Decimal to Fraction Calculator (Live Simplified & Mixed Tool)
Convert terminating and repeating decimals into simplified fractions and mixed numbers in lowest terms. Features automated Greatest Common Divisor (GCD) reduction, algebraic repeating decimal solving, and step-by-step conversion proofs.
To convert a terminating decimal to a fraction, write the decimal numbers over 1 followed by as many zeros as there are decimal places (e.g., $0.75 = 75/100$). Then, divide both the numerator and denominator by their Greatest Common Divisor (GCD) to reduce the fraction to lowest terms: 75 ÷ 25 / 100 ÷ 25 = 3/4.
The 3-Step Formula to Convert Decimals to Fractions
A terminating decimal represents a fraction whose denominator is a power of 10 (10, 100, 1,000, 10,000, etc.). To convert a terminating decimal into a simplified fraction:
- Count Decimal Places: Count how many digits appear to the right of the decimal point ($d$).
- Write as a Fraction: Place the numbers over $10^d$ (for example, $0.625$ has 3 decimal places, so it becomes $625 / 1000$).
- Reduce Using the GCD: Calculate the Greatest Common Divisor (GCD) of the numerator and denominator using Euclid's algorithm and divide both numbers to reach lowest terms ($625 \div 125 / 1000 \div 125 = \mathbf{5/8}$).
How to Convert Repeating Decimals Using Algebra
Repeating decimals (such as $0.333...$ or $0.1666...$) cannot be converted by dividing by powers of 10 because the digits continue indefinitely. Instead, mathematicians use an algebraic elimination technique:
- Set the decimal equal to a variable:
x = 0.1666... - Multiply by a power of 10 to move the non-repeating digits left of the decimal:
10x = 1.666... - Multiply by a power of 10 to move one repeating cycle left of the decimal:
100x = 16.666... - Subtract the two equations to cancel out the infinite repeating tail:
100x − 10x = 16.666... − 1.666... → 90x = 15 - Solve for $x$ and reduce:
x = 15 / 90 = 1 / 6.
Can Every Decimal Be Converted to a Fraction? (Rational vs. Irrational)
Not all decimals can be expressed as a fraction of two integers. The real number system is divided into two categories:
- Rational Numbers: Any number that terminates (e.g., $0.75 = 3/4$) or repeats in an infinite pattern (e.g., $0.333... = 1/3$) is rational and can be converted into an exact fraction $a/b$.
- Irrational Numbers: Decimals that continue indefinitely without ever repeating a pattern (such as $\pi \approx 3.14159265...$, Euler's number $e \approx 2.71828...$, or $\sqrt{2} \approx 1.41421...$) cannot be converted into an exact fraction of whole numbers.
Common Decimals to Fractions & Percentages Reference Matrix
| Decimal Value | Simplified Fraction | Decimal Classification | Percentage (%) |
|---|---|---|---|
| 0.125 | 1 / 8 | Terminating Decimal | 12.5% |
| 0.25 | 1 / 4 | Terminating Decimal | 25.0% |
| 0.333... | 1 / 3 | Repeating Decimal | 33.33% |
| 0.375 | 3 / 8 | Terminating Decimal | 37.5% |
| 0.50 | 1 / 2 | Terminating Decimal | 50.0% |
| 0.625 | 5 / 8 | Terminating Decimal | 62.5% |
| 0.666... | 2 / 3 | Repeating Decimal | 66.67% |
| 2.625 | 2 5/8 (or 21/8) | Mixed Decimal (> 1) | 262.5% |
Related Fraction & Mathematical Calculators:
- Mixed Number to Improper Fraction Calculator — Convert mixed numbers to improper fractions with step-by-step reduction proofs.
- Fraction to Percent Calculator — Convert proper fractions, improper fractions, and mixed numbers to percentages and decimals.
- Percent to Fraction Calculator — Convert percentages into simplified fractions and mixed numbers.
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- Full Percentage Calculator Hub — Master tool for general percentages, ratios, and variance.
Frequently Asked Questions
What is 0.625 as a fraction in simplest form?
0.625 as a simplified fraction is 5/8. Place 625 over 1,000 (625/1000) and divide both numerator and denominator by their Greatest Common Divisor (125) to yield 5/8.
How do you convert a repeating decimal like 0.333... into a fraction?
Use the "Repeating Decimal" tab. Set x = 0.333..., multiply by 10 (10x = 3.333...), and subtract the first equation (9x = 3). Dividing 3 by 9 gives the simplified fraction 1/3.
Can a decimal be converted to a mixed number?
Yes. Any decimal greater than 1 converts into a mixed number. For example, 2.75 equals the mixed number 2 3/4 (or the improper fraction 11/4).
What is 0.125 as a fraction?
0.125 as a simplified fraction is 1/8. Write 125 over 1,000 (125/1000) and divide both numbers by 125 to get 1/8.
Is my calculation data saved or logged?
No. All calculations run 100% locally inside your browser's runtime memory using JavaScript. No decimal values, homework entries, or engineering metrics are transmitted over an HTTP network or saved in an external database.