Percentage Calculations: The Ultimate Guide
Master every type of percentage calculation you'll encounter in daily life and business. From discounts to tips, markup to percentage change—this guide covers it all.
Percentage Basics
A percentage is a way of expressing a number as a fraction of 100. The word comes from Latin "per centum" meaning "by the hundred." Understanding this simple concept unlocks all percentage calculations.
The Core Relationship
Percentage = (Part / Whole) × 100
Part = (Percentage / 100) × Whole
Converting Between Forms
| Percentage | Decimal | Fraction |
|---|---|---|
| 1% | 0.01 | 1/100 |
| 10% | 0.10 | 1/10 |
| 25% | 0.25 | 1/4 |
| 33.33% | 0.3333 | 1/3 |
| 50% | 0.50 | 1/2 |
| 75% | 0.75 | 3/4 |
| 100% | 1.00 | 1 |
| 150% | 1.50 | 3/2 |
Percent → Decimal
Divide by 100 (move decimal 2 places left)
75% = 75 ÷ 100 = 0.75
Decimal → Percent
Multiply by 100 (move decimal 2 places right)
0.45 = 0.45 × 100 = 45%
Finding What Percent One Number Is of Another
This answers the question: "X is what percent of Y?"
Formula
Percentage = (Part / Whole) × 100
Example: Test Score
You answered 42 questions correctly out of 50. What's your percentage score?
Percentage = (42 / 50) × 100 = 0.84 × 100 = 84%
Example: Budget Portion
You spend $450 on rent from your $1,800 monthly income. What percentage goes to rent?
Percentage = (450 / 1800) × 100 = 0.25 × 100 = 25%
Finding a Percentage of a Number
This answers the question: "What is X% of Y?"
Formula
Result = (Percentage / 100) × Number
Or: Result = Percentage × Number ÷ 100
Example: Tip Calculation
What is 20% of a $75 restaurant bill?
Tip = (20 / 100) × 75 = 0.20 × 75 = $15
Example: Down Payment
You need a 15% down payment on a $25,000 car. How much?
Down payment = (15 / 100) × 25,000 = 0.15 × 25,000 = $3,750
Mental Math Shortcuts
10% Shortcut
Move the decimal one place left
10% of $85 = $8.50
5% Shortcut
Find 10% and halve it
5% of $85 = $8.50 ÷ 2 = $4.25
15% Shortcut
Find 10% + 5%
15% of $85 = $8.50 + $4.25 = $12.75
20% Shortcut
Find 10% and double it
20% of $85 = $8.50 × 2 = $17
Percentage Increase and Decrease
Percentage change tells you how much something has grown or shrunk relative to its original value.
Percentage Change Formula
% Change = ((New - Original) / Original) × 100
Positive = increase, Negative = decrease
Applying an Increase
New = Original × (1 + percent/100)
Example: $100 + 25% increase
$100 × 1.25 = $125
Applying a Decrease
New = Original × (1 - percent/100)
Example: $100 - 25% decrease
$100 × 0.75 = $75
Example: Stock Price Change
A stock goes from $50 to $62. What's the percentage increase?
% Change = ((62 - 50) / 50) × 100
= (12 / 50) × 100 = 24% increase
Example: Weight Loss
You went from 180 lbs to 162 lbs. What's the percentage decrease?
% Change = ((162 - 180) / 180) × 100
= (-18 / 180) × 100 = -10% (10% decrease)
Calculating Discounts
Discounts are one of the most common real-world percentage calculations. Here's how to handle them:
Method 1: Calculate Discount Amount First
Discount = Original Price × (Discount% / 100)
Final Price = Original Price - Discount
$80 shirt at 30% off:
Discount = $80 × 0.30 = $24
Final = $80 - $24 = $56
Method 2: Direct Calculation (Faster)
Final Price = Original × (1 - Discount%/100)
$80 shirt at 30% off:
Final = $80 × 0.70 = $56
Stacked Discounts
When multiple discounts apply, they multiply—they don't add. A 20% off coupon plus a 10% sale doesn't equal 30% off!
