📋 Table of Contents
What is a Percentage?
The word "percentage" comes from the Latin "per centum," meaning "by the hundred." A percentage is a way of expressing a number as a fraction of 100. The symbol % represents "per hundred."
Why Use Percentages?
Percentages make comparing values easier by standardizing them to a common scale. Instead of comparing "3 out of 25" versus "7 out of 50," we can compare 12% versus 14%—much simpler!
Understanding the Basics
- 100% = whole/complete: Everything, the total amount
- 50% = half: One half of the total
- 25% = quarter: One fourth of the total
- 10% = tenth: One tenth of the total
- 1% = hundredth: One hundredth of the total
Percentages can also exceed 100%. For example, 150% means one and a half times the original amount, and 200% means double the original amount.
Basic Percentage Formulas
These fundamental formulas solve most percentage problems you'll encounter.
Formula 1: Calculate Percentage
Finding What Percentage One Number Is of Another
Percentage = (Part ÷ Whole) × 100
Example: Test Score Percentage
Problem: You scored 42 out of 50 on a test. What's your percentage?
Solution:
Percentage = (42 ÷ 50) × 100
Percentage = 0.84 × 100
Answer: 84%
Formula 2: Find Percentage of a Number
Calculating a Percentage of a Value
Result = (Percentage ÷ 100) × Number
Or simplified: Result = (Percentage × Number) ÷ 100
Example: Discount Calculation
Problem: A $250 jacket is 30% off. How much is the discount?
Solution:
Discount = (30 ÷ 100) × 250
Discount = 0.30 × 250
Answer: $75
Final price = $250 - $75 = $175
Formula 3: Find the Whole from a Percentage
Finding the Original Amount
Whole = (Part ÷ Percentage) × 100
Example: Original Price
Problem: You paid $120 after a 20% discount. What was the original price?
Solution:
$120 represents 80% of the original price (100% - 20% = 80%)
Original = (120 ÷ 80) × 100
Original = 1.5 × 100
Answer: $150
Percentage Increase and Decrease
These calculations show how much a value has changed relative to its starting point.
Percentage Increase Formula
Calculating Growth Rate
Percentage Increase = [(New Value - Original Value) ÷ Original Value] × 100
Example: Salary Increase
Problem: Your salary increased from $50,000 to $55,000. What's the percentage increase?
Solution:
Increase = [(55,000 - 50,000) ÷ 50,000] × 100
Increase = [5,000 ÷ 50,000] × 100
Increase = 0.10 × 100
Answer: 10% increase
Percentage Decrease Formula
Calculating Reduction Rate
Percentage Decrease = [(Original Value - New Value) ÷ Original Value] × 100
Example: Price Reduction
Problem: A phone's price dropped from $800 to $600. What's the percentage decrease?
Solution:
Decrease = [(800 - 600) ÷ 800] × 100
Decrease = [200 ÷ 800] × 100
Decrease = 0.25 × 100
Answer: 25% decrease
Percentage of a Number
Finding a percentage of a number is useful for calculating tips, taxes, discounts, and more.
Quick Mental Math Tricks
- 10%: Move decimal one place left (10% of 450 = 45)
- 5%: Find 10%, then divide by 2 (5% of 450 = 22.5)
- 1%: Move decimal two places left (1% of 450 = 4.5)
- 25%: Divide by 4 (25% of 80 = 20)
- 20%: Find 10%, then double (20% of 450 = 90)
- 15%: Find 10%, find 5%, add them (15% of 450 = 45 + 22.5 = 67.5)
Example: Restaurant Tip
Problem: Your dinner bill is $87.50. Calculate an 18% tip.
Method 1 - Calculator:
Tip = (18 ÷ 100) × 87.50 = $15.75
Method 2 - Mental Math:
10% of $87.50 = $8.75
5% of $87.50 = $4.38 (half of 10%)
3% of $87.50 = $2.63 (roughly)
18% = 10% + 5% + 3% = $8.75 + $4.38 + $2.63 = ≈$15.75
Percentage Difference
Percentage difference compares two values by showing how different they are relative to their average.
