Convert any fraction to a decimal instantly — with repeating-decimal detection and full long-division steps. Free, fast, and mobile-friendly.
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Fractions and decimals are two languages for the same idea: parts of a whole. The fraction 3/8 and the decimal 0.375 name exactly the same quantity, just written differently. Converting between them is a core arithmetic skill used in measurement, finance, science, and everyday estimation — and our free Fraction to Decimal Calculator performs the conversion instantly, flagging whether the decimal terminates or repeats.
The conversion itself is simple division: the fraction bar means “divided by,” so a/b is just a ÷ b. But the interesting part is what comes out. Some fractions, like 1/4 = 0.25, divide evenly and terminate. Others, like 1/3 = 0.333…, never end — they repeat a digit or block of digits forever. Knowing which kind you are dealing with matters for rounding, for exactness, and for understanding the number.
This calculator does the long division for you and classifies the result. Enter 7/11 and it reports 0.636363… as repeating with the block “63”. Enter 7/8 and it reports the clean terminating value 0.875. Students can use it to check homework, professionals to convert specifications, and anyone to settle “what is that as a decimal?” in a second.
Whether a fraction terminates or repeats is decided entirely by its denominator — after simplifying. A fraction in lowest terms terminates exactly when its denominator’s prime factors are only 2s and 5s. Why? Because terminating decimals are fractions over powers of 10 (10 = 2×5, 100 = 2²×5²), so any denominator that reduces to 2s and 5s can be scaled up to a power of 10.
Examples make it concrete: 3/8 has denominator 8 = 2³, so it terminates (0.375). But 5/12 simplifies to a denominator of 12 = 2²×3 — the factor 3 means it repeats (0.41666…). Always simplify first: 6/15 looks like it has a factor of 3, but it reduces to 2/5, which terminates at 0.4.
Repeating decimals have a beautiful structure: the repeating block’s length is at most one less than the denominator. For 1/7 the block “142857” has 6 digits. Every repeating decimal is also a rational number in disguise — 0.999… is exactly 1, a famous result that follows from simple algebra (let x = 0.999…, then 10x = 9.999…, subtract to get 9x = 9, so x = 1).
In practice, repeating decimals must be rounded, which introduces tiny errors. Engineers and financial software track this carefully: 1/3 of a dollar cannot be paid exactly, so systems define rounding rules (usually to the cent). The calculator shows enough digits for you to round correctly yourself.
a/b = a ÷ b
Divide the numerator by the denominator. After simplifying, the decimal terminates iff the denominator’s prime factors are only 2 and 5.
Convert 3/8. Divide: 3 ÷ 8 = 0.375 exactly. The denominator 8 = 2³ has only prime factor 2, so the decimal terminates. Verification: 0.375 × 8 = 3.000 ✓.
Convert 2/3. Divide: 2 ÷ 3 = 0.666… The 6 repeats forever because the denominator 3 is not composed of 2s and 5s. Written with bar notation: 0.3̄. Rounded to two decimals: 0.67.
Convert 9/12. Simplify first: GCD is 3, so 9/12 = 3/4. Then 3 ÷ 4 = 0.75. Skipping the simplification would still give 0.75, but simplifying reveals why it terminates (denominator 4 = 2²).
Convert 11/4. Divide: 11 ÷ 4 = 2.75. Improper fractions convert the same way — the result is simply greater than 1. As a check: 2.75 = 2 + 0.75 = 2 + 3/4 = 11/4 ✓.
Construction plans often mix formats: a blueprint might call for a 5/16-inch reveal while a digital caliper reads in decimals. Converting 5/16 gives 0.3125 inches — a value the caliper displays directly. Carpenters who convert fluently move between tape measures (fractional) and digital tools (decimal) without errors.
In finance, interest rates and fees appear as both. A “3/8 point” mortgage fee equals 0.375 points; on a $400,000 loan that is 0.00375 × 400,000 = $1,500. Converting the fraction first makes the dollar math straightforward.
Cooking with scales introduces the same need. A recipe listing 2/3 cup of flour becomes 0.666… cups — and a kitchen scale measuring in grams needs a weight conversion anyway. Knowing that 1/3 cup ≈ 0.333 cups helps cooks estimate when scaling recipes up or down.
The 2s-and-5s rule for terminating decimals has a converse worth knowing: if the simplified denominator has any prime factor other than 2 or 5, the decimal must repeat, and the repeating block length is at most denominator minus 1. For 1/13, the block “076923” actually has 6 digits — well under the maximum of 12. For 1/17, the block is the full 16 digits. These maximal-length repetends are called cyclic numbers and have fascinated mathematicians for centuries.
