Math Tables
Fraction, Decimal and Percent Table
Fractions, decimals and percentages are three notations for one number, and the conversion between them is the single most reused piece of arithmetic in school and in work. A half is 0.5 is 50%. The trouble starts at thirds, where no decimal terminates, and at sixteenths, where the percentages are exact but unfamiliar.
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There is a useful asymmetry to keep in mind. Every fraction with a denominator made only of 2s and 5s terminates as a decimal — halves, quarters, fifths, eighths, tenths, sixteenths, twentieths, twenty-fifths. Any denominator with a prime factor of 3, 7 or 11 repeats forever. That is why 1/8 is exactly 0.125 and 1/3 is not exactly anything.
The tables below give four decimal places for repeating values, which is enough for money and for most measurements, and mark the rounded percentage separately so the difference between the exact fraction and its decimal reading stays visible. The last two tables work in the other direction, from decimal to lowest terms.
One practical warning before you use them. Rounding twice loses money. Convert the fraction to a decimal once, at the end, rather than rounding a percentage and then rounding the money. A 33.33% discount on $1,000 gives $333.30 and leaves a cent unaccounted for; one third off gives $333.33. The tables show both forms precisely so that you can pick the exact one whenever the amount matters.
Halves through eighths: fraction, decimal and percent
| Fraction | Decimal | Percent | Percent (rounded) |
|---|---|---|---|
| 1/2 | 0.5000 | 50.00% | 50% |
| 1/3 | 0.3333 | 33.33% | 33% |
| 2/3 | 0.6667 | 66.67% | 67% |
| 1/4 | 0.2500 | 25.00% | 25% |
| 3/4 | 0.7500 | 75.00% | 75% |
| 1/5 | 0.2000 | 20.00% | 20% |
| 2/5 | 0.4000 | 40.00% | 40% |
| 3/5 | 0.6000 | 60.00% | 60% |
| 4/5 | 0.8000 | 80.00% | 80% |
| 1/6 | 0.1667 | 16.67% | 17% |
| 5/6 | 0.8333 | 83.33% | 83% |
| 1/8 | 0.1250 | 12.50% | 13% |
| 3/8 | 0.3750 | 37.50% | 38% |
| 5/8 | 0.6250 | 62.50% | 63% |
| 7/8 | 0.8750 | 87.50% | 88% |
Thirds and sixths repeat forever. 1/3 is 0.3333 to four places, and no number of decimal places makes it exact — the percent column is a rounded reading, not an identity.
Tenths, twelfths and sixteenths
| Fraction | Decimal | Percent | Percent (rounded) |
|---|---|---|---|
| 1/10 | 0.1000 | 10.00% | 10% |
| 3/10 | 0.3000 | 30.00% | 30% |
| 7/10 | 0.7000 | 70.00% | 70% |
| 9/10 | 0.9000 | 90.00% | 90% |
| 1/12 | 0.0833 | 8.33% | 8% |
| 5/12 | 0.4167 | 41.67% | 42% |
| 7/12 | 0.5833 | 58.33% | 58% |
| 11/12 | 0.9167 | 91.67% | 92% |
| 1/16 | 0.0625 | 6.25% | 6% |
| 3/16 | 0.1875 | 18.75% | 19% |
| 5/16 | 0.3125 | 31.25% | 31% |
| 7/16 | 0.4375 | 43.75% | 44% |
| 9/16 | 0.5625 | 56.25% | 56% |
| 11/16 | 0.6875 | 68.75% | 69% |
| 13/16 | 0.8125 | 81.25% | 81% |
| 15/16 | 0.9375 | 93.75% | 94% |
Sixteenths are the finest division on a traditional ruler, which is why 1/16 = 6.25% and 3/16 = 18.75% come up constantly in woodworking and machining.
