What Is a Percentage?
A percentage is a way of expressing a number as a fraction of 100. The word "percent" comes from the Latin per centum, meaning "by the hundred." The symbol % is used to denote percentages.
For example, 75% means 75 out of every 100, or equivalently, the fraction 75/100 = 0.75.
The Three Core Percentage Formulas
Almost every percentage problem can be solved using one of three fundamental formulas:
1. Finding a Percentage of a Number
Formula: Part = (Percentage ÷ 100) × Whole
Example: What is 30% of 250?
- Part = (30 ÷ 100) × 250
- Part = 0.30 × 250
- Part = 75
So 30% of 250 is 75.
2. Finding What Percentage One Number Is of Another
Formula: Percentage = (Part ÷ Whole) × 100
Example: 45 is what percentage of 180?
- Percentage = (45 ÷ 180) × 100
- Percentage = 0.25 × 100
- Percentage = 25%
So 45 is 25% of 180.
3. Finding the Whole When a Percentage and Part Are Known
Formula: Whole = Part ÷ (Percentage ÷ 100)
Example: 60 is 40% of what number?
- Whole = 60 ÷ (40 ÷ 100)
- Whole = 60 ÷ 0.40
- Whole = 150
So 60 is 40% of 150.
How to Calculate Percentage Change
Percentage change measures how much a value has increased or decreased relative to its original value.
Formula: Percentage Change = ((New Value − Old Value) ÷ Old Value) × 100
A positive result means an increase; a negative result means a decrease.
Example: Percentage Increase
A product cost $80 and now costs $100. What is the percentage increase?
- Percentage Change = ((100 − 80) ÷ 80) × 100
- = (20 ÷ 80) × 100
- = 0.25 × 100
- = 25% increase
Example: Percentage Decrease
A price dropped from $200 to $150. What is the percentage decrease?
- Percentage Change = ((150 − 200) ÷ 200) × 100
- = (−50 ÷ 200) × 100
- = −0.25 × 100
- = −25% (a 25% decrease)
Percentage Tips and Tricks
The "10% Trick"
To find 10% of any number, simply move the decimal point one place to the left. For example: 10% of 350 = 35. You can then multiply or divide to find other percentages quickly:
- 5% = half of 10% → 5% of 350 = 17.5
- 20% = double of 10% → 20% of 350 = 70
- 25% = divide by 4 → 25% of 350 = 87.5
Converting Between Fractions, Decimals, and Percentages
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/3 | 0.333… | 33.3% |
| 2/3 | 0.666… | 66.7% |
| 1/5 | 0.2 | 20% |
Real-World Applications of Percentages
Sales Tax
If an item costs $45 and the sales tax rate is 8%, the total cost is:
Tax = 8% × $45 = $3.60 → Total = $45 + $3.60 = $48.60
Discounts
A jacket originally priced at $120 is on sale for 35% off:
Discount = 35% × $120 = $42 → Sale price = $120 − $42 = $78
Grade Calculations
You scored 42 out of 50 on a test. Your percentage score is:
(42 ÷ 50) × 100 = 84%
Common Mistakes to Avoid
- Confusing "percent of" with "percent more than": "20% more than 100" is 120, not 20.
- Reversing the change direction: If a price increased by 25%, it does NOT decrease by 25% to get back to the original. A 25% increase followed by a 25% decrease yields 93.75% of the original.
- Forgetting to divide by 100: 25% of 80 = 0.25 × 80 = 20, not 25 × 80 = 2000.