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Decimal to Fraction Calculator
To convert a decimal to a fraction, write it over a power of ten and reduce: 0.375 is 375/1000, which is 3/8. A repeating decimal needs subtraction: for x = 0.1(6), 100x − 10x = 16.(6) − 1.(6), so 90x = 15 and x = 1/6. This calculator does both exactly and shows the steps.
Over a power of ten, then reduce.
- Let x = 0.1(6).
- Multiply by 100 to move one whole block of repeats past the point: 100x = 16.(6).
- Multiply by 10 to move just the digits that do not repeat: 10x = 1.(6).
- Subtract the second from the first, and the repeating tails cancel: 90x = 15.
- So x = 15/90.
- Divide the top and bottom by their greatest common divisor, 15: 1/6.
| Result | Value |
|---|---|
| Fraction (lowest terms) | 1/6 |
| Check: back to a decimal | 0.1(6) |
| Closest fraction with a denominator up to 16 | 1/6 (exact) |
| Nearest 1/16 (as on a ruler) | 3/16 (off by 0.0208(3)) |
The figure above is an example: put in your own. Type a decimal such as 0.375, or a repeating one with the repeating digits in brackets, 0.1(6) for 0.1666…, or with three dots. The closest-fraction row is for rounded decimals and ruler measurements. Worked out exactly in this browser and kept there; nothing is sent anywhere.
Type a decimal, including a repeating one such as 0.1(6), and read the exact fraction in lowest terms, with the steps. For a rounded decimal, the closest simple fraction and the nearest ruler fraction come too.
Converting a decimal to a fraction writes the same value as one whole number over another. Every decimal that ends or repeats is a fraction; 0.375 is 3/8 and 0.1(6) is 1/6.
- No sign-up
- Nothing uploaded
- Shows every step
- Works offline
- Exact, no rounding errors
Key facts
| Detail | Answer |
|---|---|
| Rule | Over a power of ten, then divide by the greatest common divisor |
| Example | 0.1(6) = 15/90 = 1/6 |
| Repeating | As many 9s as repeating digits, then 0s for the rest |
| Rounded decimals | 0.333 = 333/1000 exactly; closest simple fraction 1/3 |
| Accepts | 0.375, .5, 0,5, 0.1(6), 0.333..., 2.(6), -0.75, 1.5e-3 |
| Where your numbers are kept | This browser only |
How to convert a decimal to a fraction
Terminating decimals in two steps; repeating ones in four.
Count the decimal places
0.375 has three.
Write it over that power of ten
375/1000.
Reduce
Divide by the greatest common divisor, 125: 3/8.
For a repeating decimal, let x be the decimal
x = 0.1(6).
Multiply so the repeats line up, and subtract
100x − 10x = 16.(6) − 1.(6) = 15.
Divide and reduce
x = 15/90 = 1/6.
Who converts decimals to fractions
Anyone with a decimal who needs it as a fraction to measure, cook or show working.
Students
Repeating decimals to fractions, with the algebra shown.
Woodworkers and DIY
0.3125 inch as 5/16 on the tape.
Machinists
Decimal inches to the nearest 1/64.
Cooks
0.75 cup as 3/4 cup.
Teachers
Worked examples for any decimal.
Anyone checking a calculator
Whether 0.333 really is a third.
Repeating decimals to fractions: multiply, subtract, divide
A repeating decimal is an exact fraction, and finding it takes two multiplications and a subtraction. Call the decimal x. Multiply it by powers of ten until two copies have identical repeating tails, then subtract so the tails cancel.
| Decimal | First multiple | Second multiple | Subtract | Fraction |
|---|---|---|---|---|
| 0.(3) | 10x = 3.(3) | x = 0.(3) | 9x = 3 | 1/3 |
| 0.1(6) | 100x = 16.(6) | 10x = 1.(6) | 90x = 15 | 1/6 |
| 0.(27) | 100x = 27.(27) | x = 0.(27) | 99x = 27 | 3/11 |
| 2.(6) | 10x = 26.(6) | x = 2.(6) | 9x = 24 | 8/3 = 2 2/3 |
| 0.08(3) | 1000x = 83.(3) | 100x = 8.(3) | 900x = 75 | 1/12 |
| 0.(142857) | 1,000,000x = 142857.(142857) | x = 0.(142857) | 999,999x = 142,857 | 1/7 |
The pattern has a shortcut: the denominator is as many 9s as there are repeating digits, followed by as many 0s as there are non-repeating digits after the point (90 for 0.1(6), 900 for 0.08(3)). The Common Core makes this a Grade 8 skill, to convert a decimal expansion which repeats eventually into a rational number (8.NS.A.1). The calculator writes the steps out for any repeating decimal you type, and 0.(9) comes out as 1, which it is.
0.333 is not 1/3: rounded decimals and close fractions
A decimal that has been rounded is no longer the fraction it came from. 0.333 is exactly 333/1000, and a converter that returns 1/3 for it is guessing. This calculator gives the exact fraction first, and then, separately, the closest fraction whose denominator is no bigger than the limit you choose, with how far off it is.
| Rounded decimal | Exact fraction | Closest with denominator up to 16 | Off by |
|---|---|---|---|
| 0.333 | 333/1000 | 1/3 | 0.000(3) |
| 0.67 | 67/100 | 2/3 | 0.00(3) |
| 3.14 | 157/50 | 3 1/7 (22/7) | 0.00(285714) |
| 0.618 | 309/500 | 8/13 | 0.002(615384) |
| 1.414 | 707/500 | 1 5/12 | 0.002(6) |
| 0.7 | 7/10 | 7/10 | exact |
The closest fraction is found by trying every denominator up to the limit, which is the same answer the continued-fraction method gives. 22/7 for 3.14 is the familiar approximation of pi; 8/13 and 17/12 are similarly good simple stand-ins for 0.618 and 1.414. Which to quote depends on whether the decimal was exact to begin with: a measured 0.333 may well have been a third, a price of 0.33 was not.
