Percentage is one of the most useful mathematical concepts in everyday life. From calculating discounts and taxes to understanding statistics and financial returns, percentages are everywhere. This complete guide will teach you everything you need to know about percentages, with practical examples and real-world applications.
The word "percent" comes from the Latin "per centum," meaning "per hundred." A percentage is simply a way of expressing a number as a fraction of 100. Understanding percentages is essential for making informed decisions in shopping, finance, business, and many other areas of life.
What is a Percentage?
A percentage is a way of expressing a number as a fraction of 100. The symbol % (percent) means "per hundred" or "out of 100."
Key Concepts:
- 50% means 50 out of 100, or 50/100, or 0.5
- 100% represents the whole or complete amount
- Percentages can be greater than 100%
- Percentages make comparisons easier
Essential Percentage Formulas
1. Calculate Percentage of a Number
To find what percentage of a number is:
Result = (Percentage ÷ 100) × Number
or
Result = (Percentage × Number) ÷ 100
Example:
What is 25% of 80?
Result = (25 ÷ 100) × 80
Result = 0.25 × 80
Result = 20
2. Find What Percentage One Number is of Another
To find what percentage one number represents of another:
Percentage = (Part ÷ Whole) × 100
Example:
15 is what percentage of 60?
Percentage = (15 ÷ 60) × 100
Percentage = 0.25 × 100
Percentage = 25%
3. Find the Whole When You Know the Part and Percentage
To find the whole when you know a part and its percentage:
Whole = (Part ÷ Percentage) × 100
Example:
20 is 40% of what number?
Whole = (20 ÷ 40) × 100
Whole = 0.5 × 100
Whole = 50
Converting Between Percentages, Decimals, and Fractions
Percentage → Decimal
Divide by 100
or move decimal 2 places left
Examples:
25% = 25 ÷ 100 = 0.25
7.5% = 7.5 ÷ 100 = 0.075
150% = 150 ÷ 100 = 1.5
Decimal → Percentage
Multiply by 100
or move decimal 2 places right
Examples:
0.25 = 0.25 × 100 = 25%
0.075 = 0.075 × 100 = 7.5%
1.5 = 1.5 × 100 = 150%
Fraction → Percentage
Divide, then × 100
or (numerator ÷ denominator) × 100
Examples:
1/4 = (1 ÷ 4) × 100 = 25%
3/5 = (3 ÷ 5) × 100 = 60%
7/8 = (7 ÷ 8) × 100 = 87.5%
Common Conversions Reference Table
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/10 | 0.1 | 10% |
Practical Examples
Example 1: Calculating Discounts
A $120 jacket is on sale with a 30% discount. What is the sale price?
Solution:
Step 1: Calculate the discount amount
Discount = 30% of $120
Discount = (30 ÷ 100) × 120
Discount = 0.30 × 120 = $36
Step 2: Subtract from original price
Sale Price = $120 - $36
Sale Price = $84
Quick Method:
Sale Price = 120 × (1 - 0.30) = 120 × 0.70 = $84
Example 2: Adding Sales Tax
You buy items totaling $50. The sales tax is 8%. What is the total cost?
Solution:
Step 1: Calculate the tax amount
Tax = 8% of $50
Tax = (8 ÷ 100) × 50
Tax = 0.08 × 50 = $4
Step 2: Add to original price
Total = $50 + $4
Total = $54
Quick Method:
Total = 50 × (1 + 0.08) = 50 × 1.08 = $54
Example 3: Percentage Increase
A stock price increased from $40 to $52. What is the percentage increase?
Solution:
Step 1: Find the increase amount
Increase = $52 - $40 = $12
Step 2: Calculate percentage
Percentage = (Increase ÷ Original) × 100
Percentage = (12 ÷ 40) × 100
Percentage = 0.30 × 100
Percentage = 30% increase
Example 4: Calculating Tips
Your restaurant bill is $85. You want to leave a 20% tip. How much is the tip and total?
Solution:
Calculate the tip
Tip = 20% of $85
Tip = (20 ÷ 100) × 85
Tip = 0.20 × 85 = $17
Calculate total
Total = $85 + $17
Total = $102
Real-World Applications of Percentages
Shopping & Retail
- • Calculating discounts and sales
- • Comparing prices and deals
- • Understanding sales tax
- • Loyalty program rewards
Personal Finance
- • Interest rates on loans and savings
- • Investment returns
- • Budget allocation
- • Expense tracking
Business & Economics
- • Profit margins and markups
- • Market share analysis
- • Growth rates
- • Commission calculations
Statistics & Data
- • Survey results
- • Probability calculations
- • Data analysis
- • Performance metrics
Education
- • Grade calculations
- • Test scores
- • Attendance rates
- • Academic performance
Health & Nutrition
- • Body fat percentage
- • Nutritional values
- • Medication dosages
- • Success rates
Tips for Working with Percentages
Use Mental Math Shortcuts
10% = divide by 10, 50% = divide by 2, 25% = divide by 4, 1% = divide by 100
Double-Check Your Work
Verify results make sense. A 50% discount on $100 should be $50, not $150.
Remember the Base
Always identify what the percentage is "of." 20% of 100 is different from 20% of 50.
Practice Conversions
Get comfortable converting between percentages, decimals, and fractions quickly.
Frequently Asked Questions
Can a percentage be greater than 100%?
Yes! Percentages can exceed 100%. For example, if something doubles, it increases by 100%, making it 200% of the original value. This is common in growth rates and comparisons.
What's the difference between percentage and percentage points?
Percentage points measure the arithmetic difference between percentages. If interest rates go from 5% to 8%, that's a 3 percentage point increase, but a 60% relative increase (3/5 × 100).
How do I calculate percentage decrease?
Use the formula: Percentage Decrease = [(Original - New) ÷ Original] × 100. For example, if a price drops from $100 to $80: [(100 - 80) ÷ 100] × 100 = 20% decrease.
Why can't I just add percentages together?
Percentages are relative to their base. A 10% increase followed by a 10% decrease doesn't return to the original value because the second 10% is calculated on a different base. Always calculate each percentage separately.
What's an easy way to calculate 15% for tips?
Calculate 10% (move decimal one place left), then add half of that amount. For $50: 10% = $5, half of $5 = $2.50, so 15% = $5 + $2.50 = $7.50.
How do I find the original price before a discount?
If you know the sale price and discount percentage, use: Original Price = Sale Price ÷ (1 - Discount%). For example, if sale price is $70 after 30% off: $70 ÷ 0.70 = $100.
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