What is a Percentage?
A percentage (from the Latin per centum meaning "by the hundred") is a mathematical ratio or fraction expressed as a fraction of 100. It is denoted using the percent symbol (%). Percentages allow seamless comparison of proportions regardless of the absolute starting size.
Percentage Formulas & Worked Examples
Understanding the three primary mathematical percentage formulas helps in solving everyday financial and academic calculations:
1. Basic Percentage Formula
Find a specific percentage portion of a base number.
Value = (Percentage ÷ 100) × Base
Example: 15% of $200 = (15 ÷ 100) × 200 = $30
Example: 15% of $200 = (15 ÷ 100) × 200 = $30
2. Percentage Share Formula
Determine what percent a part represents relative to a total whole.
Percentage = (Part ÷ Whole) × 100
Example: 30 out of 200 = (30 ÷ 200) × 100 = 15%
Example: 30 out of 200 = (30 ÷ 200) × 100 = 15%
3. Percentage Difference Formula
Compare two numbers without designating an initial starting baseline.
Diff = |V1 - V2| ÷ ((V1 + V2) ÷ 2) × 100
Example: 100 & 150 = |50| ÷ 125 × 100 = 40%
Example: 100 & 150 = |50| ÷ 125 × 100 = 40%
4. Percentage Change Formula
Calculate the resulting total after increasing or decreasing by a percentage.
Final = Base × (1 ± (Percentage ÷ 100))
Example: $100 + 20% = 100 × 1.20 = $120
Example: $100 + 20% = 100 × 1.20 = $120
Common Real-World Applications
- Retail Shopping Discounts: Calculate final sale prices during store promotions (e.g. 25% off a $80 jacket = $60 final price).
- Sales Tax & Gratuity (Tips): Compute total bill costs including state sales tax or restaurant tipping (e.g. 18% tip on a $50 dinner = $9 tip).
- Academic Test Scores: Convert raw test scores into percentages to calculate course grades and GPAs.
- Financial Interest Rates: Determine yield returns, credit card APR fees, and loan interest accrual over time.
Frequently Asked Questions
How do I calculate a percentage of a number?+
To calculate a percentage of a number, convert the percentage into a decimal by dividing by 100, then multiply by the number. For example, to find 15% of 200: (15 ÷ 100) × 200 = 0.15 × 200 = 30.
What is the difference between Percentage Difference and Percentage Change?+
Percentage Change is used when comparing an initial starting value to a final resulting value over time (such as price increases or population growth). Percentage Difference is used when comparing two values of equal status where neither is considered the baseline starting point, taking the absolute difference relative to their average.
How do I calculate percentage increase or decrease?+
To increase a value by X%, multiply the base number by (1 + X/100). For example, $100 increased by 20% is 100 × 1.20 = $120. To decrease a value by X%, multiply by (1 - X/100). For example, $100 decreased by 20% is 100 × 0.80 = $80.
How do I find what percent one number is of another?+
Divide the first number by the second number, then multiply by 100. For example, to find what percent 30 is of 200: (30 ÷ 200) × 100 = 0.15 × 100 = 15%.
How do I calculate reverse percentage (finding the original price)?+
To find the original price before a discount or tax was applied, divide the final price by the percentage factor. For example, if an item costs $80 after a 20% discount, divide $80 by 0.80 (100% - 20%) to find the original price of $100.