How to Convert Repeating Decimals to Simplified Fractions with Step-by-Step GCD Working

That 0.8333... on your digital caliper isn't going to set itself

You are standing at the workbench with a digital caliper in hand and the readout says 0.8333 inches. You need to transfer that measurement to a marking gauge, but the gauge is marked in fractions — eighths, sixteenths, thirty-seconds. You know 0.8333 is not 3/4 (that is 0.75) and not 7/8 (that is 0.875). It sits in between, and those threes keep going.

That endless run of 3s is not a glitch. It is a repeating decimal, and underneath it is a perfectly clean fraction that has an exact, finite numerator and denominator. The problem is not that the fraction is hard to find. It is that most conversion charts handle only terminating decimals, so a repeating one gets rounded away, and rounding is exactly what you cannot afford when the number is going onto a marking gauge.

Here is the method that lands on the exact simplified fraction every time, with the greatest common divisor work shown so you can verify the result rather than trust it.

Why repeating decimals break fraction charts

Digital tools speak in decimals. Router bits, drill indexes, tape measures and fractional calipers speak in fractions. Most of the time the conversion is a lookup: a chart on the wall maps 0.75 to 3/4 and 0.9375 to 15/16.

The chart works because those decimals terminate. A terminating decimal has a finite number of digits, so it can always be written as an integer over a power of ten, and the chart has an entry for each entry in that family. A repeating decimal never terminates, so there is no obvious denominator to grab. Put 0.8333 over 10000 and you get 8333/10000, which is not wrong but is also not useful — no gauge on earth is divided into ten-thousandths, and 8333 and 10000 share no common factor, so the fraction cannot be reduced further in that form.

The trick is to work with the decimal as an infinite object rather than as a string of digits. Multiplying by a power of ten shifts the repeating part, and subtracting the two versions of the number cancels the infinite tail.

The power-of-ten subtraction method, step by step

The method has four moves, and they are the same for every repeating decimal:

1. Name the number. Write x = 0.8333... so you have something to work with algebraically. 2. Shift past the repeating block. Multiply by ten raised to the length of the repeating block. One repeating digit means multiply by 10; two repeating digits, by 100; three, by 1000. 3. Subtract. Subtract the original equation from the shifted one. The infinite tails are identical, so they cancel, and what is left is a plain integer equation. 4. Reduce. Simplify the resulting fraction by dividing numerator and denominator by their greatest common divisor, found with the Euclidean algorithm.

If the decimal has a non-repeating prefix before the repeating block — 0.1666... rather than 0.666... — you shift twice, once to clear the prefix and once to clear the repeating block, then subtract. That is the only variation.

Worked example: 0.8333... to 5/6, with the GCD shown

The caliper reading is 0.8333 with the 3 repeating, so the repeating block is one digit long:

x = 0.8333...

10x = 8.3333...

100x = 83.3333...

Subtract the first shifted equation from the second, so that the infinite tails cancel:

100x − 10x = 90x = 83.3333... − 8.3333... = 75

x = 75 / 90

Now reduce. Run the Euclidean algorithm on 75 and 90: divide the larger by the smaller, 90 ÷ 75 = 1 with remainder 15. Then divide the previous divisor by that remainder: 75 ÷ 15 = 5 with remainder 0. The last non-zero remainder is the GCD:

GCD(75, 90) = 15

75 / 90 ÷ 15 = 5 / 6

The caliper is reading five sixths of an inch. Check it by dividing: 5 ÷ 6 = 0.83333..., which matches the readout.

The three shapes of repeating decimal

Every repeating decimal falls into one of three cases, and the difference is only how many times you multiply by ten.

Pure, one repeating digit. 0.333... Multiply by 10: 10x = 3.333... and x = 0.333..., so 9x = 3 and x = 3/9. GCD(3, 9) = 3, giving 1/3.

Pure, repeating block of two or more digits. 0.4545... The block "45" is two digits, so multiply by 100: 100x = 45.4545... minus x = 0.4545... gives 99x = 45, so x = 45/99. GCD(45, 99) = 9, giving 5/11.

Mixed, with a prefix that does not repeat. 0.1666... The digit 1 never repeats; only the 6s do. Multiply by 10 to move the prefix out of the way (10x = 1.666...) and by 1000 to clear it entirely (1000x = 166.666...). Subtracting: 990x = 165, so x = 165/990. GCD(165, 990) = 165, giving 1/6.

Notice the pattern in the denominators: 9 for one repeating digit, 99 for two, 999 for three. That is where the shortcut "put the repeating digits over as many 9s as there are digits" comes from, and it is the same arithmetic with the subtraction step skipped.

Common repeating decimals and their exact fractions

The GCD column is the part most people skip, and it is the part that decides whether the answer is usable. 75/90 is correct but tells you nothing about which gauge to reach for. 5/6 tells you immediately that no sixteenths or thirty-seconds marking will land on it exactly.

What the calculator does with your input

The decimal-to-fraction calculator on this site takes the digits you type, builds the exact fraction over the matching power of ten, and then reduces it. Type 0.75 and it forms 75/100, finds GCD(75, 100) = 25, and reports 3/4. Type 0.9375 and it forms 9375/10000, finds GCD = 625, and reports 15/16.

That mechanism has one boundary worth knowing about. The calculator treats the input as a terminating decimal, because a text field cannot contain an infinite tail. If you type 0.8333, it will faithfully build 8333/10000 and report that the GCD is 1, because 8333 and 10000 genuinely share no factor. It is answering the question you asked. The repeating decimal and the four-digit truncation of it are different numbers, and only you know which one you have.

So the workflow is: use the algebra above to get from the repeating decimal to the exact raw fraction — 75/90 in the caliper case — then use the calculator to confirm the reduction step, or to handle a terminating measurement such as 0.9375. The subtraction is the part that needs a human. The reduction is the part worth automating, because a single arithmetic slip in the Euclidean algorithm turns 5/6 into something else entirely.

One more limit to keep in mind: the input field expects plain digits with an optional decimal point. An ellipsis, a vinculum, or a parenthesised repeating block will be rejected, which is a fair trade — it also means a typo cannot silently change the interpretation of your number.

Choosing the nearest fraction a tool can actually mark

The exact answer for the caliper reading is 5/6 inch, and 5/6 cannot be marked on a gauge divided into sixteenths or thirty-seconds. That leaves you choosing the closest available increment and knowing the error you just accepted:

Halving the increment halves the worst-case error, which is the general rule: the finest scale you own sets the floor on how close you can get. On a thirty-second gauge the reading is 27/32, and the error is a quarter of a millimeter — acceptable for joinery, not acceptable for a bearing seat.

For completeness, the difference between the true value and the four digits the caliper displayed is far smaller than either of those errors: 5/6 minus 0.8333 is 1/30000 of an inch, about 0.85 micrometres. The display resolution is not your limiting factor. The scale on the gauge is.

Checking the answer without redoing the division

Two cheap checks catch most errors. First, multiply the simplified fraction back out: 5 × 1 and 6 × 1, then divide, and confirm the decimal repeats as expected. Second, sanity-check the magnitude — 5/6 must sit between 3/4 and 7/8, since 0.8333 sits between 0.75 and 0.875. If a reduction produces a fraction outside the interval the original decimal occupied, the GCD was wrong.

The reason this matters more in a workshop than in a classroom is that a fraction error has a physical consequence. A rounded decimal is at least honest about being approximate; a wrongly reduced fraction looks exact, gets marked, and gets cut.