Percentage Difference Formula
Comparing Two Values
Percentage Difference = [|Value1 - Value2| ÷ ((Value1 + Value2) ÷ 2)] × 100
Note: |x| means absolute value (always positive)
Example: Comparing Prices
Problem: Store A sells a laptop for $900, Store B for $1,100. What's the percentage difference?
Solution:
Difference = [|900 - 1100| ÷ ((900 + 1100) ÷ 2)] × 100
Difference = [200 ÷ 1000] × 100
Difference = 0.20 × 100
Answer: 20% difference
Percentage Difference vs. Percentage Change
| Aspect | Percentage Change | Percentage Difference |
|---|---|---|
| Use Case | Comparing old vs. new value | Comparing two independent values |
| Reference Point | Original value | Average of both values |
| Direction Matters | Yes (increase vs. decrease) | No (absolute difference) |
| Example | Price went from $100 to $120 | Product A costs $100, B costs $120 |
Real-World Examples
Let's apply percentage calculations to common everyday situations.
Example 1: Sales Tax
Shopping with Tax
Scenario: Item costs $45.00, sales tax is 7.5%. What's the total?
Solution:
Tax amount = (7.5 ÷ 100) × 45 = $3.375 ≈ $3.38
Total = $45.00 + $3.38 = $48.38
Quick method: Calculate 107.5% of $45
Total = (107.5 ÷ 100) × 45 = $48.38
Example 2: Successive Discounts
Multiple Discounts
Scenario: A $200 item has a 20% store discount, then an additional 10% coupon. Final price?
Solution:
After first discount (20%): $200 × 0.80 = $160
After second discount (10%): $160 × 0.90 = $144
Note: 20% + 10% ≠ 30% total discount!
Actual total discount: (200 - 144) ÷ 200 × 100 = 28%
Example 3: Interest Calculation
Simple Interest
Scenario: You invest $5,000 at 4% annual interest for 3 years. How much interest earned?
Solution:
Annual interest = 4% of $5,000 = $200
Total interest (3 years) = $200 × 3 = $600
Final balance = $5,000 + $600 = $5,600
Example 4: Grade Weighting
Weighted Average
Scenario: Final grade = 40% homework (90%), 30% midterm (85%), 30% final (88%). What's your grade?
Solution:
Homework contribution: 0.40 × 90 = 36 points
Midterm contribution: 0.30 × 85 = 25.5 points
Final contribution: 0.30 × 88 = 26.4 points
Total = 36 + 25.5 + 26.4 = 87.9%
Example 5: Population Growth
Year-Over-Year Growth
Scenario: A city's population was 500,000 in 2024 and 535,000 in 2026. What's the growth rate?
Solution:
Growth = [(535,000 - 500,000) ÷ 500,000] × 100
Growth = [35,000 ÷ 500,000] × 100
Growth = 0.07 × 100 = 7% increase
Common Percentage Mistakes
Avoid these frequent errors when working with percentages.
Mistake 1: Adding Percentages Directly
Reality: $100 + 20% = $120. Then $120 - 20% = $96. You lost $4!
Why: Percentages are calculated on different base values.
Mistake 2: Using the Wrong Base
Reality: (100 - 80) ÷ 100 × 100 = 20% decrease (correct!)
But: To go back from $80 to $100 requires a 25% increase, not 20%
Why: (100 - 80) ÷ 80 × 100 = 25%
Mistake 3: Confusing Percentage Points with Percentages
Percentage point change: 8 - 5 = 3 percentage points
Percentage change: (8 - 5) ÷ 5 × 100 = 60% increase
Key Difference: These are NOT the same thing!
Mistake 4: Successive Percentage Changes
Reality: $100 → 40% off = $60 → 20% off = $48
Actual discount: 52%, not 60%
Why: Second percentage applies to already-reduced price
Mistake 5: Percentage of What?
If you paid $80: 25% more = $80 + (0.25 × $80) = $100
If original was $100: 25% more = $125
Lesson: Always clarify the reference amount!