Quick answer
To convert a decimal to a fraction: count the decimal places (n), write the digits as the numerator and 10n as the denominator, then simplify by the GCD. For 0.75: 75/100 ÷ GCD(75,100) = 75/100 ÷ 25 = 3/4.
- Terminating decimals: count the decimal places (n), write digits/10n, then divide both by the GCD. No algebra needed.
- Repeating decimals need the algebraic subtraction method: multiply by 10n (n = repeating block length), subtract to cancel the infinite tail, solve for x.
- Mixed repeating decimals: numerator = all digits minus non-repeating digits; denominator = one 9 per repeating digit followed by one 0 per non-repeating digit.
- 0.999... = 1 exactly - not a rounding. The repeating formula gives 9/9 = 1, and no real number exists between 0.999... and 1.
- Irrational numbers cannot be converted: only terminating and repeating decimals are rational. Pi, sqrt(2), and e have infinite non-repeating decimal expansions.
How to use our decimal to fraction converter
Enter any decimal number - positive, negative, terminating, or repeating. The calculator returns the exact fraction in lowest terms, the mixed number (if applicable), and a step-by-step explanation of the conversion method used.
Explained to a beginner
Think of a decimal as a stack of coins. 0.75 means 75 cents - literally 75 out of 100. Write that as 75/100. Can you repack those coins into larger groups without getting change? Both 75 and 100 divide evenly by 25, so yes: 3 groups of 25 cents out of 4 groups of 25 cents = 3/4. The GCD is simply the size of the largest group that fits evenly into both the numerator and denominator.
Repeating decimals need a different trick. Let x = 0.333... Multiply both sides by 10: now 10x = 3.333... Both still carry the same endless "3" tail. Subtract: 10x - x = 3.333... - 0.333... = 3 exactly. So 9x = 3 and x = 1/3. The infinite tails cancel because they are identical - and that algebraic move works for any repeating decimal.
Terminating decimals
A terminating decimal has a finite number of digits after the decimal point. The conversion is direct:
$$\text{decimal} \times 10^n = \text{integer} \implies \frac{\text{integer}}{10^n} \xrightarrow{\div \gcd} \text{simplified fraction}$$
Count the decimal places (n). Write the decimal digits as the numerator. Use 10n as the denominator. Simplify using the GCD:
$$0.75 = \frac{75}{100} = \frac{3}{4}$$
0.75 has two decimal places, so the denominator is 100. 75/100, simplified with GCD(75,100) = 25, gives 3/4.
Repeating decimals
A repeating (recurring) decimal has a block of digits that repeats infinitely. The algebraic method:
$$x = 0.\overline{d_1 d_2 \ldots d_n} \implies 10^n x - x = d_1 d_2 \ldots d_n \implies x = \frac{d_1 d_2 \ldots d_n}{10^n - 1}$$
Let x = the decimal. Multiply by 10n (where n = length of the repeating block) to shift one full cycle. Subtract the original x to cancel the repeating part. Solve for x:
$$0.\overline{3} = \frac{3}{9} = \frac{1}{3}$$
0.333… → 10x = 3.333… → 10x − x = 3 → 9x = 3 → x = 3/9 = 1/3.
| Repeating decimal | Repeating block | Formula | Fraction |
|---|---|---|---|
| 0.1̄ = 0.111… | 1 | 1/(10−1) = 1/9 | 1/9 |
| 0.3̄ = 0.333… | 3 | 3/9 | 1/3 |
| 0.6̄ = 0.666… | 6 | 6/9 | 2/3 |
| 0.9̄ = 0.999… | 9 | 9/9 | 1 |
| 0.142857̄ | 142857 | 142857/999999 | 1/7 |
| 0.0̄9̄ = 0.0909… | 09 | 9/99 | 1/11 |
The 0.9̄ = 1 row in that table is the one I always linger on. The algebraic proof is airtight - 9/9 = 1 - but the intuitive argument lands better: there is no real number you can place between 0.999... and 1. In mathematics, two distinct numbers always have a value between them. If there's no gap, there's no distinction.
Mixed repeating decimals
When non-repeating digits precede the repeating block, use a two-step subtraction:
$$0.41\overline{6} = \frac{416 - 41}{900} = \frac{375}{900} = \frac{5}{12}$$
For 0.416̄6̄… (repeating 6): multiply by 1000 to get 416.666… and by 10 to get 4.166… Subtract: 1000x − 10x = 412, so 990x = 412 and x = 412/990 = 206/495. Alternatively: numerator = all digits − non-repeating part; denominator = 9s for the repeating digits followed by 0s for the non-repeating digits.