Common decimals written as fractions in lowest terms
| Decimal | Fraction | Percent | Comparison |
|---|---|---|---|
| 0.05 | 1/20 | 5.00% | under a quarter |
| 0.1 | 1/10 | 10.00% | under a quarter |
| 0.125 | 1/8 | 12.50% | under a quarter |
| 0.15 | 3/20 | 15.00% | under a quarter |
| 0.2 | 1/5 | 20.00% | under a quarter |
| 0.25 | 1/4 | 25.00% | under a half |
| 0.3 | 3/10 | 30.00% | under a half |
| 0.375 | 3/8 | 37.50% | under a half |
| 0.4 | 2/5 | 40.00% | under a half |
| 0.5 | 1/2 | 50.00% | over a half |
| 0.6 | 3/5 | 60.00% | over a half |
| 0.625 | 5/8 | 62.50% | over a half |
| 0.75 | 3/4 | 75.00% | over three quarters |
| 0.8 | 4/5 | 80.00% | over three quarters |
| 0.875 | 7/8 | 87.50% | over three quarters |
| 0.95 | 19/20 | 95.00% | over three quarters |
Lowest terms means the numerator and denominator share no factor. 0.375 is 375/1000, which reduces to 3/8 by dividing both by 125.
Common percentages as fractions and decimals
| Percent | Decimal | Fraction | Worth of $1 |
|---|---|---|---|
| 5% | 0.0500 | 1/20 | $0.05 |
| 10% | 0.1000 | 1/10 | $0.1 |
| 12.5% | 0.1250 | 1/8 | $0.125 |
| 15% | 0.1500 | 3/20 | $0.15 |
| 20% | 0.2000 | 1/5 | $0.2 |
| 25% | 0.2500 | 1/4 | $0.25 |
| 30% | 0.3000 | 3/10 | $0.3 |
| 40% | 0.4000 | 2/5 | $0.4 |
| 50% | 0.5000 | 1/2 | $0.5 |
| 60% | 0.6000 | 3/5 | $0.6 |
| 75% | 0.7500 | 3/4 | $0.75 |
| 80% | 0.8000 | 4/5 | $0.8 |
| 90% | 0.9000 | 9/10 | $0.9 |
| 100% | 1.0000 | 1/1 | $1 |
12.5% is one eighth, the discount that shows up on every "1/8 off" promotion. 33.3% is not exactly one third, which is why a third off a price rounded to the nearest cent does not always match a 33.3% discount.
How to Use These Tables
- Fraction to percent: divide the numerator by the denominator and multiply by 100. 3/8 is 0.375, which is 37.5%.
- Percent to fraction: write the percentage over 100 and reduce. 60% is 60/100, which is 3/5.
- Decimal to fraction: count the decimal places, put the digits over 10, 100 or 1,000, then divide both by their largest common factor. 0.625 is 625/1,000, which reduces to 5/8.
- Cooking and measuring: use the sixteenths table when halving a recipe written in cups — 3/8 cup halved is 3/16, not 1/6.
- Discounts: 12.5% off is one eighth off, so divide the price by 8 rather than multiplying by 0.125 and rounding twice.
Common Questions
Why does 1/3 never become exact as a decimal?
Because 3 does not divide any power of 10. A terminating decimal is a fraction with a denominator of 10, 100, 1,000 and so on, and those denominators only have factors of 2 and 5. Since 3 is neither, the division never settles.
Which fractions terminate?
Those whose denominator in lowest terms has no prime factors other than 2 and 5. So 1/8, 1/16, 1/20 and 1/25 terminate; 1/3, 1/6, 1/7 and 1/12 do not.
Is 33.3% the same as one third?
No. One third is 33.333...%, so 33.3% is slightly less. On $100 the difference is under a cent; on $1,000,000 it is $333. Using 33.33% still leaves a gap of about $3 per million.
How do I convert a fraction to a percentage without a calculator?
Scale the fraction so the denominator becomes 100. 3/5 becomes 60/100 by multiplying both parts by 20, so it is 60%. This works whenever the denominator divides 100 — halves, quarters, fifths, tenths, twentieths, twenty-fifths and fiftieths.
Why are sixteenths so common in the US?
Because the inch is traditionally divided in half repeatedly: 1/2, 1/4, 1/8, 1/16, 1/32, 1/64. Those are the markings on a tape measure, so trade work is written in sixteenths even where decimals would be simpler.
What is 0.875 as a fraction?
Seven eighths. 0.875 is 875/1,000, and dividing both by 125 gives 7/8. It is also 87.5%, the value behind the "7/8 inch" wrench size.
More Math Tables
Sources
- Common Core State Standards for Mathematics — number system, grades 6-7 — Common Core State Standards Initiative
- NIST Handbook 44 — tolerances and decimal notation in commerce — National Institute of Standards and Technology
- Fractions and decimals — mathematical reference — National Council of Teachers of Mathematics
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