Decimal to Fraction Calculator

How to convert decimals, percents, and fractions

A terminating decimal is a fraction over a power of 10. OpenStax Prealgebra 2e (Section 5.1) says to convert a decimal by identifying the place value of the farthest right digit: in 0.03, the 3 is in the hundredths place, so the denominator is 100. Then divide top and bottom by their greatest common factor. Common Core standard 4.NF.C.6 gives the two-place case: "rewrite 0.62 as 62/100," which reduces by 2 to 31/50. To find that factor for any pair of whole numbers, open the GCF and LCM calculator.

A percent is a ratio over 100. OpenStax Prealgebra 2e (Section 6.1) defines it: "A percent is a ratio whose denominator is 100." NIST Special Publication 811 (Chapter 7, section 7.10.2) notes that the symbol % stands for the number 0.01. Dividing the decimal fraction by 100 turns 25% into 25/100 = 1/4, and 35% into 35/100 = 7/20.

A repeating decimal has a tail of digits that recurs without end. OpenStax Prealgebra 2e (Section 5.3) defines it: "A repeating decimal is a decimal in which the last digit or group of digits repeats endlessly." Suppose k fixed decimal digits sit before p repeating digits. Multiplying by 10^k and by 10^(k+p) and subtracting cancels the tail. That leaves x = (all digits − fixed digits) ÷ (10^k × (10^p − 1)). OpenStax Intermediate Algebra 2e (Section 12.3) reaches the same fraction by summing an infinite geometric series with S = a_1 / (1 − r).

Going the other way, OpenStax Prealgebra 2e (Section 5.3) notes: "To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction." Common Core standard 7.NS.A.2.d states what happens next: "the decimal form of a rational number terminates in 0s or eventually repeats." In lowest terms, a fraction stops only when its bottom number has no prime factors other than 2 and 5, because a terminating decimal is a fraction over a power of 10 and 10 = 2 × 5. Otherwise the division repeats: it can leave only d − 1 different non-zero remainders, so one must come back within d − 1 steps. This follows from the steps above; it is not a quotation.

Decimal to fraction and repeating decimal formulas

terminating (k places): x = digits / 10^k
repeating (p digits after k fixed): x = (all − fixed) / (10^k × (10^p − 1))
percent: divide the fraction by 100
lowest terms: divide numerator and denominator by gcd(n, d)
ruler mark (scale S): round(x × S) / S; error (mils) = (mark − x) × 1,000

Worked examples: terminating, repeating, and unit fractions

Terminating decimal and mixed number: 0.125 has three decimal places, so it is 125/1000. Dividing by gcd(125, 1000) = 125 gives 1/8 (12.5000%). For 1.75, the raw fraction is 175/100. Dividing by gcd(175, 100) = 25 gives 7/4, or 1 3/4 as a mixed number.

Mixed repeating decimal: 0.8333… has 1 fixed digit (8) and 1 repeating digit (3). Enter 0.83 with 1 repeating digit: x = (83 − 8) ÷ (10 × 9) = 75/90. Dividing by gcd(75, 90) = 15 gives 5/6, which expands to 0.8(3).

Pure repeating decimal: 1/7 expands by long division to 0.(142857) because remainder 1 comes back after 6 steps. Enter 0.142857 with 6 repeating digits to get 142857/999999. Dividing by gcd(142857, 999999) = 142857 gives 1/7.

Assumptions, input bounds, and rounding limits

• In decimal and percent modes, the digits you type are read one by one, never as a rounded machine number. A non-zero value must be at least 0.000001 in size and smaller than 10^15. It can have at most 12 decimal places and at most 15 digits in all, not counting leading zeros; a longer entry is refused with a message rather than rounded. Trailing zeros after a decimal point are dropped, so 0.830 is read as 0.83.
• The repeating digits are the last ones you type, and they repeat forever: 0.83 with 1 repeating digit means 0.8333…, which is exactly 5/6, not 0.83. Type the block once in full at the end of the number. For 0.4166…, type 0.416 with 1 repeating digit to get 5/12. Typing 0.4 with 1 repeating digit means 0.(4) = 4/9. The links to the inch and feet pages carry the value of the fraction to 12 significant digits, 0.833333333333 for 5/6, whatever the mode, so 25% goes on as 0.25.
• In fraction mode, the top number can be up to 1,000,000,000,000 in size. The bottom number must be from 1 to 100,000. A repeating cycle for 1/d has at most d − 1 digits. Cycles longer than 40 digits show their first 40 digits plus an ellipsis (…) and list the full cycle length in the steps.
• The equality 0.999… = 1 is exact: entering 0.9 with 1 repeating digit gives (9 − 0)/9 = 9/9 = 1.
• The best approximation is the closest fraction with a denominator no larger than your limit, found with the continued-fraction method that Python's limit_denominator uses. When two fractions are exactly as close, this page keeps the convergent that method reaches first; with a limit of 1, the value 1/2 shows 0 rather than 1. This page settles ties that way by its own choice.

How to choose exact fractions or ruler marks

Check first if your decimal is exact or rounded. A price, tax rate, or recipe amount like 0.375 is exact and reduces to 3/8. A four-digit readout like 0.3333 is a rounded third. With 0 repeating digits it gives 3333/10000, while 1 repeating digit gives 1/3.

For shop work, use the binary ruler table instead of the base-10 fraction. A caliper reading of 0.400 in is 2/5 in. On a tape, the nearest 1/16-inch mark is 3/8 in (0.375 in, −25.00 mils). At both the 1/32-inch and 1/64-inch scales, the closest mark is 13/32 in (0.40625 in, +6.25 mils).

Use the max denominator box to pull a clean ratio from a long decimal. A cap of 100 turns 0.492983 into 35/71 (error −0.000025). Raising the cap to 1,000 finds 281/570 (error −0.000001).

Copy the worked steps when you need to show homework or shop notes. Each line lists the raw fraction, the Euclidean greatest common divisor (such as gcd(18, 24) = 6 for 3/4), the lowest-terms fraction, and the mixed number.

Decimal to fraction questions