The Percentage Compendium
Percentage means "per hundred" and denotes a ratio or a fraction as a fraction of 100. It is often denoted using the percent sign, "%". Percentages are used widely in finance, retail, and science to represent proportions, growth, and probability.
The Fundamental Formula
The basic formula for percentage is (Value / Total) × 100 = Percentage. Conversely, to find a value from a percentage, use (Percentage / 100) × Total = Value.
Growth & Decay
Percentage change is used to track the growth or decline of a value over time. It is calculated by dividing the absolute difference by the original value: ((New - Old) / Old) × 100.
Common Equivalents
| Percent | Decimal | Fraction |
|---|---|---|
| 1% | 0.01 | 1/100 |
| 10% | 0.1 | 1/10 |
| 20% | 0.2 | 1/5 |
| 25% | 0.25 | 1/4 |
| 50% | 0.5 | 1/2 |
| 75% | 0.75 | 3/4 |
| 100% | 1.0 | 1/1 |
Real-World Applications
Sales tax and discounts are always calculated as percentages of the original price. For example, a 20% discount on a $100 item saves you $20.
Interest rates (APR) represent the cost of borrowing or the return on investment as a percentage of the principal amount.
Polls and survey results are expressed as percentages to represent the proportion of the population that holds a certain view.
Daily recommended intake values and body fat percentages are crucial metrics used in medicine and nutrition.
Worked Examples
| Question | Calculation | Result |
|---|---|---|
| What is 18% of 250? | 0.18 × 250 | 45 |
| 60 is what percent of 80? | 60 / 80 × 100 | 75% |
| $48 rises to $60 | (60 − 48) / 48 × 100 | 25% increase |
| $64 after 20% off | 64 / 0.80 | $80 original |
| 8.25% tax on $80 | 80 × 0.0825 | $6.60 tax, $86.60 total |
1. Percent of a number
18% of 250 means 18 per 100, or 0.18. Multiply: 0.18 × 250 = 45. The same shortcut works in reverse: 18% of 250 equals 250% of 18, and both are 45.
2. A number as a percent of another
To ask “60 is what percent of 80?”, divide the part by the whole and multiply by 100: 60 / 80 × 100 = 75%. The denominator is the original or whole amount, not the part.
3. Percentage change
Percentage change uses the starting value as the base. A price moving from $48 to $60 changes by $12. 12 / 48 × 100 = 25%. Going from $60 back to $48 is a $12 change on a $60 base, which is 20%, not 25%. The direction changes the base.
4. Price before a discount
A sale price is what remains after the discount. At 20% off, the buyer pays 80% of the original price. If the tag says $64, the original price is 64 / 0.80 = $80. Dividing by the percent off (64 / 0.20) answers a different question and is not the original price.
Sales tax is added the other way. At 8.25%, tax on $80 is 80 × 0.0825 = $6.60, so the register total is $86.60. The sales tax calculator and discount calculator apply these two cases on their own.
Frequently Asked Questions
How do I find a percentage of a number?▼
How do I calculate percentage change?▼
How do I find the original price before a discount?▼
How do I add sales tax to a price?▼
Pro Tip: Reverse Percentages
Percentages are reversible! 8% of 50 is the same as 50% of 8. Both equal 4. This mental shortcut can make quick estimations significantly easier when the original numbers are complex.