Decimal inches to the nearest sixteenth
Tape measures and drill sets work in halves, quarters, eighths and sixteenths of an inch, and drawings and calipers in decimal inches. The “nearest 1/16” row of the calculator rounds a decimal to the nearest ruler mark, which is different from the closest fraction of any kind: 0.33 inch is nearest 5/16 on a ruler, even though 1/3 is closer.
| Decimal inch | Fraction | Decimal inch | Fraction |
|---|---|---|---|
| 0.0625 | 1/16 | 0.5625 | 9/16 |
| 0.125 | 1/8 | 0.625 | 5/8 |
| 0.1875 | 3/16 | 0.6875 | 11/16 |
| 0.25 | 1/4 | 0.75 | 3/4 |
| 0.3125 | 5/16 | 0.8125 | 13/16 |
| 0.375 | 3/8 | 0.875 | 7/8 |
| 0.4375 | 7/16 | 0.9375 | 15/16 |
| 0.5 | 1/2 | 1 | 1 |
Choose 32 or 64 as the limit for finer work. Every sixteenth ends as a decimal because 16 is a power of 2. For metric parts, one inch is exactly 25.4 mm by the 1959 international yard and pound agreement, so a decimal inch times 25.4 gives millimetres with no rounding.
What this calculator does not do
The exact fraction is always the decimal as typed. The calculator reads terminating decimals, repeating decimals in brackets, decimals ending in three dots (it says which digits it takes to repeat, and you can type brackets to change that), negative numbers and powers of ten such as 1.5e-3. It does not convert an irrational number such as pi or the square root of 2 exactly, since no fraction equals them; a decimal of one gives a close fraction only. A value typed as a fraction is refused with a pointer to the fraction to decimal calculator. For the arithmetic of fractions, the fraction calculator shows each operation.
This calculator or an answer-only one
What this page does that a typical answer-only widget does not.
| This page | A typical calculator | |
|---|---|---|
| Repeating decimals in | 0.1(6) and 0.1666... accepted | Rarely, or a separate tool |
| Rounded decimals | Exact fraction, then the closest simple one | Guesses 1/3 for 0.333, or gives 333/1000 only |
| Ruler fractions | Nearest 1/16, 1/32, 1/64 | Not offered |
| Shows the steps | Including the subtraction | Answer only |
| Exact arithmetic | Whole numbers of any size | Floating point |
| Your numbers leave the browser | Never | Varies |
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Frequently asked questions
How do you convert a decimal to a fraction?
Write the digits after the point over 10, 100, 1000 and so on (one zero per decimal place), then reduce. 0.375 has three places, so 375/1000; dividing both by 125 gives 3/8. For 2.25, 225/100 reduces to 9/4, which is 2 1/4.
How do you convert a repeating decimal to a fraction?
Let x be the decimal and multiply by powers of ten so two copies have the same repeating tail, then subtract. For x = 0.(27): 100x = 27.(27), so 100x − x = 27, 99x = 27 and x = 3/11. Type 0.(27) into the calculator and it shows each step.
What is 0.333 as a fraction?
Exactly, 333/1000. It is close to 1/3, which is 0.(3), but not equal: the gap is 0.000333…. If 0.333 was a rounded third, use 1/3; the calculator gives both, labelled. Type 0.(3) to mean the repeating decimal, which is exactly 1/3.
Is 0.999... equal to 1?
Yes. Let x = 0.(9); then 10x = 9.(9), and subtracting gives 9x = 9, so x = 1. Another way: 1/3 = 0.(3), and three times that is 0.(9), which must equal 3/3 = 1. The calculator gives 1 for 0.(9).
How do I convert a decimal to a fraction of an inch?
Multiply by 16 (or 32, or 64) and round to a whole number; that is the numerator over 16. 0.33 × 16 = 5.28, so 5/16. Choose the limit in the calculator and the “nearest 1/16” row does it, with how far off the ruler mark is.
What is 0.625 as a fraction?
5/8. 0.625 is 625/1000, and the greatest common divisor of 625 and 1000 is 125, which leaves 5/8. All the eighths end in 25 or 75 or 5: 0.125, 0.375, 0.625 and 0.875. Each is a whole number of eighths, because 0.125 is 1/8, so the calculator always finds a denominator of 8 for them.
Can every decimal be written as a fraction?
Every decimal that ends or repeats can, and those are exactly the rational numbers. Decimals that go on for ever without repeating, such as pi or the square root of 2, cannot; any fraction for them, like 22/7, is an approximation.
How do you turn a decimal greater than 1 into a mixed number?
Keep the whole number and convert the decimal part: 3.75 is 3 + 0.75 = 3 + 3/4 = 3 3/4. Or convert the whole thing, 375/100 = 15/4, and divide: 15 ÷ 4 = 3 remainder 3, so 3 3/4.
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