Decimal to Fraction Calculator
This decimal to fraction calculator turns any decimal into a fraction in its lowest terms, and handles recurring decimals properly rather than approximating them. You get the fraction, the mixed number, the percentage, the number it was simplified by, and whether the match is exact.
0.8333 above and 1 here.How to use the decimal to fraction calculator
- Type the decimal into the box. Negatives and numbers above 1 are fine.
- Leave the switch on terminating for a decimal that ends, such as 0.375 or 2.6.
- Switch to recurring if the digits repeat forever, then say how many of the trailing digits are the repeating block.
- Read the fraction at the top, with the mixed number beneath it and the full working below that.
The results update as you type. In recurring mode, write the repeating block once at the end of the number: for 0.8333… type 0.83 and set the repeat length to 1, so the calculator knows the final 3 is the part that goes on forever. Setting a repeat length longer than the number of decimal places you typed gets you a message instead of an answer.
The stats panel reports what the fraction was simplified by. “÷ 125” means the first, unsimplified fraction was cancelled down by 125. “Already lowest terms” means the raw place-value fraction could not be reduced.
The place-value method
A terminating decimal is already a fraction; the point is just shorthand for a denominator. One decimal place means tenths, two means hundredths, three means thousandths. So the conversion has only two steps: write the digits over the matching power of ten, then cancel.
Counting places is the whole first step. 0.7 is 7/10. 0.07 is 7/100. 0.075 is 75/1000. Count the digits after the point, including zeros, and put that many zeros after the 1 in the denominator.
The second step is finding the greatest common divisor of the top and bottom, then dividing both by it. That single division is what puts the fraction in lowest terms. Doing it in stages works just as well — halving twice does the same job as dividing by 4 — but the greatest common divisor gets you there in one move.
Example: 0.375 to a fraction
Three decimal places, so the denominator is 1,000.
Check it the other way: 3 ÷ 8 = 0.375.
Example: 0.44 and 0.0625
The 0.0625 case shows why the leading zero matters. Four places means ten thousandths, not thousandths. Read it as 625/1000 and you get 5/8, which is ten times too big.
Recurring decimals and the 9s rule
A recurring decimal cannot be written over a power of ten, because its digits never run out. Treating 0.3333 as a terminating decimal gives 3333/10000, which is a perfectly exact fraction for the four digits you typed but is not one third. That is the trap: the naive method answers a slightly different question and gives no sign it has done so.
The proper method is algebra. Call the number x, multiply it by whatever power of ten shifts the repeating block exactly one period to the left, then subtract. The infinite tails are identical, so they cancel and leave whole numbers behind.
Example: 0.8333… the algebra way
Both multiples end in the same endless run of 3s, so the subtraction wipes the tail out completely.
Run that argument in general and it collapses to a rule you can apply by sight. For a block of r repeating digits sitting after k non-repeating ones, the denominator is r nines followed by k zeros. The numerator is the number formed by all the decimal digits, minus the number formed by the non-repeating ones alone.
The nines do the repeating and the zeros hold the non-repeating digits out of the way. Nothing else is going on. This is also why every recurring decimal is a fraction: the rule always produces one, so a decimal that goes on forever without any pattern cannot be written as a fraction at all.
0.166 and set 1 repeating digit, or enter 0.16 and set 1. Both give 1/6, because the calculator only needs to see where the block starts. What it must not do is guess — leave the switch on terminating and you get 166/1000 = 83/500 instead.Negatives, and decimals bigger than 1
A minus sign travels along untouched. Convert the size of the number first, then put the sign back on the front. −0.85 becomes 85/100, which cancels to 17/20, so the answer is −17/20. Writing 17/−20 means the same value but is not how a fraction is normally left; the sign belongs on the numerator, or in front of the whole fraction.
A decimal above 1 gives a top-heavy fraction, where the numerator is larger than the denominator. That is an improper fraction, and it is a correct answer, not an unfinished one. The mixed number is the same value with the whole part pulled out to the front.
To split an improper fraction yourself, divide the numerator by the denominator. 17 ÷ 5 is 3 remainder 2, so 17/5 is 3 whole parts and 2/5 left over. For a negative, do the split on the size of the number and apply the sign to the result: −2 3/4 means −(2 + 3/4), not −2 + 3/4. The improper fraction calculator and the mixed number calculator handle that conversion in both directions.
