Percentages are everywhere. Sales tax, restaurant tips, exam scores, interest rates, discounts, statistical data — all expressed as percentages. Despite how common they are, many people still reach for a calculator the moment a percentage calculation appears. This guide teaches you how to solve all common percentage problems, with worked examples for each type.
What Is a Percentage?
A percentage is simply a fraction expressed as parts per hundred. The word “percent” literally comes from the Latin per centum — “per hundred.” So 35% means 35 out of 100, or the fraction 35/100 = 0.35.
Converting between forms:
- 35% → decimal: divide by 100 → 0.35
- 35% → fraction: simplify → 7/20
- 0.35 → percent: multiply by 100 → 35%
- 7/20 → percent: divide and multiply by 100 → 35%
Type 1: Percentage of a Number (X% of Y)
Formula: Result = (X ÷ 100) × Y
This is the most common type — finding a specific percentage of a number.
Examples:
- 20% of 150 = (20 ÷ 100) × 150 = 0.20 × 150 = 30
- 15% tip on a $45 bill = 0.15 × 45 = $6.75
- 8.5% sales tax on $299 = 0.085 × 299 = $25.42
- 7% annual interest on $10,000 = 0.07 × 10,000 = $700
Mental math shortcut: For 10%, simply move the decimal point one place left. Then halve it for 5%, double for 20%.
Type 2: What Percentage Is X of Y?
Formula: Percentage = (X ÷ Y) × 100
Use this when you want to express one number as a percentage of another.
Examples:
- What % is 45 out of 200? = (45 ÷ 200) × 100 = 22.5%
- You scored 72/90 on an exam: (72 ÷ 90) × 100 = 80%
- You’ve paid $15,000 of a $65,000 loan: (15,000 ÷ 65,000) × 100 = 23.1%
Type 3: Percentage Change (Increase or Decrease)
Formula: % Change = ((New Value − Old Value) ÷ Old Value) × 100
A positive result = increase; a negative result = decrease.
Examples:
- Price went from $80 to $100: ((100 − 80) ÷ 80) × 100 = +25%
- Temperature dropped from 86°F to 72°F: ((72 − 86) ÷ 86) × 100 = −16.3%
- Stock rose from $142 to $178: ((178 − 142) ÷ 142) × 100 = +25.4%
Important: Percentage change is asymmetric. A 50% increase followed by a 50% decrease does NOT return you to the original value. A 50% increase on $100 = $150; a 50% decrease on $150 = $75. You need a 100% increase to recover from a 50% loss.
Type 4: Reverse Percentage (Find the Original)
Formula: Original = Amount ÷ (Percentage ÷ 100)
Use this when you know a percentage result and want to find the original base number.
Examples:
- 45 is 15% of what number? = 45 ÷ 0.15 = 300
- After a 20% discount, an item costs $56. What was the original price? = 56 ÷ 0.80 = $70
- After a 7.5% raise, your salary is $53,750. What was it before? = 53,750 ÷ 1.075 = $50,000
Type 5: Adding or Subtracting a Percentage
Add percentage (markup): Result = Base × (1 + P/100)
Subtract percentage (discount): Result = Base × (1 − P/100)
Examples:
- Add 20% markup to a $50 wholesale item: $50 × 1.20 = $60
- 30% discount on an $89.99 item: $89.99 × 0.70 = $62.99
- 10% sales tax on a $45 meal: $45 × 1.10 = $49.50
- 25% tip on a $80 dinner: $80 × 0.25 = $20 tip; $80 × 1.25 = $100 total
Common Percentage Reference Table
| Percentage | Decimal | Fraction | Example ($100 base) |
|---|---|---|---|
| 1% | 0.01 | 1/100 | $1.00 |
| 5% | 0.05 | 1/20 | $5.00 |
| 10% | 0.10 | 1/10 | $10.00 |
| 15% | 0.15 | 3/20 | $15.00 |
| 20% | 0.20 | 1/5 | $20.00 |
| 25% | 0.25 | 1/4 | $25.00 |
| 33.3% | 0.333 | 1/3 | $33.33 |
| 50% | 0.50 | 1/2 | $50.00 |
| 75% | 0.75 | 3/4 | $75.00 |
| 100% | 1.00 | 1/1 | $100.00 |
| 150% | 1.50 | 3/2 | $150.00 |
Percentage vs. Percentage Point: A Critical Distinction
This is one of the most commonly confused concepts in statistics and media reporting.
- A percentage is a relative measure
- A percentage point is an absolute difference between two percentages
Example: If interest rates rise from 3% to 5%, that is a 2 percentage point increase, but a 66.7% relative increase (because 2/3 = 0.667).
Media often conflate these, which is why financial and statistical literacy matters. When someone says “the unemployment rate fell 2%,” ask: 2 percentage points (from 5% to 3%) or 2% of 5% (from 5.0% to 4.9%)?
Using Our Percentage Calculator
Our free Percentage Calculator handles all five calculation types with a single tool:
- % of a number (What is 20% of 150?)
- X is what % of Y (45 is what % of 200?)
- % change (increase or decrease between two values)
- Reverse percentage (find the original before a % change)
- Add/subtract % (markup, discount, tax)
- Discount calculator (sale price after % off)
Enter your values, choose your calculation type, and get an instant result with the formula shown — so you understand exactly how it was calculated.