Converting a decimal to a fraction depends on one key distinction: does the decimal terminate (end), or does it repeat forever? The two cases use genuinely different methods.
**A quick note before starting: which method to use depends entirely on recognizing whether the decimal terminates or repeats first — this single identification step determines the entire rest of the approach, so it's worth pausing to check for a repeating pattern (or knowing from context, like a fraction's denominator, that one exists) before committing to either method below.**
**Terminating decimals.** Write the decimal over the power of 10 that matches how many digits follow the decimal point, then simplify. A decimal with one digit after the point goes over 10; two digits go over 100; three digits go over 1,000, and so on. For 0.375 (three decimal places): write it as 375/1000. Then simplify by dividing both by their GCF. GCF(375, 1000) = 125, so 375÷125 = 3 and 1000÷125 = 8. Result: 3/8.
**Worked example: convert 0.64 to a fraction.** Two decimal places, so write it as 64/100. GCF(64, 100) = 4. Divide: 64÷4=16, 100÷4=25. Result: 16/25.
**Repeating decimals.** These need an algebraic trick, because you can't write "infinitely many digits" over a finite power of 10 directly. Let x equal the repeating decimal. Multiply both sides by a power of 10 that shifts the decimal point exactly one full repeating cycle to the right. Subtract the original equation from the shifted one — the repeating parts cancel out, leaving a solvable equation.
**Worked example: convert 0.777... to a fraction.** Let x = 0.777... Multiply by 10 (one digit repeats): 10x = 7.777... Subtract the first equation from the second: 10x − x = 7.777... − 0.777..., giving 9x = 7. Solve: x = 7/9.
**Worked example: convert 0.454545... to a fraction (a two-digit repeating block).** Let x = 0.454545... Multiply by 100 this time, since two digits repeat: 100x = 45.454545... Subtract: 100x − x = 45.454545... − 0.454545..., giving 99x = 45. Solve: x = 45/99, which simplifies (GCF(45,99)=9) to 5/11.
**Mixed decimals — a non-repeating part followed by a repeating part.** These need one extra step: first multiply to shift past the non-repeating digits, then apply the same subtraction technique. For 0.1666... (the 1 doesn't repeat, only the 6 does): let x = 0.1666..., multiply by 10 to get 10x = 1.666..., multiply by 100 to get 100x = 16.666... Subtracting 10x from 100x: 90x = 15, so x = 15/90 = 1/6.
**Why the multiplier choice matters.** The power of 10 you multiply by must shift the decimal exactly one full repeat-cycle to line up the repeating digits for cancellation. Using the wrong power (multiplying by 10 for a two-digit repeat, for instance) leaves some of the repeating pattern un-cancelled, and the subtraction won't produce a clean, solvable equation.
**Common mistakes.** The most frequent error with terminating decimals is misreading the place value — writing 0.64 as 64/1000 instead of 64/100 (0.64 has two decimal places, not three), which changes the entire answer. With repeating decimals, the most common mistake is choosing the wrong power of 10 to multiply by, especially with multi-digit repeating blocks, or forgetting to simplify the final fraction after the subtraction step produces one that isn't yet in lowest terms.
**Worked example: convert 0.125 to a fraction.** Three decimal places, so write it as 125/1000. GCF(125,1000)=125. Divide: 125÷125=1, 1000÷125=8. Result: 1/8.
**Worked example: convert 2.5 (a decimal greater than 1) to a fraction.** Separate the whole-number part (2) from the decimal part (0.5). Convert 0.5 to a fraction: 5/10, simplifies to 1/2. Combine: 2 1/2, or as an improper fraction, (2×2+1)/2 = 5/2.
**Worked example: convert a longer repeating decimal, 0.181818..., to a fraction.** Let x = 0.181818... The repeating block is two digits ("18"), so multiply by 100: 100x = 18.181818... Subtract: 100x − x = 18.181818... − 0.181818..., giving 99x = 18, so x = 18/99, which simplifies (GCF=9) to 2/11.
**Double-checking a repeating-decimal conversion.** Divide the resulting fraction back out by hand and confirm it reproduces the original repeating decimal — 2÷11 = 0.181818..., matching the original. This is worth doing whenever the algebra feels uncertain, since a wrong choice of multiplier is easy to make silently.
**A quick check worth running on any decimal-to-fraction conversion: divide the resulting fraction back out and confirm it reproduces the original decimal, exactly as demonstrated for the repeating-decimal case above — the identical habit applies equally well to the terminating-decimal cases covered earlier in this guide.**
**Worked example: convert a decimal greater than 1 with a repeating part, 1.333...** Separate the whole number (1) from the repeating decimal part (0.333...). Convert 0.333... using the standard method: let x=0.333..., 10x=3.333..., 9x=3, x=1/3. Combine: 1 + 1/3 = 4/3.
**Worked example: convert a repeating decimal where the repeating block starts immediately but spans three digits, 0.123123123...** Let x = 0.123123... Multiply by 1000 (three-digit block): 1000x = 123.123123... Subtract: 999x = 123, so x = 123/999, which simplifies (GCF=3) to 41/333.
**Worked example: convert a decimal with more decimal places, 0.0625, to a fraction.** Four decimal places, so write it over 10,000: 625/10000. GCF(625,10000)=625. Divide: 625÷625=1, 10000÷625=16. Result: 1/16.
**Converting a percentage directly to a fraction, a closely related skill.** A percentage converts to a fraction by placing it over 100 and simplifying, exactly like a two-decimal-place terminating decimal: 60% = 60/100, which simplifies (GCF=20) to 3/5. This is really the same terminating-decimal method above, just starting from the percentage form rather than the decimal form.
**A final quick reference worth keeping in mind: the same terminating/repeating distinction that governs fraction-to-decimal conversion (covered in the companion guide) governs this conversion too, just approached from the opposite direction** — recognizing which case you're in in either direction relies on the identical underlying fact about a denominator's prime factors.
For the reverse conversion — fraction to decimal, including why some fractions terminate and others repeat — see How to Convert a Fraction to a Decimal, and for reducing the resulting fraction, see How to Simplify a Fraction.