Percentage Calculator

Quick Reference: Common Percentages

Percentage	Decimal	Fraction	Example (of 100)
1%	0.01	1/100	1
5%	0.05	1/20	5
10%	0.10	1/10	10
12.5%	0.125	1/8	12.5
20%	0.20	1/5	20
25%	0.25	1/4	25
33.33%	0.3333	1/3	33.33
50%	0.50	1/2	50
66.67%	0.6667	2/3	66.67
75%	0.75	3/4	75
100%	1.00	1/1	100

Example Problems

Understanding Percentages

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 % means "per 100" or "out of 100."

Basic Percentage Formulas

1. Finding a Percentage of a Number

Formula: Result = (Percentage ÷ 100) × Number

Example: 25% of 80 = (25 ÷ 100) × 80 = 0.25 × 80 = 20

2. Finding What Percentage One Number is of Another

Formula: Percentage = (Part ÷ Whole) × 100

Example: 15 is what % of 60? = (15 ÷ 60) × 100 = 25%

3. Finding the Whole When You Know the Part and Percentage

Formula: Whole = (Part ÷ Percentage) × 100

Example: 20 is 25% of what? = (20 ÷ 25) × 100 = 80

4. Percentage Increase

Formula: New Value = Old Value × (1 + Percentage/100)

Example: Increase 50 by 20% = 50 × (1 + 20/100) = 50 × 1.20 = 60

5. Percentage Decrease

Formula: New Value = Old Value × (1 - Percentage/100)

Example: Decrease 50 by 20% = 50 × (1 - 20/100) = 50 × 0.80 = 40

6. Percentage Change

Formula: % Change = [(New Value - Old Value) ÷ Old Value] × 100

Example: From 50 to 75 = [(75 - 50) ÷ 50] × 100 = 50% increase

7. Percentage Difference

Formula: % Difference = [|Value1 - Value2| ÷ Average] × 100

Where Average = (Value1 + Value2) ÷ 2

Example: Between 50 and 70 = [|50 - 70| ÷ 60] × 100 = 33.33%

Percentage Change vs. Percentage Difference

• Percentage Change: Compares a new value to an original value (has direction - increase or decrease)
• Percentage Difference: Compares two values without a reference point (no direction - always positive)

Percentage Points vs. Percentage

Important Distinction:

• Percentage Points: Absolute difference between percentages
• Percentage: Relative change

Example: Interest rate increases from 3% to 5%

• Increase of 2 percentage points (5% - 3% = 2 points)
• Increase of 66.67% [(5-3) ÷ 3 × 100 = 66.67%]

Common Percentage Applications

Finance & Shopping

• Sales tax and discounts
• Interest rates and APR
• Investment returns
• Price changes
• Tips and gratuities

Statistics & Science

• Probability and chance
• Experimental error
• Population changes
• Survey results
• Concentration and purity

Everyday Life

• Test scores and grades
• Completion progress
• Battery levels
• Nutritional information
• Sports statistics

Tips for Working with Percentages

Mental Math Shortcuts

• 10%: Move decimal point left one place (10% of 450 = 45)
• 1%: Move decimal point left two places (1% of 450 = 4.5)
• 5%: Half of 10% (5% of 450 = 22.5)
• 20%: Double 10% (20% of 450 = 90)
• 25%: Divide by 4 (25% of 400 = 100)
• 50%: Divide by 2 (50% of 450 = 225)

Converting Between Forms

• Percentage to Decimal: Divide by 100 (45% = 0.45)
• Decimal to Percentage: Multiply by 100 (0.45 = 45%)
• Fraction to Percentage: Divide and multiply by 100 (3/4 = 0.75 = 75%)
• Percentage to Fraction: Put over 100 and simplify (60% = 60/100 = 3/5)

Common Mistakes to Avoid

1. Mixing Up Percentage and Percentage Points

Wrong: "Taxes increased from 5% to 7%, a 2% increase"

Right: "Taxes increased from 5% to 7%, a 2 percentage point increase (40% increase)"

2. Applying Sequential Percentages Incorrectly

Wrong: 50% off + 20% off = 70% off total

Right: First discount 50%, then 20% off remaining = 60% off total

Calculation: $100 - 50% = $50, then $50 - 20% = $40 (60% off original)

3. Using Wrong Base for Percentage Change

When calculating percentage change, always divide by the ORIGINAL (old) value, not the new value.

4. Forgetting Order of Operations

When increasing by a percentage, multiply by (1 + percentage/100), not add the percentage.

Wrong: 100 + 20% = 120

Right: 100 × (1 + 0.20) = 120

Practice Tips

• Estimate before calculating to catch errors
• Check if your answer makes sense
• Verify calculations with different methods
• Use real-world examples to build intuition
• Remember that percentages are just another way of expressing fractions