Long division is the engine: each step produces one decimal digit and a remainder, and remainders smaller than the divisor must eventually repeat (pigeonhole principle), forcing the digits to cycle. The calculator mimics this remainder-tracking to detect the repeating block, which is why it can name the exact repeating digits rather than just showing a rounded approximation.
There is a deep connection to base representation: “terminating” depends on the base. In base 10, 1/3 repeats; in base 3, it terminates as 0.1; in base 12, 1/3 = 0.4 terminates. The 2s-and-5s rule is really “prime factors of the base” — in base 10 those are 2 and 5. Computers, working in base 2, terminate only fractions whose denominators are powers of 2.
Simplify: GCD(5,6) = 1, already simplest. Divide: 5 ÷ 6 = 0.8333… The 8 does not repeat — only the 3 does (0.83̄). The non-repeating lead digits come from the factors of 2 in the denominator (6 = 2×3); the 3 causes the eventual repeat. The calculator identifies exactly which digits repeat.
Denominator 25 = 5² — only 5s. Scale to a power of 10: multiply top and bottom by 4 to get 44/100 = 0.44. Any denominator of only 2s and 5s can be scaled to 10^n this way, which is the constructive proof that such fractions terminate.
The decimal expansion of a fraction also settles “which is bigger” disputes mechanically: compare 5/9 vs 11/20 by converting – 0.555… vs 0.55, so 5/9 wins. Decimals linearize fractions onto one number line, which is why calculators, spreadsheets, and computers convert internally. The fraction is the exact value; the decimal is its address on the number line.
Convert 3/16 to a decimal. 3 divided by 16 = 0.1875. Terminates (16 = 2^4). Check: 0.1875 x 16 = 3.
Convert 5/12 and identify the repeating part. 5 divided by 12 = 0.41666… The 6 repeats; the 41 is the non-repeating lead from the factor 4 in 12.
Which is larger: 7/11 or 0.64? 7 divided by 11 = 0.6363… < 0.64, so 0.64 is larger. The decimal comparison is immediate once converted.
Convert 9/40. Denominator 40 = 8 x 5 – only 2s and 5s, so it terminates. 9 divided by 40 = 0.225. Via scaling: 9/40 = 225/1000 = 0.225.
The Fraction to Decimal Calculator converts a/b to a decimal by division (a ÷ b), displaying the value rounded to 10 significant decimals. It detects terminating decimals (simplified denominator has only prime factors 2 and 5) versus repeating decimals, and identifies the repeating digit block via remainder tracking.
Divide the numerator by the denominator. For 3/4, compute 3 ÷ 4 = 0.75. The fraction bar literally means division. The calculator performs this division and classifies the result.
1 ÷ 8 = 0.125. The denominator 8 = 2³, so the decimal terminates. Terminating decimals are exactly the fractions whose simplified denominators have only 2s and 5s.
0.333… with the 3 repeating forever (written 0.3̄). It never terminates because the denominator 3 has a prime factor other than 2 or 5. This is the classic example of a repeating decimal.
Simplify the fraction, then factor the denominator. If the only prime factors are 2 and 5, it terminates; otherwise it repeats. Example: 7/20 terminates (20 = 2²×5 → 0.35), but 7/21 = 1/3 repeats. The calculator applies the 2s-and-5s rule to explain the classification.
5 ÷ 6 = 0.8333… — the 3 repeats (0.83̄). Rounded to two decimal places: 0.83. It equals 0.8333… with the 3 repeating.
Use bar notation (a bar over the repeating digits, like 0.6̄) or an ellipsis (0.666…). For calculations, keep the fraction form for exactness. Bar notation (0.6̄) is the standard compact way to write it.
Yes — every fraction has a decimal expansion, but it may repeat forever. Fractions correspond exactly to terminating or repeating decimals; non-repeating, non-terminating decimals (like π) are irrational and cannot be written as fractions. This is the definition of rational numbers.
7 ÷ 2 = 3.5. Improper fractions convert the same way; the result is just greater than 1. The calculator shows the division explicitly.
Cite this page as: Fraction to Decimal Calculator. Sigma Calculator. Retrieved October 9, 2026, from https://sigmacalculator.net/math-calculators/fraction-to-decimal-calculator/
Last updated: October 9, 2026