Example: Stacked Discounts
$100 item with 20% sale + additional 10% coupon:
- After 20% sale: $100 × 0.80 = $80
- After 10% coupon: $80 × 0.90 = $72
- Total discount: ($100 - $72) / $100 = 28% (not 30%)
Finding Original Price from Discounted Price
If you know the final price and discount percentage:
Original = Final Price / (1 - Discount%/100)
Example: Final price is $56 after 30% off.
Original = $56 / 0.70 = $80
Tips, Tax, and Gratuity
Adding percentages on top of a base amount is the opposite of discounts:
Adding Percentage Formula
Total = Original × (1 + percent/100)
Example: Restaurant Bill with Tip
Your bill is $65 and you want to leave a 20% tip:
Tip amount = $65 × 0.20 = $13
Total = $65 + $13 = $78
Or directly: $65 × 1.20 = $78
Example: Price with Sales Tax
Item costs $45, sales tax is 8.25%:
Tax amount = $45 × 0.0825 = $3.71
Total = $45 + $3.71 = $48.71
Or directly: $45 × 1.0825 = $48.71
Standard Tip Percentages
| Service | Typical Range |
|---|---|
| Restaurant (US) | 15-20% |
| Delivery | 15-20% |
| Haircut | 15-20% |
| Taxi/Rideshare | 15-20% |
| Hotel Housekeeping | $2-5 per night |
Markup vs Margin
These two terms are often confused but have different meanings in business:
Markup
Percentage added to cost to get price
Markup% = ((Price - Cost) / Cost) × 100
Cost: $50, Price: $75
Markup = (25/50) × 100 = 50%
Margin
Percentage of price that is profit
Margin% = ((Price - Cost) / Price) × 100
Cost: $50, Price: $75
Margin = (25/75) × 100 = 33.3%
Key insight: For the same product, markup percentage is always higher than margin percentage. A 100% markup = 50% margin. A 50% markup = 33.3% margin.
Conversion Between Markup and Margin
Markup to Margin: Margin = Markup / (1 + Markup)
Margin to Markup: Markup = Margin / (1 - Margin)
Note: Use decimal forms (50% = 0.50)
Compound Percentage Changes
When percentages change multiple times, they compound—meaning each change is applied to the result of the previous one.
The "Return to Original" Trap
A 50% loss followed by a 50% gain does NOT get you back to your original amount!
- Start: $100
- After 50% loss: $100 × 0.50 = $50
- After 50% gain: $50 × 1.50 = $75 (NOT $100!)
To recover from a 50% loss, you need a 100% gain.
Recovery Percentages
| If You Lose... | You Need to Gain... |
|---|---|
| 10% | 11.1% |
| 20% | 25% |
| 25% | 33.3% |
| 33% | 50% |
| 50% | 100% |
| 75% | 300% |
| 90% | 900% |
Compound Annual Growth Rate (CAGR)
When something grows at different rates each year, the average growth rate is found using CAGR:
CAGR = (Ending Value / Beginning Value)^(1/years) - 1
Example: Investment grows from $10,000 to $15,000 over 5 years:
CAGR = (15,000/10,000)^(1/5) - 1 = 8.45% annual growth
Common Mistakes to Avoid
❌ Adding stacked percentages
20% off + 10% off ≠ 30% off. They multiply!
❌ Assuming equal gains recover equal losses
A 20% drop requires a 25% gain to recover, not 20%.
❌ Confusing "percent OF" vs "percent MORE"
"25% OF 80" = 20. "25% MORE than 80" = 100.
❌ Mixing up markup and margin
50% markup ≠ 50% margin. Know which you need.
❌ Using the wrong base
Percent change uses the ORIGINAL as the denominator, not the new value.
Summary: Key Formulas
- What percent is A of B: (A/B) × 100
- X% of Y: Y × (X/100)
- Increase by X%: Original × (1 + X/100)
- Decrease by X%: Original × (1 - X/100)
- Percent change: ((New - Old) / Old) × 100
- Markup: ((Price - Cost) / Cost) × 100
- Margin: ((Price - Cost) / Price) × 100
Calculate Percentages Instantly
Use our free percentage calculator to solve any percentage problem quickly and accurately.