Decimal to Fraction Calculator

Convert any decimal number to a fraction in its simplest form. Supports terminating decimals (0.75) and repeating decimals (0.333..., 0.142857...).

Enter a decimal. For repeating decimals, append ... or use the repeating block field below.

Decimal Number

Append "..." to indicate a repeating decimal (e.g. 0.333...)

Repeating Block (optional)

Leave blank for terminating decimals.

What is a Decimal to Fraction Converter?

A decimal to fraction converter transforms a decimal number into its equivalent rational fraction in lowest terms — the form where the numerator and denominator share no common factors other than 1. For example, 0.75 becomes 3/4 and 0.333... becomes 1/3. This conversion is essential in mathematics, engineering, cooking, carpentry, and any field where fractional measurements are required or preferred over decimal notation.

The conversion method differs depending on whether the decimal terminates or repeats. Terminating decimals (those that end, like 0.125) are converted by multiplying by a power of 10 to eliminate the decimal point, then simplifying by the greatest common factor (GCF). Repeating decimals require an algebraic technique: assigning the decimal to a variable, multiplying by 10 raised to the power of the repeating block length, and subtracting to eliminate the repeating portion — a technique formalised in 17th-century number theory.

Every rational number — any number expressible as a ratio of two integers — is either a terminating or repeating decimal. Numbers like π (3.14159...) and √2 (1.41421...) are irrational: their decimal expansions neither terminate nor repeat, and they cannot be expressed as fractions. The Fundamental Theorem of Arithmetic guarantees that any terminating decimal has a denominator that, in lowest terms, contains only the prime factors 2 and 5.

How to Convert Repeating Decimals

For a purely repeating decimal like 0.333...:

Let x = 0.333...

10x = 3.333...

10x − x = 3 → 9x = 3 → x = 3/9 = 1/3

How the Decimal to Fraction Calculator Works

Formula, assumptions, and calculation steps for this math tool.

Methodology

Math calculators apply the relevant arithmetic, algebraic, geometric, or numeric rule to the values entered and simplify the result where possible.

Calculation Steps

Read the values and operation selected.
Normalize signs, decimals, fractions, or units if needed.
Apply the mathematical rule or formula.
Format the answer and any intermediate values for checking.

Assumptions and Limits

Inputs must be within the supported domain of the operation.
Decimal answers may be rounded for readability.
Symbolic simplification is limited to the calculator scope.

Frequently Asked Questions

What is a terminating decimal?

A terminating decimal has a finite number of digits after the decimal point, such as 0.25 or 1.375. These always convert to fractions with denominators that are powers of 2 and/or 5.

What is a repeating decimal?

A repeating decimal has one or more digits that repeat infinitely, like 0.333... = 1/3 or 0.142857142857... = 1/7. Every rational number is either terminating or repeating.

How do you simplify a fraction?

Divide both the numerator and denominator by their GCF (Greatest Common Factor). For example, 6/8 → GCF(6,8)=2 → 3/4.

What is a mixed number?

A mixed number combines a whole number and a proper fraction, such as 1 3/4. It is used when the decimal value is greater than 1.

Can every decimal be written as a fraction?

Only rational numbers can be written as fractions. Irrational numbers like π (3.14159...) and √2 (1.41421...) have non-terminating, non-repeating decimals and cannot be expressed as fractions.

Real-World Applications

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Arithmetic Homework

Convert decimal answers back to the fraction form required by maths textbooks.

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Recipe Scaling

Convert 0.75 cups to 3/4 cups for cookbooks and recipe measurements.

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Engineering / Machining

Express decimal measurements as fractional inches for drill bit or wrench sizing.

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Construction

Convert 0.375 inches to 3/8" for lumber cuts and material specifications.

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Exam Prep

Verify fraction conversions in SAT, ACT, and GCSE maths practice tests.

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Financial Ratios

Express ratios like 0.333... as 1/3 for clear presentation in financial reports.

Common Mistakes

Not simplifying the fraction

0.75 = 75/100, but the simplified form is 3/4 — always divide numerator and denominator by their GCD.

Misidentifying the repeating block

0.142857142857... has a 6-digit repeating block. Using only 3 digits (0.142) gives an inaccurate fraction.

Handling the integer part of mixed decimals

2.75 = 2 + 3/4 = 11/4 as an improper fraction — the integer multiplies the denominator, not just the numerator.

Applying the repeating method to terminating decimals

Terminating decimals (0.5, 0.125) use the power-of-10 method, not the algebraic 9s method.

Not applying GCD when simplifying

Finding GCD incorrectly leaves an unsimplified fraction. The Euclidean algorithm provides the guaranteed GCD.

Common Decimal-to-Fraction Reference

Decimal	Fraction	Type
0.5	1/2	Terminating
0.25	1/4	Terminating
0.75	3/4	Terminating
0.333...	1/3	Repeating
0.666...	2/3	Repeating
0.142857...	1/7	Repeating (6-digit block)
0.125	1/8	Terminating
0.1	1/10	Terminating

References

Euclid. Elements. c. 300 BC (T.L. Heath translation, Dover, 1956).
Niven, Ivan et al. An Introduction to the Theory of Numbers. Wiley, 1991.
Stewart, James. Precalculus: Mathematics for Calculus. Cengage Learning, 2015.
National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. NCTM, 2000.
Hardy, G.H. & Wright, E.M. An Introduction to the Theory of Numbers. Oxford University Press, 2008.