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What is X% of Y?
Finding a percentage of a number:
What is 15% of $85.00 (restaurant tip)?
Formula: (15 / 100) × 85 = 12.75
Answer: $12.75
What is 8.5% of $1,200 (sales tax)?
Formula: (8.5 / 100) × 1,200 = 102
Answer: $102.00
What is 25% of 240 (test score)?
Formula: (25 / 100) × 240 = 60
Answer: 60 points
X is What Percent of Y?
Finding what percentage one number is of another:
45 is what percent of 180 (test score)?
Formula: (45 / 180) × 100 = 25%
Answer: 25%
$340 is what percent of $850 (budget used)?
Formula: (340 / 850) × 100 = 40%
Answer: 40% of budget used
72 out of 80 questions correct:
Formula: (72 / 80) × 100 = 90%
Answer: 90% score
Percentage Change
How much did a value increase or decrease?
Revenue: $120,000 last year → $156,000 this year
Formula: ((156,000 - 120,000) / 120,000) × 100
= (36,000 / 120,000) × 100
= 30%
Answer: +30% increase
Stock price: $85 → $68
Formula: ((68 - 85) / 85) × 100
= (-17 / 85) × 100
= -20%
Answer: -20% decrease
Website traffic: 5,200 → 4,940 visitors
Formula: ((4,940 - 5,200) / 5,200) × 100
= (-260 / 5,200) × 100
= -5%
Answer: -5% decrease
Percentage Increase/Decrease
Applying a percentage change to a starting value:
Price after 20% discount on $250:
Discount amount: 20% × $250 = $50
Sale price: $250 - $50 = $200
Or: $250 × (1 - 0.20) = $250 × 0.80 = $200
Salary after 7% raise on $65,000:
Raise amount: 7% × $65,000 = $4,550
New salary: $65,000 + $4,550 = $69,550
Or: $65,000 × 1.07 = $69,550
Price after 8% tax on $45:
Tax amount: 8% × $45 = $3.60
Total: $45 + $3.60 = $48.60
Reverse Percentage — Finding the Original Value
The calculation that trips people up most often:
Sale price is $140 after a 30% discount. What was the original price?
The sale price is 70% of the original (100% - 30% = 70%)
Formula: $140 / 0.70 = $200
Answer: Original price was $200
WRONG approach: $140 + 30% = $140 + $42 = $182 ✗
(This adds 30% of the sale price, not the original price)
Price is $108 after 8% tax. What was the pre-tax price?
The total is 108% of the original
Formula: $108 / 1.08 = $100
Answer: Pre-tax price was $100
Compound Percentage Changes
Why percentages don't simply add and subtract:
Investment: +50% then -50%
Start: $1,000
After +50%: $1,000 × 1.50 = $1,500
After -50%: $1,500 × 0.50 = $750
Net result: -25% (not 0%!)
Investment: -20% then +20%
Start: $1,000
After -20%: $1,000 × 0.80 = $800
After +20%: $800 × 1.20 = $960
Net result: -4% (not 0%!)
Rule: To recover from an X% loss, you need more than X% gain.
After -20% loss, need +25% gain to break even
After -50% loss, need +100% gain to break even
Percentage Difference vs Percentage Change
Percentage Change (before → after):
Old: 80, New: 100
Change: ((100 - 80) / 80) × 100 = +25%
(Relative to the starting value)
Percentage Difference (comparing two values):
Value A: 80, Value B: 100
Difference: (|100 - 80| / ((80 + 100) / 2)) × 100
= (20 / 90) × 100 = 22.2%
(Relative to the average of both values — symmetric)
Use percentage change when tracking over time.
Use percentage difference when comparing two groups.
Business Metrics Examples
Conversion rate:
1,200 visitors, 48 purchases
Rate: (48 / 1,200) × 100 = 4%
Gross margin:
Revenue: $500,000, COGS: $300,000
Gross profit: $200,000
Margin: (200,000 / 500,000) × 100 = 40%
Customer churn rate:
1,000 customers at start, 50 left during month
Churn: (50 / 1,000) × 100 = 5%
Year-over-year growth:
Q1 2023: $2.1M, Q1 2024: $2.73M
Growth: ((2.73 - 2.1) / 2.1) × 100 = 30%
Tip Calculator
Restaurant bill: $87.50
15% tip: 0.15 × $87.50 = $13.13 → Total: $100.63
18% tip: 0.18 × $87.50 = $15.75 → Total: $103.25
20% tip: 0.20 × $87.50 = $17.50 → Total: $105.00
25% tip: 0.25 × $87.50 = $21.88 → Total: $109.38
Split 4 ways at 20% tip:
Total: $105.00 / 4 = $26.25 per person