Math Tables
Percentage Conversion Table
A percentage is a fraction with a denominator of 100, which makes it convenient to compare and awkward to compute with. The three representations — fraction, decimal, percent — describe the same number, and moving between them is where most everyday arithmetic errors begin: 1/3 is not exactly 33%, a 20% rise followed by a 20% fall does not return you to where you started, and "5% more" is a different calculation from "5 percentage points more".
The first table converts common fractions to decimals and percentages. The repeating thirds are worth noting: 1/3 is 0.333… and 33.33% is only an approximation, which is why three payments of 33.33% leave a cent behind on a $100 bill.
The third table covers the asymmetry of percentage change, and it is the one most people need. A 50% loss requires a 100% gain to recover, because the loss is measured against the larger starting value and the gain against the smaller one. The same asymmetry means a 10% discount followed by a 10% increase leaves you 1% below the original price, not back at it.
Fractions to decimals and percentages
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333333 | 33.3333% |
| 2/3 | 0.666667 | 66.6667% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 2/5 | 0.4 | 40% |
| 3/5 | 0.6 | 60% |
| 4/5 | 0.8 | 80% |
| 1/6 | 0.166667 | 16.6667% |
| 5/6 | 0.833333 | 83.3333% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
| 5/8 | 0.625 | 62.5% |
| 7/8 | 0.875 | 87.5% |
| 1/10 | 0.1 | 10% |
| 1/12 | 0.083333 | 8.3333% |
| 1/16 | 0.0625 | 6.25% |
| 1/20 | 0.05 | 5% |
| 1/25 | 0.04 | 4% |
Thirds are repeating decimals: 1/3 is 0.3333…, so 33.33% is an approximation, not an equality.
Percentages to fractions and decimals
| Percentage | Fraction | Decimal | Multiply a value by |
|---|---|---|---|
| 1% | 1/100 | 0.01 | 0.01 |
| 5% | 1/20 | 0.05 | 0.05 |
| 10% | 1/10 | 0.1 | 0.1 |
| 12.5% | 1/8 | 0.125 | 0.125 |
| 15% | 3/20 | 0.15 | 0.15 |
| 20% | 1/5 | 0.2 | 0.2 |
| 25% | 1/4 | 0.25 | 0.25 |
| 30% | 3/10 | 0.3 | 0.3 |
| 33.33% | 1/3 | 0.3333 | 0.3333 |
| 40% | 2/5 | 0.4 | 0.4 |
| 50% | 1/2 | 0.5 | 0.5 |
| 60% | 3/5 | 0.6 | 0.6 |
| 66.67% | 2/3 | 0.6667 | 0.6667 |
| 75% | 3/4 | 0.75 | 0.75 |
| 80% | 4/5 | 0.8 | 0.8 |
| 90% | 9/10 | 0.9 | 0.9 |
| 100% | 1 | 1 | 1 |
| 150% | 3/2 | 1.5 | 1.5 |
| 200% | 2 | 2 | 2 |
The last column is the multiplier you apply — 25% of 80 is 80 × 0.25. Adding 25% to 80 is 80 × 1.25, which is a different multiplier.
Applying a percentage change to 100
| Change | Result | Multiplier | Change back needed to return to 100 |
|---|---|---|---|
| +5% | 105 | 1.05 | -4.76% |
| +10% | 110 | 1.1 | -9.09% |
| +15% | 115 | 1.15 | -13.04% |
| +20% | 120 | 1.2 | -16.67% |
| +25% | 125 | 1.25 | -20% |
| +33.33% | 133.33 | 1.3333 | -25% |
| +50% | 150 | 1.5 | -33.33% |
| +100% | 200 | 2 | -50% |
| -10% | 90 | 0.9 | +11.11% |
| -20% | 80 | 0.8 | +25% |
| -25% | 75 | 0.75 | +33.33% |
| -33.33% | 66.67 | 0.6667 | +49.99% |
| -50% | 50 | 0.5 | +100% |
The last column is the asymmetry that trips people up: a 50% loss needs a 100% gain to recover, and a 20% loss needs a 25% gain.
How to Use These Tables
- Percent of a number: convert to a decimal and multiply. 15% of 240 is 240 × 0.15 = 36.
- Increase by a percent: multiply by (1 + rate). Decrease: multiply by (1 − rate). Never add the percentage to the number.
- Percentage change: divide the change by the original value, not by the new one. Going from 80 to 100 is a 25% increase; going from 100 to 80 is a 20% decrease.
- Percentage points versus percent: moving from 4% to 5% is a 1 percentage point rise but a 25% increase. Both are correct, and mixing them is how interest rate discussions go wrong.
Common Questions
Is 1/3 exactly 33%?
No. 1/3 is 33.333…%, so 33% is slightly less and 33.33% is still slightly less. Three lots of 33.33% make 99.99%, which is why splitting an amount three ways and rounding each part can leave a cent unaccounted for.
Why is a 20% fall not undone by a 20% rise?
Because the two percentages apply to different bases. A 20% fall on 100 gives 80; a 20% rise on 80 gives 96. To get back to 100 from 80 you need a 25% rise.
What is the difference between percent and percentage points?
A percentage point is the arithmetic difference between two percentages. Going from 4% to 5% is +1 percentage point, but it is a 25% increase in the rate. Loan and election coverage mix these up constantly.
How do I find the original value after a percentage change?
Divide by the multiplier. If a price after a 20% increase is 96, the original was 96 ÷ 1.2 = 80. Dividing by 1.2 is not the same as subtracting 20% from 96, which would give 76.8.
Is a 100% increase the same as doubling?
Yes — a 100% increase multiplies by 2. A 200% increase multiplies by 3, not by 2, which is a common misreading in marketing copy.
Sources
- NIST Special Publication 811 — Guide for the Use of the International System of Units (percent and fraction) — National Institute of Standards and Technology
- Consumer price index — calculating percentage changes — U.S. Bureau of Labor Statistics
- Fractions and percentages — adult numeracy guidance — U.S. Department of Education, LINCS
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