Percentages run your money, your grades, your shopping, and your health stats — yet most people only half-remember the formulas. This pillar guide covers everything: what percentages really mean, all 5 core formulas with worked examples, increase/decrease, reverse percentages, and every real-world use from tips to taxes. Bookmark it.
What Is a Percentage, Really?
“Percent” comes from the Latin per centum — “per hundred.” A percentage is simply a standardized way to express a fraction: 45% = 45/100 = 0.45. The genius of percentages is comparability. Saying “36 out of 80 students passed” is hard to compare; saying “45% passed” instantly benchmarks against any other group. That standardization is why percentages dominate finance, statistics, retail, and education. (See Investopedia’s percentage overview for the finance angle.)
Three forms, same value — learn to flip between them instantly:
| Percentage | Fraction | Decimal |
|---|---|---|
| 1% | 1/100 | 0.01 |
| 10% | 1/10 | 0.1 |
| 25% | 1/4 | 0.25 |
| 50% | 1/2 | 0.5 |
| 75% | 3/4 | 0.75 |
| 100% | 1 | 1.0 |
| 150% | 3/2 | 1.5 |
The 5 Core Formulas (With Examples)
Every percentage problem is one of these five. Master them and nothing surprises you.
1. What is X% of Y?
Formula: (X ÷ 100) × Y. Example: 20% of 150 = 0.20 × 150 = 30. Use it for: discounts, tips, tax, commissions.
2. X is what % of Y?
Formula: (X ÷ Y) × 100. Example: 30 is what % of 150? 30 ÷ 150 = 0.2 × 100 = 20%. Use it for: grades (45/60 = 75%), completion rates, market share.
3. X is Y% of what number?
Formula: X ÷ (Y ÷ 100). Example: 30 is 20% of what? 30 ÷ 0.20 = 150. Use it for: reverse-engineering originals (more in the reverse section below).
4. Percentage increase
Formula: ((New − Old) ÷ Old) × 100. Example: salary $50k → $55k: (5,000 ÷ 50,000) × 100 = 10% raise.
5. Percentage decrease
Formula: ((Old − New) ÷ Old) × 100. Example: $80 jacket → $60: (20 ÷ 80) × 100 = 25% off.
Try them instantly with our Percentage Calculator — all three modes in one tool.
Percentage Increase: Deep Dive
Increase answers “how much did it grow, relative to where it started?” The denominator is always the original value — that’s the rule people break most.
Worked example — investment: You invest $2,000; it grows to $2,600. Increase = (2,600 − 2,000) ÷ 2,000 = 600 ÷ 2,000 = 30% gain.
Worked example — compounding trap: A price rises 10% then falls 10%. Back to start? No! $100 → $110 → $99. The fall applies to the new base ($110), so you end 1% down. Asymmetric bases are why “up 50% then down 50%” leaves you at 75%, not 100%.
Mental shortcut: to add X%, multiply by (1 + X/100). Add 15% → ×1.15. Add 8% tax → ×1.08.
Percentage Decrease: Deep Dive
Decrease is the mirror: ((Old − New) ÷ Old) × 100. Retail lives on this formula — and on shoppers misreading it.
Worked example — sale: $120 headphones at 30% off. Discount = 0.30 × 120 = $36. You pay $84. Shortcut: multiply by (1 − 0.30) = ×0.70 directly.
Worked example — stacked discounts: “Extra 20% off sale items!” A $100 item at 30% off, then 20% off the sale price: $100 × 0.70 = $70, then $70 × 0.80 = $56. That’s 44% off total — NOT 50%. Discounts multiply; they never add.
Check any deal with our Discount Calculator.
Reverse Percentages: Finding the Original
The classic: “I paid $84 after a 30% discount — what was the original price?” Most people compute 30% of $84 ($25.20) and add it → $109.20. Wrong. The 30% was taken off the original, not the sale price.
