In This Article
Key Takeaways
- Percentage means "per hundred" — 45% means 45 out of every 100.
- To find a % of a number: (% ÷ 100) × number.
- To find what % one number is of another: (part ÷ whole) × 100.
- Percentage change: ((new − old) ÷ old) × 100.
- To add X% to a number, multiply by (1 + X/100). To subtract X%, multiply by (1 − X/100).
What is a Percentage?
A percentage is a number or ratio expressed as a fraction of 100. The word "percent" comes from the Latin per centum, meaning "by the hundred." When we say 45%, we mean 45 out of every 100, or 45/100 = 0.45 as a decimal.
Percentages are one of the most practical math concepts — you encounter them in sales discounts, interest rates, test scores, statistics, nutrition labels, tax rates, and countless other everyday situations. Mastering the three core percentage formulas puts you in control of these calculations.
Formula 1: Percentage of a Number
The most common percentage question is: "What is X% of Y?"
Or equivalently: Value = Number × (Percentage / 100)
Example: What is 30% of 250?
Value = (30 ÷ 100) × 250 = 0.30 × 250 = 75
Example: What is 15% tip on a $64 restaurant bill?
Tip = 0.15 × 64 = $9.60
Example: A product is discounted 25% off its $120 price. What is the discount?
Discount = 0.25 × 120 = $30. Sale price = 120 − 30 = $90
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Formula 2: What Percent is X of Y?
This formula answers: "X is what percentage of Y?" You're finding the percentage that one number represents relative to another.
Example: 18 is what percent of 72?
Percentage = (18 ÷ 72) × 100 = 0.25 × 100 = 25%
Example: You scored 43 out of 50 on a test. What percentage did you get?
Score = (43 ÷ 50) × 100 = 0.86 × 100 = 86%
Example: A class has 28 students; 7 are absent. What percentage is absent?
Absent % = (7 ÷ 28) × 100 = 25%
Formula 3: Percentage Change
Percentage change measures how much a value has increased or decreased relative to its starting point. It's used for price changes, growth rates, population changes, and more.
Positive result = increase. Negative result = decrease.
Example: A stock went from $40 to $52. What is the percentage change?
% Change = ((52 − 40) ÷ 40) × 100 = (12 ÷ 40) × 100 = +30% (increase)
Example: A town's population dropped from 12,000 to 10,200. What is the percentage change?
% Change = ((10,200 − 12,000) ÷ 12,000) × 100 = (−1,800 ÷ 12,000) × 100 = −15% (decrease)
Percentage Increase and Decrease
When you want to increase or decrease a number by a percentage directly, use the multiplier method — it's faster than calculating the percentage separately and adding/subtracting.
Decrease by X%: New = Original × (1 − X/100)
| Operation | Example | Calculation | Result |
|---|---|---|---|
| Increase by 20% | $150 + 20% | 150 × 1.20 | $180 |
| Decrease by 15% | $200 − 15% | 200 × 0.85 | $170 |
| Add 8% tax | $90 + 8% tax | 90 × 1.08 | $97.20 |
| Apply 30% discount | $250 − 30% | 250 × 0.70 | $175 |
| Salary raise 5% | $60,000 + 5% | 60,000 × 1.05 | $63,000 |
Reverse Percentage (Finding the Original)
Reverse percentage works backwards: given a final value after a percentage change, find the original value. This comes up when a price already includes tax, or when a "sale price" is shown and you want to know the original.
Original = Final ÷ (1 − Percentage/100) [for a decrease]
Example: A jacket costs $85 after a 15% discount. What was the original price?
Original = 85 ÷ (1 − 0.15) = 85 ÷ 0.85 = $100
Example: A phone costs $432 including 8% sales tax. What was the pre-tax price?
Original = 432 ÷ 1.08 = $400
Real-World Examples
Percentages appear constantly in daily life. Here are the formulas applied to common scenarios:
Sales tax: If a $55 item has 7% sales tax: $55 × 1.07 = $58.85 total.
Restaurant tip: For a 20% tip on $78: $78 × 0.20 = $15.60 tip; $78 + $15.60 = $93.60 total.
Grade calculation: 37 correct out of 45 questions: (37 ÷ 45) × 100 = 82.2%.
Investment return: Portfolio grew from $10,000 to $12,350: ((12,350 − 10,000) ÷ 10,000) × 100 = 23.5% return.
Calorie tracking: A 2,000-calorie diet has 600 calories from fat: (600 ÷ 2,000) × 100 = 30% of calories from fat.
Pay raise: Salary increased from $52,000 to $55,640: ((55,640 − 52,000) ÷ 52,000) × 100 = 7% raise.