Percentage Calculator

Understanding Percentages

A percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign "%". Percentages are used to express proportions, rates, and changes in a standardized way that's easy to understand and compare.

Basic Percentage Formula

The fundamental percentage formula is:

Percentage = (Value / Total Value) × 100

Common Percentage Calculations

1. Finding a Percentage of a Number

Formula: (Percentage ÷ 100) × Number

Example: What is 20% of 150? (20 ÷ 100) × 150 = 30

2. Finding What Percentage One Number Is of Another

Formula: (Part ÷ Whole) × 100

Example: 30 is what percent of 150? (30 ÷ 150) × 100 = 20%

3. Percentage Increase/Decrease

Formula: [(New Value - Original Value) ÷ Original Value] × 100

Example: Price increased from $100 to $120: [(120 - 100) ÷ 100] × 100 = 20% increase

4. Reverse Percentage (Finding Original Value)

Formula: Final Value ÷ (1 + Percentage/100)

Example: Price after 20% increase is $120: 120 ÷ (1 + 20/100) = $100 original

Real-World Applications of Percentages

Finance and Shopping

• Discounts: Calculate sale prices and savings
• Tips: Determine appropriate gratuity amounts
• Tax Calculations: Compute sales tax or VAT
• Interest Rates: Understand loan and investment returns
• Profit Margins: Calculate business profitability

Academic and Professional

• Grades: Calculate test scores and final grades
• Statistics: Analyze survey results and data
• Science: Express concentrations and compositions
• Business: Track growth rates and market shares

Health and Fitness

• Nutrition: Calculate daily value percentages on food labels
• Body Composition: Track fat percentage changes
• Medication: Understand dosage calculations

Percentage Tricks and Shortcuts

Mental Math Shortcuts

10%: Move decimal one place left (10% of 250 = 25)

5%: Find 10% then halve it (5% of 250 = 12.5)

1%: Move decimal two places left (1% of 250 = 2.5)

50%: Divide by 2 (50% of 250 = 125)

25%: Divide by 4 (25% of 250 = 62.5)

Percentage Change Shortcut

For small percentage changes, you can approximate using multiplication:

Increase by 15% ≈ Multiply by 1.15

Decrease by 15% ≈ Multiply by 0.85

Common Percentage Errors to Avoid

1. Percentage Point vs. Percent

Percentage Point: Absolute difference between percentages (5% to 7% = 2 percentage points)

Percent: Relative difference (5% to 7% = 40% increase)

2. Base Value Confusion

Always identify the correct base value. A 50% increase followed by a 50% decrease does NOT return to the original value.

3. Cumulative Percentages

Successive percentage changes are multiplicative, not additive. A 20% increase followed by another 20% increase results in a 44% total increase, not 40%.

Advanced Percentage Concepts

Compound Percentage Changes

When applying multiple percentage changes consecutively, use the formula:

Final Value = Initial Value × (1 ± p₁/100) × (1 ± p₂/100) × ...

Weighted Averages

When combining percentages from groups of different sizes, calculate the weighted average:

Weighted % = Σ(Value × Weight) ÷ Σ(Weights)

Percentage of Percentage

To find what percentage one percentage is of another, treat them as regular numbers:

Result = (Percentage A ÷ Percentage B) × 100