The Three Formulas
"Percent" means "per hundred", so 15% is 15/100, or 0.15. Nearly every everyday percentage question is one of three:
| Question | Formula |
|---|---|
| What is X% of Y? | Y × X ÷ 100 |
| X is what percent of Y? | X ÷ Y × 100 |
| Percent change from old to new | (new − old) ÷ old × 100 |
The first two are the same relationship rearranged. The third is where most errors happen, because the answer depends on which number you divide by.
Percent Of and "What Percent"
15% of $80 is 80 × 0.15 = $12. A 15% discount on an $80 item therefore brings the price to $68. A shortcut for the reverse is to multiply by what is left: 80 × 0.85 = $68.
What percent is 12 of 80? 12 ÷ 80 = 0.15, times 100 = 15%. The number after "of" is always the base you divide by.
Percent Change: Always Divide by the Starting Value
The U.S. Bureau of Labor Statistics, which publishes the Consumer Price Index, describes the method in three steps: subtract the earlier value from the later one, divide that difference by the earlier value, and multiply by 100.
Worked Example: A Year of Inflation (BLS CPI-U)
The CPI-U, U.S. city average for all items, was 278.802 in December 2021 and 296.797 in December 2022.
- Difference: 296.797 − 278.802 = 17.995 index points
- Divide by the earlier value: 17.995 ÷ 278.802 = 0.0645
- Multiply by 100: 6.5% (rounded, as BLS publishes it)
BLS adds two cautions: a 12-month change must compare the same month a year apart (January to December is only 11 months), and a 12-month change is not the sum of the 12 monthly changes.
Percent vs. Percentage Points
When the thing being compared is already a percentage, such as an interest rate or a tax rate, there are two different ways to describe a change, and they are easy to mix up:
- Percentage points: the simple difference. A rate that moves from 4% to 5% rose by 1 percentage point.
- Percent change: the relative change. The same move is (5 − 4) ÷ 4 = 25% higher.
Both statements are true. Headlines that say a rate "jumped 25%" and others that say it "rose one point" are describing the same event, so check which one you are reading.
The Up-Then-Down Trap
Percent changes do not cancel out, because each one uses a different base. A $100 price that rises 50% becomes $150; a 50% cut from $150 takes it to $75, not back to $100. To undo a percent increase of r, you need a decrease of r ÷ (1 + r): undoing a 25% increase takes a 20% cut.
The same logic explains why a 10% gain followed by a 10% loss leaves you at 99% of where you started, and why BLS warns against adding monthly changes to get a yearly one.
Reverse Percentages: Finding the Original Number
Sometimes you know the result and the percentage but not the starting value. If a jacket costs $68 after a 15% discount, the original price is not $68 plus 15% of $68 ($78.20). The $68 is 85% of the original, so divide: 68 ÷ 0.85 = $80.
The same move works for anything that has had a percentage added. A total of $107 that includes a 7% charge came from 107 ÷ 1.07 = $100. The rule: divide by the multiplier you would have used going forward (1 − the cut, or 1 + the increase), never subtract or add the percentage of the final number.
Quick Mental Math
- 10%: move the decimal one place left ($80 → $8). 5% is half of that; 20% is double.
- 1%: move it two places ($80 → $0.80), then multiply.
- Swap trick: X% of Y equals Y% of X, so 8% of 50 is the same as 50% of 8, which is 4.
Frequently Asked Questions
How do I calculate a percentage of a number?
Multiply the number by the percentage and divide by 100. For example, 15% of 80 is 80 × 15 ÷ 100 = 12.
How do I work out percent change?
Subtract the old value from the new value, divide by the old value and multiply by 100. This is the method the Bureau of Labor Statistics uses for CPI changes.
What is the difference between percent and percentage points?
Percentage points are the simple difference between two percentages; percent change is the relative difference. A rate going from 4% to 5% is up 1 percentage point, or 25%.
Why does a 50% drop not cancel a 50% gain?
Each change is measured against a different starting value. $100 up 50% is $150, and $150 down 50% is $75.
How do I find what percent one number is of another?
Divide the part by the whole and multiply by 100. 12 out of 80 is 12 ÷ 80 × 100 = 15%.
Sources
- U.S. Bureau of Labor Statistics, "Calculating percent changes" (bls.gov/cpi/factsheets/calculating-percent-changes.htm), method and CPI-U December 2021 and December 2022 index values.
- U.S. Bureau of Labor Statistics, CPI News Release Technical Note, "Calculating Index Changes" (bls.gov/cpi/technical-notes/).
This guide is for education. Figures above are the BLS published examples; check bls.gov for the latest CPI data.