How do you convert a decimal to a fraction?

To convert a terminating decimal to a fraction, count the number of decimal places (n), then write the decimal digits as the numerator and 10ⁿ as the denominator. For example, 0.75 has 2 decimal places, so the numerator is 75 and the denominator is 100. Then reduce by the greatest common divisor: 75/100 ÷ 25/25 = 3/4. For repeating decimals, use the algebraic method described in the FAQ below.

What is a repeating decimal?

A repeating decimal is a decimal in which one or more digits repeat infinitely. For example, 1/3 = 0.3333..., 2/3 = 0.6666..., and 1/7 = 0.142857142857.... Any fraction whose denominator has prime factors other than 2 and 5 will produce a repeating decimal. Repeating decimals are written with a bar over the repeating block (e.g., 0.̄3) or with ellipsis notation (0.333...).

How do you convert 0.333... to a fraction?

Use the algebraic method: let x = 0.333.... Multiply both sides by 10: 10x = 3.333.... Subtract the first equation from the second: 10x − x = 3.333... − 0.333..., so 9x = 3, therefore x = 3/9 = 1/3. This technique works for any repeating decimal — multiply by a power of 10 that shifts exactly one full repeating block, subtract to eliminate the repeating part, then solve.

How do you simplify a fraction?

To simplify (reduce) a fraction, divide both the numerator and denominator by their greatest common divisor (GCD). For example, 75/100: the GCD of 75 and 100 is 25, so 75 ÷ 25 = 3 and 100 ÷ 25 = 4, giving 3/4. A fraction is in lowest terms when the GCD of numerator and denominator is 1 — meaning no integer other than 1 divides both evenly.

What is the difference between terminating and repeating decimals?

A terminating decimal has a finite number of digits after the decimal point (e.g., 0.5, 0.25, 0.125). A repeating decimal has a digit or block of digits that repeats infinitely (e.g., 0.333..., 0.142857142857...). Every fraction (rational number) produces either a terminating or repeating decimal — never a random non-repeating one. Terminating decimals occur when the fraction's denominator, in lowest terms, has only 2 and 5 as prime factors.

How do you convert a fraction to a decimal?

Divide the numerator by the denominator using long division. For example, 3/8: 3 ÷ 8 = 0.375 (terminating). For 1/3: 1 ÷ 3 = 0.333... (repeating). If the division never reaches a remainder of 0, the decimal repeats. The repeating block length is at most denominator − 1 digits long. In practice, any calculator can perform this division directly.

When are fractions preferred over decimals?

Fractions are preferred when exact ratios matter. In cooking, 3/4 cup is intuitive; 0.75 cup is not on most measuring cups. In woodworking and construction, lumber and hardware dimensions use fractions of an inch (5/8", 3/4", 7/8") — decimal equivalents require memorization. In algebra and calculus, fractions avoid rounding errors and produce cleaner answers. Decimals are better for comparison and when working with calculators or computers.

What is a repeating decimal and how do I convert one to a fraction?

A repeating decimal has a digit or group of digits that repeats endlessly, like 0.333... = 1/3 or 0.142857142857... = 1/7. To convert: set x equal to the decimal, multiply by 10ⁿ (where n is the length of the repeating block) to shift one full cycle, then subtract the original equation to cancel the repeating part. Solve for x and simplify. For example: x = 0.272727..., 100x = 27.272727..., 99x = 27, x = 27/99 = 3/11.

Decimal to Fraction Calculator — Convert & Simplify

Converts any decimal to a fraction in lowest terms, or any fraction to a decimal — handles repeating decimals.

How to Use the Decimal to Fraction Calculator

This decimal to fraction calculator makes conversions instant — just choose a conversion mode using the toggle at the top. In Decimal → Fraction mode, type any decimal number — terminating (like 0.75) or repeating (like 0.333...) — and the calculator instantly returns the fraction in lowest terms, the mixed number form, and a decimal verification. In Fraction → Decimal mode, enter the numerator and denominator separately to get the exact decimal value and identify whether it terminates or repeats.

For fraction arithmetic and simplification, see our simplify fractions calculator, which reduces any fraction to lowest terms step by step.

How to Convert a Decimal to a Fraction

Converting a terminating decimal to a fraction is a three-step process:

Count decimal places. For 0.75, there are 2 decimal places.
Write as a fraction. Place the digits over 10ⁿ (where n is the number of decimal places): 75/100.
Reduce by GCD. GCD(75, 100) = 25, so 75/100 = 3/4.

For repeating decimals like 0.333..., use the algebraic method: set x = 0.333..., multiply by 10 to get 10x = 3.333..., then subtract to get 9x = 3, so x = 3/9 = 1/3.

AdvertisementResponsive Ad

Terminating vs. Repeating Decimals

Every fraction (rational number) produces either a terminating or repeating decimal — never a random non-repeating sequence. The type depends entirely on the denominator:

Terminating: The denominator (in lowest terms) has only 2 and 5 as prime factors. Examples: 1/2 = 0.5, 1/4 = 0.25, 3/8 = 0.375, 1/5 = 0.2, 7/20 = 0.35.
Repeating: The denominator has any prime factor other than 2 or 5. Examples: 1/3 = 0.333..., 1/7 = 0.142857..., 5/12 = 0.41666..., 2/9 = 0.222....

Non-terminating, non-repeating decimals (like π = 3.14159...) are irrational numbers — they cannot be written as fractions. Our rounding calculator can round any decimal to a specified number of places.

Common Decimal to Fraction Conversions

These are the most frequently needed conversions in everyday math:

0.1 = 1/10 — one tenth
0.125 = 1/8 — one eighth (common in recipes and measurements)
0.25 = 1/4 — one quarter
0.333... = 1/3 — one third (repeating)
0.375 = 3/8 — three eighths
0.5 = 1/2 — one half
0.625 = 5/8 — five eighths
0.666... = 2/3 — two thirds (repeating)
0.75 = 3/4 — three quarters
0.875 = 7/8 — seven eighths

Mixed Numbers and Improper Fractions

When the numerator is larger than the denominator, the fraction is called improper. It can be rewritten as a mixed number — a whole number plus a proper fraction. For example, 7/4 = 1¾ and 11/3 = 3⅔. Mixed numbers are easier to visualize (you can immediately see they are between two consecutive whole numbers), while improper fractions are easier to use in multiplication and division.

To convert an improper fraction to a mixed number: divide the numerator by the denominator, use the quotient as the whole number, and the remainder as the new numerator over the original denominator. Our mixed number calculator handles arithmetic with mixed numbers directly, and our remainder calculator finds the remainder from any integer division in one step.

AdvertisementResponsive Ad

The Greatest Common Divisor and Simplification

Reducing a fraction to lowest terms requires finding the greatest common divisor (GCD) of the numerator and denominator — the largest integer that divides both evenly. The most efficient algorithm is the Euclidean algorithm: repeatedly replace the larger number with the remainder of dividing the larger by the smaller until the remainder is 0. The last non-zero remainder is the GCD.

Example — GCD of 36 and 48: 48 ÷ 36 = 1 remainder 12; 36 ÷ 12 = 3 remainder 0. GCD = 12. So 36/48 = (36 ÷ 12)/(48 ÷ 12) = 3/4. For ratio problems involving fractions, see our proportion calculator.

Sources & References

Fractions — Math Is Fun — Math Is Fun
Repeating Decimal — Khan Academy — Khan Academy
Greatest Common Divisor — Britannica — Britannica

Decimal to Fraction Calculator