Percentages show up in three disguises: "what is X% of Y?", "X is what percent of Y?", and "what is the percent change from X to Y?". Once you can spot which of the three a question is, the arithmetic is one line. This guide works through each with the situations they actually appear in - a sale rack, a salary letter, a restaurant bill - and then covers the two mistakes that trip up even numerate people.
The Three Formulas, With Real Numbers
| Question | Formula | Example |
|---|---|---|
| What is X% of Y? | Y × X ÷ 100 | 18% tip on $84 = 84 × 0.18 = $15.12 |
| X is what % of Y? | X ÷ Y × 100 | 45 of 60 = 45 ÷ 60 × 100 = 75% |
| % change from X to Y? | (Y − X) ÷ X × 100 | $60 → $45 = (45 − 60) ÷ 60 × 100 = −25% |
Example 1: The $60 → $45 sale
A sweater drops from $60 to $45. The change is 45 − 60 = −15, and −15 ÷ 60 = −0.25, so the price fell 25%. Notice you divide by 60, the original price - the starting point is always the reference. Divide by 45 by mistake and you would get 33%, which is a different (and wrong) claim.
Example 2: A 5% raise on $52,000
This is a "percent of" question: 5% of 52,000 = 52,000 × 0.05 = $2,600, so the new salary is $54,600. A useful shortcut: a p% increase means multiplying by (1 + p/100), so 52,000 × 1.05 gets you there in one step - handy when stacking changes, like a 5% raise followed by a 3% raise (× 1.05 × 1.03, which is +8.15%, not +8%).
Example 3: Tip on an $84 bill
For 18% on $84: 84 × 0.18 = $15.12, total $99.12. In your head, use the 10% anchor: 10% is $8.40, so 20% is $16.80, and 18% is 20% minus 2% (2% = $1.68), giving $15.12 exactly.
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Open the Percentage Calculator →The Two Mistakes Everyone Makes
Mistake 1: Confusing percentage points with percent
If a mortgage rate moves from 4% to 6%, it rose 2 percentage points - but as a relative change it rose 50% ((6 − 4) ÷ 4). Both statements are true; they answer different questions. Be suspicious of headlines like "risk increased 50%" - if the risk went from 2 in 1,000 to 3 in 1,000, that is one twentieth of a percentage point.
Mistake 2: Assuming +50% and −50% cancel out
Increases and decreases are not symmetric, because the base changes. $100 up 50% is $150; $150 down 50% is $75. A stock that drops 20% needs a 25% gain to recover, and one that halves needs to double. The general rule: after a p% drop, the recovery required is p ÷ (100 − p) × 100 percent.
Mental-Math Tricks Worth Keeping
- Everything starts at 10%. Move the decimal one place left: 10% of $73 is $7.30. Build the rest from there - 5% is half of it, 20% is double, 15% is one-and-a-half times.
- 1% is the decimal moved two places. 1% of 5,200 is 52 - which is why 5% of 52,000 ($2,600) takes three seconds: 1% is 520, times 5.
- Flip the percentage when it helps. X% of Y always equals Y% of X. 8% of 50 feels awkward; 50% of 8 is obviously 4. Same number.
- Successive discounts multiply, not add. "30% off, then an extra 20% off" is 0.70 × 0.80 = 0.56 - a 44% total discount, not 50%.
For anything with money and time - loan growth, savings, investment returns - the percentage question becomes a compounding question. That is a topic of its own, covered in our compound interest guide with its companion calculator.
Frequently Asked Questions
What is the formula for percentage change?▼
Percentage change = (new value − old value) ÷ old value × 100. A positive result is an increase, a negative result is a decrease. The most common error is dividing by the new value instead of the old one - always divide by where you started.
Why is a 50% increase not undone by a 50% decrease?▼
Because the second percentage is taken of a different (larger) base. $100 + 50% = $150, but 50% of $150 is $75, so $150 − 50% = $75, not $100. To exactly undo a 50% increase you only need a 33.3% decrease. In general, undoing a p% increase requires a decrease of p ÷ (100 + p) × 100 percent.
What is the difference between percentage points and percent?▼
Percentage points measure the simple gap between two percentages; percent measures relative change. If an interest rate rises from 4% to 6%, it went up 2 percentage points - but that is a 50% increase, because (6 − 4) ÷ 4 = 0.5. Headlines regularly confuse the two, which can make small changes sound dramatic.
How do I calculate a price before a discount was applied?▼
Divide by (1 − discount rate), don’t multiply back up. If a jacket costs $68 after a 20% discount, the original price is 68 ÷ 0.80 = $85. Adding 20% to $68 gives $81.60, which is wrong - the 20% was taken off the original $85, not off $68.
How do I quickly work out a tip without a calculator?▼
Anchor on 10%: just move the decimal point one place left. On an $84 bill, 10% is $8.40. For 15%, add half of that again ($8.40 + $4.20 = $12.60). For 20%, double it ($16.80). This 10%-anchor trick handles almost any everyday percentage in your head.
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