Decimal to Fraction Calculator
Convert any decimal number to a simplified fraction in lowest terms, with step-by-step working. Handles terminating and long decimals.
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How to use this calculator
Write the decimal as a fraction with denominator 10^n (where n = number of decimal places), then simplify using the Greatest Common Divisor.
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Enter the decimal number you want to convert.
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The fraction in lowest terms is shown with a three-step explanation.
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If the fraction is improper (numerator > denominator), the mixed number is also displayed.
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The decimal verification confirms the fraction equals your original input.
Frequently asked questions
How do you convert a decimal to a fraction manually?
Write the decimal over 1 (e.g., 0.625/1). Multiply numerator and denominator by 10 for each decimal place (0.625 has 3 places → multiply by 1000: 625/1000). Find the GCD of 625 and 1000: GCD = 125. Divide both by 125: 5/8. So 0.625 = 5/8.
What if my decimal has many places?
The same method works but the intermediate fraction has a large denominator before simplification. For 0.142857 (≈ 1/7): 142857/1000000; GCD = 142857, giving 1/7. This calculator handles up to about 8–10 significant decimal places before floating-point precision becomes a factor.
Can repeating decimals be converted exactly?
Repeating decimals (like 0.333…) require a different algebraic approach: let x = 0.333…; 10x = 3.333…; 10x − x = 3; 9x = 3; x = 1/3. This calculator works with the finite decimal you enter — for repeating decimals, enter a sufficient number of digits and the result will closely approximate the exact fraction.
What is a mixed number?
A mixed number combines a whole number and a proper fraction: 1¾ instead of 7/4. It is easier to interpret for quantities greater than 1 — "one and three-quarters cups" is clearer than "seven-quarter cups." Improper fractions (numerator > denominator) can always be rewritten as mixed numbers.
Decimal to fraction — simplify with GCD and show mixed number
The GCD method explained
The key to decimal-to-fraction conversion is the GCD (Greatest Common Divisor). After writing the decimal as a ratio (e.g., 625/1000), the GCD of numerator and denominator tells us the largest factor that divides both — dividing by it gives the simplest form. The Euclidean algorithm finds the GCD efficiently: GCD(625, 1000) = GCD(625, 375) = GCD(375, 250) = GCD(250, 125) = GCD(125, 0) = 125. So 625/1000 = 5/8.
Terminating vs. repeating: which decimals are fractions?
Every terminating decimal is a fraction (rational number). Repeating decimals are also rational — 0.142857142857… = 1/7. Non-terminating, non-repeating decimals (like π = 3.14159… or √2 = 1.41421…) are irrational numbers and cannot be expressed as fractions. If a decimal terminates or repeats, it is guaranteed to have a fractional equivalent.
When fractions are more useful than decimals
Fractions preserve exactness in arithmetic. 1/3 + 1/6 = 1/2 exactly; as decimals: 0.3333… + 0.1666… = 0.5 (requires rounding). In baking and carpentry, fractions (3/4 cup, 5/8 inch) are the standard — not because of tradition but because rulers and measuring cups are divided into halves, quarters, and eighths. Fractions also reveal relationships: 0.333… is just "a number," but 1/3 shows it is exactly one part of three equal parts.
Learn more from an authoritative source:
WikipediaSimplify Fractions Calculator
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Compare and order up to four fractions from smallest to largest, showing decimal equivalents and the LCD. Leave unused fields at 0.
Equivalent Fractions Calculator
Generate a list of equivalent fractions for any given fraction. Shows the simplified form and up to 10 equivalents.
LCD — Least Common Denominator
Find the Least Common Denominator (LCD) of up to four denominators, with prime factorization method shown step-by-step.
Long Division Calculator
Divide any two integers with step-by-step long division. Shows quotient, remainder, decimal result, and the full long division process as text.
Scientific Calculator
Evaluate mathematical expressions including trigonometry, logarithms, exponents, and more.
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