Where do these numbers come from?
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X% of Y = X × Y ÷ 100. Percent change = (new − old) ÷ old × 100.
The three percentage questions
- What is X% of Y? Multiply: 15% of 80 = 0.15 × 80 = 12.
- X is what percent of Y? Divide: 12 ÷ 80 = 0.15 = 15%.
- Percent change from old to new: (new − old) ÷ old × 100. From 80 to 92: 12 ÷ 80 = +15%.
Reverse percentages: finding the original
If a price after a 15% increase is $92, the original is $92 ÷ 1.15 = $80, not $92 − 15% ($78.20). After a 20% discount to $64, the original was $64 ÷ 0.80 = $80. Divide by (1 ± the rate) to undo a percentage change.
Why gains and losses are not symmetric
| Fall | Rise needed to recover |
|---|---|
| −10% | +11.1% |
| −20% | +25.0% |
| −25% | +33.3% |
| −33.3% | +49.9% |
| −50% | +100.0% |
| −75% | +300.0% |
A 50% fall halves the base, so you need a 100% rise to get back. The same logic explains why two successive changes of +10% and −10% leave you at 99% of where you started.
Percent vs percentage points
If an interest rate goes from 4% to 5%, it rose by 1 percentage point but by 25 percent. News reports and loan offers often mix the two, so check which one is meant.
Three common questions
What is X% of Y? Multiply: 15% of 80 = 0.15 × 80 = 12. X is what percent of Y? Divide: 12 ÷ 80 = 15%. Percent change? (new − old) ÷ old: from 80 to 92 is +15%.
Percent vs percentage points
If a rate goes from 4% to 5%, it rose 1 percentage point but 25 percent. Loan and savings rates are usually compared in percentage points. A 50% drop followed by a 50% rise does not get you back to the start: 100 → 50 → 75.
Frequently asked questions
What is 15% of 80?
12.
How do I calculate a percentage increase?
Subtract the old value from the new, divide by the old value and multiply by 100.
What is 20% off $45?
$45 × 0.20 = $9 off, so you pay $36.
How do I add 20% to a price?
Multiply by 1.20.
How do I remove 20% that was already added?
Divide by 1.20, not multiply by 0.80.