Correct method: $84 = 70% of original → original = 84 ÷ 0.70 = $120. General rule: Original = Final ÷ (1 ± rate). Use + for “after an increase,” − for “after a decrease.”
More examples: Salary after 10% raise is $55k → was 55,000 ÷ 1.10 = $50k. Phone bill $46.20 includes 5% tax → pre-tax 46.20 ÷ 1.05 = $44.
Real-World Uses
Tipping
US standard: 15–20% of the pre-tax bill. $60 bill at 18%: 0.18 × 60 = $10.80 tip, $70.80 total. Shortcut: 10% is moving the decimal ($6), double it for 20% ($12), adjust. Our Tip Calculator splits it per person too.
Sales tax
$200 laptop, 8% tax: 200 × 1.08 = $216. Remember tax applies to the discounted price, not the sticker — $200 at 25% off with 8% tax: 150 × 1.08 = $162.
Grades
42/50 on a test: 42 ÷ 50 × 100 = 84%. Weighted: homework 90% (×30% weight) + exam 80% (×70%): 0.9×0.3 + 0.8×0.7 = 0.27 + 0.56 = 83%.
Finance
APR 6% on $10,000 for one year (simple): $600 interest. Compounded monthly: 10,000 × (1 + 0.06/12)¹² = $10,616.78 — an extra $16.78 the “6%” headline hides. Try our Compound Interest Calculator. For the formal math, Khan Academy’s exponential growth lesson is excellent.
7 Mistakes Everyone Makes
- Adding percentages: 30% off + 20% off ≠ 50% off (it’s 44%).
- Wrong base: percent change always divides by the ORIGINAL value.
- Percentage points vs percent: 10% → 12% is +2 points but +20 percent.
- Reverse math: $84 after 30% off was $120, not $109.20.
- Forgetting tax-on-discount: tax applies after discounts.
- “Up 50%, down 50% = even”: it’s 75%. Bases differ.
- Rounding too early: keep decimals until the final step.
Conclusion
Percentages look simple — per hundred, what could be easier? — but they punish sloppy thinking like nothing else in everyday math. The denominator is everything: an increase and a decrease are never mirror images, a percentage point is not a percent, and the “original” hides until you divide instead of multiply. Master the five core formulas in this guide, sanity-check every result against the size of the change, and the numbers in sales, salaries, grades and interest rates stop being traps and start being information. When the stakes are real, run it through the percentage calculator and trust the formula, not the feeling.
Frequently Asked Questions
How do I calculate percentages quickly in my head?
Learn the 10% anchor: 10% = move the decimal one place. Then build: 20% = double the 10%, 5% = half the 10%, 15% = 10% + 5%. For 18% of $60: 10% is $6, 5% is $3, 1% is $0.60 (×3 = $1.80) → $6+$3+$1.80 = $10.80.
What is the difference between percent and percentage points?
A change from 10% to 12% is a 2-percentage-point rise, but a 20% relative increase. News about interest rates and elections uses points; growth stats use percent. Mixing them up exaggerates or shrinks the real change.
Can percentages be over 100%?
Yes — 150% just means 1.5× the base. “200% growth” = tripled. Over-100% is normal for growth, margins vs markup, and efficiency ratios.
How do I reverse a percentage increase?
Divide by (1 + rate): $55k after a 10% raise → 55,000 ÷ 1.10 = $50k. For decreases divide by (1 − rate).
Why do stacked discounts never add up?
Each discount applies to the already-reduced price — a shrinking base. 30% then 20%: ×0.70 × 0.80 = ×0.56, i.e., 44% off total, not 50%.
What is a good tip percentage?
15–20% of the pre-tax bill is standard in the US; 20%+ for great service. Our Tip Calculator handles the split.
Related Tools & Guides
- Percentage Calculator — all 3 modes
- How to Calculate Percentage Increase — the deep-dive companion guide
- Tip Calculator — with bill splitting
- Discount Calculator — sale price + savings
- Sales Tax Calculator — with-tax totals
- Compound Interest Calculator — growth over time