = (20/100) × 500 = 100
- 45 is what % of 180? (45/180) × 100 = 25%
- Change from 200 to 250: ((250-200)/200) × 100 = 25% increase
- 36 is 15% of what? 36 / 0.15 = 240
- % Difference between 75 and 90: 18.18%
- 250 + 12%: 280
- $336 after 12% added, original: $300
When to Use Each Calculator Mode
- X% of Y: Discounts, tax calculations, tips
- X is what %: Test scores, proportions, market share
- % Change: Price changes, growth rates, performance metrics
- Find Whole: "I saved $36 which was 15% off — what was the original price?"
- % Difference: Comparing two measurements where neither is the "baseline"
- Add %: Markup pricing, tax-inclusive totals
- Reverse %: Finding pre-tax price, removing markup, reversing compound interest
Mental Math Shortcuts
- 10%: Move the decimal one place left. 10% of $85 = $8.50
- 5%: Find 10%, then halve it. 5% of $85 = $4.25
- 20%: Find 10%, then double it. 20% of $85 = $17.00
- 1%: Move the decimal two places left. 1% of $85 = $0.85
- Any %: Combine the above. 15% = 10% + 5%. 7% = 5% + 1% + 1%
Handy trick: X% of Y equals Y% of X. So 8% of 50 = 50% of 8 = 4. Use whichever direction is easier to compute.
Percent vs Percentage Points
These terms are often confused but mean different things:
- Percentage points: The absolute difference between two percentages. Going from 5% to 7% is a 2 percentage point increase
- Percent change: The relative change. Going from 5% to 7% is a 40% increase (2/5 × 100)
This distinction matters in news, finance, and statistics. "Unemployment rose 2 percentage points" (5% to 7%) is very different from "unemployment rose 2 percent" (5% to 5.1%).