Common decimals as fractions
A handful of decimals come up constantly. Learning these means you can convert most everyday numbers without any working at all, and spot when an answer is wrong.
| Decimal | Fraction | Percentage | Type |
|---|---|---|---|
| 0.125 | 1/8 | 12.5% | terminating |
| 0.2 | 1/5 | 20% | terminating |
| 0.25 | 1/4 | 25% | terminating |
| 0.333… | 1/3 | 33.333…% | recurring |
| 0.375 | 3/8 | 37.5% | terminating |
| 0.5 | 1/2 | 50% | terminating |
| 0.625 | 5/8 | 62.5% | terminating |
| 0.666… | 2/3 | 66.666…% | recurring |
| 0.75 | 3/4 | 75% | terminating |
| 0.8 | 4/5 | 80% | terminating |
| 0.875 | 7/8 | 87.5% | terminating |
The eighths run 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875 — each one 0.125 more than the last. The fifths run in steps of 0.2. Only the thirds recur here, and that is because 3 is not a factor of any power of ten. That test tells you in advance which case you are in: cancel the fraction, and if the denominator contains anything other than 2s and 5s, the decimal repeats. The fraction to decimal calculator runs the same table in reverse and marks the repeating blocks.
Percentages sit one short step away. A percentage is a decimal multiplied by 100, so 0.375 is 37.5% and 3/8, all three the same number. The percent to decimal calculator covers that shift on its own.
Common mistakes
- Leaving the fraction unsimplified. 375/1000 is a right answer, but nobody wants it. Cancel by the greatest common divisor.
- Miscounting the decimal places. 0.0625 has four, not three. Count zeros as digits or your answer comes out ten times too large.
- Converting a recurring decimal as if it terminated. 0.3333 typed as a terminating decimal gives 3333/10000, not 1/3. Use the recurring switch.
- Getting the repeat length wrong. In 0.8333… only the 3 repeats, so the repeat length is 1. Saying 2 claims the 33 block repeats and describes a different number.
- Rounding before converting. Converting a value you already rounded to 0.33 gives 33/100, and 33/100 is not 1/3.
- Dropping the whole part. 2.6 is 13/5, not 3/5. The 2 is worth 10/5.
- Putting the minus sign on the denominator. Write −17/20, not 17/−20.
- Assuming every decimal is a fraction. Only decimals that terminate or repeat can be written as one. A never-repeating decimal such as π cannot.
Frequently asked questions
What is 0.375 as a fraction?
0.375 = 3/8. Three decimal places puts it over 1,000, giving 375/1000. The greatest common divisor of 375 and 1,000 is 125, and dividing both by it leaves 3/8. As a percentage that is 37.5%.
What is 0.625 as a fraction?
0.625 = 5/8. Written over its place value it is 625/1000, and both numbers divide by 125 to give 5/8. It sits exactly between 0.5 (1/2) and 0.75 (3/4), which is a quick way to sanity-check the answer.
What is 0.333… as a fraction?
0.3333… = 1/3. One repeating digit means a denominator of 9, so it is 3/9, which cancels to 1/3. Typed as a terminating decimal instead, 0.3333 converts to 3333/10000 — close, but a different number. Use the recurring setting.
How do you convert a repeating decimal to a fraction?
Put the repeating block over that many nines, then add a zero to the denominator for each non-repeating digit after the point. 0.2727… is 27/99 = 3/11. 0.1666… is (166 − 16)/900 = 150/900 = 1/6. The subtraction removes the digits that do not repeat.
Why does the 9s rule work?
It is the algebra written short. For x = 0.8333…, multiplying gives 100x = 83.333… and 10x = 8.333…. The endless tails are identical, so subtracting leaves 90x = 75 and x = 75/90 = 5/6. The nines in the denominator come from that subtraction of two powers of ten.
What is 2.6 as a fraction?
2.6 = 13/5, or 2 3/5 as a mixed number. One decimal place gives 26/10, and halving both parts leaves 13/5. Since 13 is bigger than 5 the fraction is improper, which is expected for any decimal above 1.
How do you convert a negative decimal?
Convert the size of the number and put the minus back afterwards. −0.85 becomes 85/100, which cancels to 17/20, so the answer is −17/20. As a mixed number a negative is read as one piece: −2 3/4 means −(2 + 3/4), which is −2.75.
What does “closest match” mean in the results?
It appears when the decimal you typed cannot be stored exactly and the fraction shown is the nearest one to it. Terminating decimals normally come back marked exact, and any decimal converted with the recurring setting is exact by construction, because the algebra gives the true value rather than an approximation.
Can every decimal be written as a fraction?
Only those that stop or repeat. Both cases have a method: place value for the first, the 9s rule for the second. A decimal whose digits go on forever with no repeating block, such as π or the square root of 2, is irrational and has no exact fraction — 22/7 is only an approximation of π.
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