Percentage Calculator
Percentage Fundamentals
A percentage represents a fraction of 100, derived from the Latin “per centum” meaning “by the hundred.” When we express a value as a percentage, we’re stating how many parts out of 100 that value represents.
Percentage Applications
Financial Applications
- Interest calculations: Simple interest = Principal × Rate × Time
- Discounts: Sale price = Original price × (1 − Discount percentage ÷ 100)
- Tax calculations: Total price = Base price × (1 + Tax rate ÷ 100)
- Profit margin: Margin = (Revenue − Cost) ÷ Revenue × 100
- Return on investment: ROI = (Gain − Cost) ÷ Cost × 100
Statistical Applications
- Percentiles: Values that divide a dataset so that a specific percentage falls below that value
- Percentage points: The arithmetic difference between two percentages
- Growth rates: Percentage change in values over time
- Relative frequencies: The proportion of observations in each category
- Probability: Often expressed as percentages to indicate likelihood
Scientific Applications
- Concentration: Percentage of solute in a solution
- Efficiency: Actual output ÷ Theoretical maximum output × 100
- Error rate: (Observed value − Expected value) ÷ Expected value × 100
- Purity levels: Amount of desired substance ÷ Total amount × 100
- Yield percentage: Actual yield ÷ Theoretical yield × 100
Practical Percentage Calculations
1. Calculating Discounts
To calculate a discounted price, multiply the original price by one minus the discount percentage (as a decimal).
2. Calculating Percentage Change
To calculate the percentage change between two values, divide the absolute difference by the original value and multiply by 100.
3. Finding the Original Value Before a Percentage Change
To find the original value before a percentage change, divide the current value by one plus the percentage change (as a decimal).
Common Percentage Mistakes to Avoid
Sequential Percentage Changes
A common error is assuming that percentage changes are additive. For example, a 10% increase followed by a 10% decrease does not return to the original value:
Starting value: 100
After 10% increase: 100 × 1.1 = 110
After 10% decrease: 110 × 0.9 = 99
Result: 1% loss, not 0% change
Percentage Points vs. Percentages
Confusing percentage points with relative percentages leads to misinterpretation of data:
Change from 10% to 15%:
Absolute change: 5 percentage points
Relative change: (15 − 10) ÷ 10 × 100 = 50% increase
Percentage Change from Zero
Calculating percentage change when the original value is zero is mathematically undefined:
Change from 0 to 30:
Percentage change = (30 − 0) ÷ 0 × 100 = undefined
Solution: In such cases, use absolute change or other appropriate metrics.
Reversing Percentage Changes
To reverse a percentage increase, the decrease percentage needs to be calculated differently:
If a value increases by 25% (× 1.25), to return to the original:
Decrease percentage = (1 − 1 ÷ 1.25) × 100 = 20%
100 × 1.25 = 125, then 125 × 0.8 = 100
Reference Table: Common Percentage Conversions
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33.33…% |
| 1/4 | 0.25 | 25% |
| 1/5 | 0.2 | 20% |
| 1/6 | 0.166… | 16.66…% |
| 1/8 | 0.125 | 12.5% |
| 1/10 | 0.1 | 10% |
| 1/12 | 0.083… | 8.33…% |
| 1/20 | 0.05 | 5% |
| 1/25 | 0.04 | 4% |
| 1/50 | 0.02 | 2% |
| 1/100 | 0.01 | 1% |