Percentages are everywhere — sale discounts, tax rates, test scores, interest rates, nutrition labels, poll results, and sports statistics. Despite being simple mathematically, many people get tripped up on the different types of percentage problems. This guide covers every formula you will ever need.
What Is a Percentage?
A percentage is a number or ratio expressed as a fraction of 100. The word comes from Latin "per centum" meaning "per hundred." When we say 75%, we mean 75 out of 100, or 0.75 as a decimal.
Three numbers are always in play with percentages:
- The Base — the whole amount (e.g., 200)
- The Rate — the percentage (e.g., 25%)
- The Part — the result (e.g., 50)
Know any two and you can find the third.
How to Find X% of a Number
This is the most common percentage calculation — finding a percentage of a given value.
Examples:
- What is 15% of 200? → 0.15 × 200 = 30
- What is 8.5% of $350? → 0.085 × 350 = $29.75
- What is 120% of 50? → 1.20 × 50 = 60
→ Use our Percentage Calculator for instant results with all four percentage types.
What Percent Is X of Y?
This finds what percentage one number is of another.
Examples:
- What percent is 30 of 200? → (30/200) × 100 = 15%
- What percent is 45 of 60? → (45/60) × 100 = 75%
- You scored 47 out of 60 on a test. What percent? → (47/60) × 100 = 78.3%
Percentage Change: Increase and Decrease
Percentage change measures how much a value has changed relative to its original value.
Examples:
- Sales went from $80,000 to $100,000: ((100,000−80,000)/80,000) × 100 = +25%
- Price dropped from $50 to $35: ((35−50)/50) × 100 = −30%
- Population changed from 10,000 to 9,200: ((9200−10000)/10000) × 100 = −8%
The Asymmetry Trap: Why 50% Down ≠ 50% Up
One of the most important — and most misunderstood — properties of percentages:
A 50% loss followed by a 50% gain does NOT bring you back to the starting point.
To recover from a 50% loss, you need a 100% gain. To recover from a 30% loss, you need a 42.9% gain. This asymmetry is why avoiding large losses matters more than chasing large gains in investing.
| Loss | Gain needed to recover |
|---|---|
| 10% | 11.1% |
| 20% | 25% |
| 33% | 50% |
| 50% | 100% |
| 75% | 300% |
Reverse Percentage: Finding the Original Value
If you know the result after a percentage change, you can find the original value.
Examples:
- A shirt costs $75 after a 25% discount. Original price? → $75 / 0.75 = $100
- A price of $115 includes 15% tax. Pre-tax price? → $115 / 1.15 = $100
- After a 20% raise, salary is $60,000. Original salary? → $60,000 / 1.20 = $50,000
→ Calculate discounts and original prices instantly.
Mental Math Tips for Percentages
Quick percentage shortcuts for everyday use:
- 10%: Move decimal one place left. 10% of 350 = 35
- 5%: Find 10%, then halve it. 5% of 350 = 17.5
- 20%: Find 10%, double it. 20% of 350 = 70
- 1%: Move decimal two places left. 1% of 350 = 3.50
- 15% tip: Find 10% + half of 10%. 10% of $48 = $4.80; half = $2.40; 15% tip = $7.20
- 25%: Divide by 4. 25% of 200 = 50
- 33%: Divide by 3 (approximately). 33% of 90 ≈ 30