Negative decimals
Convert the absolute value to a fraction, then apply the negative sign:
$$-0.25 = -\frac{25}{100} = -\frac{1}{4}$$
−0.25 → 25/100 → GCD = 25 → 1/4 → −1/4.
Common decimal–fraction reference table
| Decimal | Fraction | Percentage |
|---|---|---|
| 0.1 | 1/10 | 10% |
| 0.125 | 1/8 | 12.5% |
| 0.2 | 1/5 | 20% |
| 0.25 | 1/4 | 25% |
| 0.333… | 1/3 | 33.33…% |
| 0.375 | 3/8 | 37.5% |
| 0.4 | 2/5 | 40% |
| 0.5 | 1/2 | 50% |
| 0.6 | 3/5 | 60% |
| 0.625 | 5/8 | 62.5% |
| 0.666… | 2/3 | 66.67% |
| 0.75 | 3/4 | 75% |
| 0.8 | 4/5 | 80% |
| 0.875 | 7/8 | 87.5% |
The table above covers the conversions that come up most often in tests and measurement contexts. When I work with fractions regularly, I find it worth memorizing the eighths column specifically: 1/8 = 0.125, 3/8 = 0.375, 5/8 = 0.625, 7/8 = 0.875.
Quarters and thirds feel intuitive, but eighths catch people by surprise - and they appear constantly in engineering, cooking, and unit conversions.
Worked examples for converting decimals to fractions
Example 1: 0.125
Three decimal places → denominator 1000. 125/1000. GCD(125, 1000) = 125. Result: 1/8.
Example 2: 1.6
One decimal place → denominator 10. 16/10. GCD(16, 10) = 2. Result: 8/5. As a mixed number: 1 3/5.
Example 3: 0.272727…
Repeating block "27" (length 2). x = 0.272727…, 100x = 27.2727…, 99x = 27, x = 27/99 = 3/11.
Example 4: 0.8333…
Non-repeating part "8", repeating part "3". Numerator = 83 − 8 = 75. Denominator = 90 (one 9 for repeating digit, one 0 for non-repeating). 75/90 = 5/6.
Common mistakes for decimal to fraction calculation
Forgetting to simplify
0.6 = 6/10, not the final answer - simplify with GCD(6,10) = 2 to get 3/5. Always divide by the GCD to reach the lowest terms.
Treating 0.999… as less than 1
0.999… is exactly equal to 1. Using the repeating formula: 9/9 = 1. This is a proven mathematical identity, not an approximation.
Using the wrong power of 10 for repeating decimals
For 0.121212…, the repeating block is "12" (length 2), so multiply by 10² = 100, not 10. 100x − x = 12, 99x = 12, x = 12/99 = 4/33.
FAQs about how to convert decimals to fractions
How do you convert a decimal to a fraction?
Count the decimal places (n). Write the digits as the numerator and 10n as the denominator. Simplify by dividing both by their GCD. Example: 0.75 → 75/100 ÷ 25 = 3/4.
How do you convert a repeating decimal to a fraction?
Let x equal the decimal. Multiply by 10n (n = repeating block length). Subtract x from the result to eliminate the repeating part. Solve for x and simplify.
What is 0.5 as a fraction?
0.5 = 5/10 = 1/2. One decimal place → denominator 10 → simplify with GCD(5,10) = 5.
What is 0.333… as a fraction?
0.333… = 1/3. The repeating block is "3": 9x = 3, x = 3/9 = 1/3.
Can every decimal be written as a fraction?
Every terminating and repeating decimal is a rational number expressible as a fraction. Irrational numbers (π, √2, e) have infinite non-repeating decimal expansions and cannot be expressed as exact fractions.
Quiz: how well do you know decimal to fraction?
1. To convert 0.75 to a fraction, what are the correct steps?
2. Worked example 4 on the page converts 0.8333... to a fraction. What is the result?
3. For 0.333..., what algebraic move eliminates the infinite repeating tail?
4. According to the common decimal-fraction reference table, what fraction equals 0.625?
5. Which of the following cannot be expressed as an exact fraction?