How to Convert a Decimal to Fraction (Including Repeating Decimals)

Every decimal you can write down with a finite number of digits, and every decimal that settles into a repeating pattern, is secretly a fraction. That is the whole game. A decimal is just a fraction whose denominator happens to be a power of ten, and once you see that, turning 0.75 into 3/4 stops being a memory exercise and becomes two small steps you can always reproduce. This guide walks through the method by hand, covers the trickier case of repeating decimals, and shows where the skill earns its keep outside of homework.

The core idea: read the digits over a power of ten

Take any terminating decimal and ignore the point for a second. The digits after the point tell you the numerator; the number of those digits tells you the denominator. One digit means a denominator of 10, two digits means 100, three digits means 1000, and so on.

So 0.7 is 7/10. 0.75 is 75/100. 0.075 is 75/1000. The pattern is mechanical, which is exactly why it is reliable.

The reason this works comes straight from how place value is defined. The first slot after the point is tenths, the second is hundredths, the third is thousandths. Writing 0.75 as "7 tenths plus 5 hundredths" is the same as 70/100 + 5/100, which is 75/100. Nothing is being approximated here, so the fraction is exact.

Reduce it: divide top and bottom by the GCD

A raw fraction like 75/100 is correct but not finished. Most teachers, and most real uses, want the fraction in lowest terms. You get there by dividing the numerator and denominator by their greatest common divisor, the largest number that divides both cleanly.

For 75/100 the GCD is 25. Divide both: 75 ÷ 25 = 3 and 100 ÷ 25 = 4, so the answer is 3/4. Here is the full worked example end to end:

0.375
→ read digits over power of ten:  375 / 1000
→ greatest common divisor of 375 and 1000:  125
→ divide both by 125:  375 ÷ 125 = 3,  1000 ÷ 125 = 8
→ reduced fraction:  3/8

If you forget the reduction step you end up writing 375/1000, which is the same value but will be marked wrong on most worksheets. When I am checking my own work I run the number through the decimal to fraction converter because it prints the GCD it used, so I can copy the reasoning line into my notes rather than just the final answer. That habit caught me more than once writing 50/100 when I meant 1/2.

For values above 1, convert to an improper fraction first and then split off the whole part. 2.25 becomes 225/100, reduces to 9/4, and 9/4 is 2 with a remainder of 1 over 4, written as the mixed number 2 1/4.

Repeating decimals: the let-x-equal trick

Terminating decimals are the easy half. The interesting case is a decimal that never stops but does repeat, like 0.333... or 0.142857142857.... These are still fractions, and there is a clean piece of algebra that finds them.

Set the decimal equal to x. Multiply by a power of ten large enough to shift one full repeat past the point, then subtract the original to cancel the infinite tail.

For 0.333...:

x      = 0.333...
10x    = 3.333...
10x - x = 3.333... − 0.333...
9x     = 3
x      = 3/9 = 1/3

The subtraction is the whole trick: the endless 0.333... tails line up and cancel, leaving a plain whole number on the right. For a two-digit repeat like 0.(45) you multiply by 100 instead, giving 99x = 45, so x = 45/99 = 5/11. A mixed case such as 0.1(6), where only the 6 repeats, handles the non-repeating 1 separately and lands on 1/6.

A common trap here: 0.3 and 0.(3) are not the same number. 0.3 is 3/10; 0.(3) means 0.333... and is 1/3. Only put the digits that genuinely recur inside the repeat notation.

According to the standard result in elementary number theory, a real number is rational exactly when its decimal expansion either terminates or eventually repeats — see the Wikipedia entry on repeating decimals for the proof. That is why this method always works: if a decimal repeats, a fraction is guaranteed to exist.

Going the other direction, 1/7 expands to 0.142857142857... with a six-digit period, written 0.(142857). You find the period by long division: track each remainder, and the first time a remainder you have already seen comes back, the digits in between are the repeating block. That is more honest than reading 0.142857 off a calculator, which silently rounds and drops the rest of the cycle.

Where decimal-to-fraction actually gets used

This is not only a classroom drill. A few places it shows up in daily work:

• Imperial measurements. Drill charts and fastener sizes are labeled in fractions, but calipers and CAD files report decimals. 0.3125 inch is 5/16, 0.0625 inch is 1/16, and 0.75 inch is 3/4. Converting on the spot means grabbing the right socket on the first try.
• Cooking and scaling recipes. A recipe scaled to 0.625 of a batch is far easier to measure as 5/8 of a cup than as a decimal your measuring cups do not mark.
• Math and proofs. Showing a class that 0.999... = 1, or that 0.(6) flips to 2/3 and back, makes the idea that repeating decimals are rational concrete rather than something to take on faith.
• Ratios and odds. A decimal like 0.4 often reads more clearly as the ratio 2/5. If you are working the other way between parts and percentages, a ratio calculator pairs naturally with this conversion.

Quick reference and pitfalls

A short table of conversions worth knowing by heart:

| Decimal | Fraction | Note | |---|---|---| | 0.5 | 1/2 | reduce 5/10 | | 0.375 | 3/8 | 375/1000 ÷ 125 | | 0.3125 | 5/16 | common drill size | | 0.(3) | 1/3 | repeating | | 0.(142857) | 1/7 | six-digit period | | 2.25 | 2 1/4 | mixed number |

Three mistakes account for most wrong answers. First, skipping the reduction so 0.5 comes back as 5/10 instead of 1/2. Second, miscounting decimal places, which turns 0.075 (75/1000) into 75/100 and shifts the value tenfold. Third, mixing up 0.3 with 0.(3). Counting the digits carefully and marking only the truly repeating block fixes all three.

Once the method is in your hands, the conversion is fast: read the digits over a power of ten, reduce by the GCD, and for repeating decimals let x equal the number and subtract. When you want to skip the arithmetic or check a tricky period, the converter does every step and shows its work.

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Made by Toolora · Updated 2026-06-13