Eight types of % calculations with step-by-step explanations. Tips, discounts, tax, change, reverse, and more.
🎯
What is X% of Y?
Percentage
%
Of this number
$
Result17
🔍
X is What % of Y?
X (the part)
$
Y (the whole)
$
Result12.5%
📈
Percentage Change
From (original)
$
To (new value)
$
Percentage Change+25%
🍽️
Tip Calculator
Bill Amount
$
Tip %
%
Split between
people
$0Tip Amount
$0Total Bill
🏷️
Discount Calculator
Original Price
$
Discount
% off
$0Sale Price
$0You Save
🧾
Sales Tax Calculator
Pre-tax Price
$
Tax Rate
%
$0Tax Amount
$0Total Price
↔️
Percentage Difference
Value A
Value B
Percentage Difference40%
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Reverse Percentage
"If X is Y% of a number, what is that original number?"
Known Value
$
Is this % of original
%
Original Number$120
The Mathematical Definition of a Percentage
A percentage is a ratio expressed as a fraction of 100. The word derives from the Latin "per centum" — "by the hundred." To convert any fraction to a percentage: divide the numerator by the denominator and multiply by 100.
Example: 17 out of 68 = 17 ÷ 68 = 0.25 = 25%. To go the other direction — converting a percentage to a decimal for use in multiplication — divide by 100: 25% = 0.25. To find 25% of 80: 80 × 0.25 = 20.
Percentage vs. Percentile: A Common Mix-Up
These terms are frequently confused. A percentage is a direct ratio — "you scored 85% on the test" means you got 85 out of 100 questions right. A percentile describes your position relative to a group — "you scored in the 85th percentile" means you did better than 85% of all test-takers.
You can score 85% on a test (percentage) but land in the 60th percentile if the test was easy and most other people also scored high. The two measures are independent.
How to Calculate a Tip Quickly
Mental tip calculation is a useful everyday skill:
20% tip — Move the decimal left one place (= 10%), then double it. On a $55 bill: $5.50 × 2 = $11.00.
15% tip — Find 10%, then add half of that. On $55: $5.50 + $2.75 = $8.25.
18% tip — Find 20% and subtract 2%. On $55: $11.00 − $1.10 = $9.90.
25% tip — Divide the bill by 4. $55 ÷ 4 = $13.75.
Our tip calculator also handles bill splitting, showing how much each person owes including their share of the tip.
Understanding Sale Prices and Discounts
A "30% off" sale means the sale price is 70% of the original (100% − 30% = 70%). To find the sale price: multiply the original by (1 − discount). A $120 item at 25% off = $120 × 0.75 = $90.
To reverse the calculation — finding the original price from a sale price — divide by (1 − discount). A $90 item after 25% off has an original price of $90 ÷ 0.75 = $120. Many people incorrectly add the discount percentage back, arriving at $90 × 1.25 = $112.50, which is wrong.
Sales Tax Rates Across the United States
Sales tax varies significantly by state and locality. Five states have no statewide sales tax: Oregon, Montana, Delaware, New Hampshire, and Alaska (though Alaska allows local taxes). The highest combined state and local rates are in Tennessee (~9.55%), Louisiana (~9.45%), and Arkansas (~9.47%).
To calculate a tax-inclusive price: multiply by (1 + tax rate). A $50 item with 8.5% tax = $50 × 1.085 = $54.25. To find the pre-tax price from a total: divide by (1 + rate). $54.25 ÷ 1.085 = $50.00.
Percentage Change in Finance and Investing
Percentage change = ((new − old) ÷ old) × 100. A stock rising from $40 to $52 is a +30% change. A key asymmetry to understand: percentage losses require larger percentage gains to recover.
A 25% loss requires a 33% gain to recover.
A 50% loss requires a 100% gain to recover.
A 75% loss requires a 300% gain to recover.
This is why investors focus heavily on avoiding large drawdowns — they are mathematically much harder to overcome than they appear.
Reverse Percentage: Working Backwards
A reverse percentage solves: "After applying X%, the result is Y — what was the original?" This comes up constantly in real life: a price after tax, a salary after deducting a percentage, a value after a discount.
Example: After a 20% discount, a shirt costs $64. What was the original price? Many people instinctively add 20% to $64 and get $76.80 — but that's wrong. The $64 represents 80% of the original (100% − 20%), so the original = $64 ÷ 0.80 = $80. Our reverse percentage calculator handles this correctly and shows the formula.
Compound Percentage Changes
When percentage changes apply repeatedly, they compound — and the result is not simply the sum of the individual percentages. Two 10% gains do not equal a 20% total gain: $100 → +10% → $110 → +10% → $121 (a 21% total gain, not 20%).
More importantly: a 10% gain followed by a 10% loss returns you to $99, not $100. The order matters less than the asymmetry. This compounding behavior is why understanding percentage math is fundamental to evaluating investment returns, loan interest, and price changes over time.
What is the difference between percentage change and percentage difference?
Percentage change measures how a value changed from a defined starting point (before → after). Percentage difference compares two values without a clear starting point, using their average as the base — useful when neither value is inherently "before" the other.
How do I find the original price after a discount is applied?
Divide the sale price by (1 minus the discount as a decimal). If a shirt costs $75 after a 25% discount, the original price was $75 ÷ 0.75 = $100. Simply adding the discount percentage back to the sale price gives the wrong answer.
What's the easiest way to calculate a 20% tip in your head?
Move the decimal one place to the left to get 10%, then double it. On a $47 bill: 10% = $4.70, doubled = $9.40 for a 20% tip.
If something drops 50% and then gains 50%, do you break even?
No. A 50% drop from $100 leaves $50. A 50% gain on $50 brings you to $75 — a net loss of 25%. To break even after a 50% loss, you need a 100% gain.
Which US states have no sales tax?
Five states: Oregon, Montana, Delaware, New Hampshire, and Alaska. Alaska has no statewide tax but allows local jurisdictions to impose their own.