Percentages appear in every corner of life: discounts, tips, taxes, grades, interest rates, polls. Yet most people reach for a phone calculator because the mental math feels fragile. This guide fixes that — here are the three problems that cover almost everything, with formulas you can do in your head.
What a percentage actually is
A percentage is a fraction with a denominator of 100:
25% = 25/100 = 0.25So “25% of 200” is just “0.25 × 200”. Everything else is variation on this single idea.
Problem 1: Percent of a number
result = (percent ÷ 100) × numberExample: 15% of $80. 15% = 0.15, and 0.15 × 80 = $12.
Mental shortcuts:
- 10% = move the decimal one place left → 10% of 80 = 8.
- 5% = half of 10% → 5% of 80 = 4.
- 15% = 10% + 5% → 8 + 4 = 12.
This makes tipping fast: 20% of a $46 bill ≈ 4.60 × 2 = $9.20.
Problem 2: What percent is A of B?
percent = (A ÷ B) × 100Example: You answered 43 of 50 questions correctly. 43 ÷ 50 = 0.86, × 100 = 86%.
Useful whenever you have a part and a whole: scores, savings rates, battery percentages, progress bars.
Problem 3: Percentage change
change % = (new − old) ÷ old × 100Example: A salary rises from $50,000 to $55,000. Change = (55,000 − 50,000) ÷ 50,000 × 100 = 10%.
Two traps to remember:
- Always divide by the old value. A fall from 100 to 80 is a 20% decrease, not 25%.
- Percentages don’t round-trip. A 50% increase followed by a 50% decrease leaves you at 75% of the original — because the second 50% applies to a larger number.
Percentages that stack: the discount trap
“50% + 20% off” sounds like 70%, but the second discount applies to the already-reduced price:
final = original × (1 − 0.50) × (1 − 0.20)
= original × 0.50 × 0.80 = original × 0.40It’s a 60% total discount, not 70%. Always multiply the multipliers.
Adding percentages: the tax example
Adding a 20% VAT and then a 5% surcharge is not 25%:
- Price $100 → +20% → $120 → +5% → $126. Total effective = 26%.
When percentages are applied sequentially, they multiply, not add. When they apply to the same base, they do add: 5% sales tax plus 3% local tax on a $100 item = $8 total tax.
Common real-world applications
| Situation | Formula | Example |
|---|---|---|
| Discount | original × (1 − %) | $80 × 0.75 = $60 (25% off) |
| Tip | bill × tip% | $46 × 0.18 ≈ $8.28 |
| Grade | correct ÷ total × 100 | 43 ÷ 50 = 86% |
| Tax | price × rate | $100 × 1.13 = $113 |
| Growth | (new−old) ÷ old × 100 | (55−50) ÷ 50 = 10% |
FAQ
How do I add a percentage to a number? Multiply by (1 + percent/100). Adding 20% to 100: 100 × 1.20 = 120.
How do I take a percentage off a number? Multiply by (1 − percent/100). Removing 25% from 80: 80 × 0.75 = 60.
What’s the difference between percent and percentage point? A rate rising from 4% to 6% is +2 percentage points but a +50% relative increase. Context matters — central banks and media often deliberately blur this.
How do I convert a fraction to a percent? Divide top by bottom, then multiply by 100: 3/4 = 0.75 = 75%.
Why is 10% + 10% not always 20%? If the percentages apply to different bases sequentially (like stacking discounts), they multiply. Only percentages sharing one base can be added directly.