mathematics calculator

Fraction Calculator

Simplify, calculate, compare, convert, and find common denominators with exact integer arithmetic.

Result

Enter fractions and choose Calculate.

Formula version 1.0.0 · Exact fraction arithmetic uses integers.

Exact arithmetic

Fraction formulas and worked examples

Denominators cannot be zero. Results use a positive denominator and are reduced by the greatest common divisor.

Add or subtract

a/b ± c/d = (ad ± cb)/bd

Example: 1/3 + 1/6 = 3/6 = 1/2.

Multiply or divide

a/b × c/d = ac/bd

a/b ÷ c/d = ad/bc

Example: 3/4 ÷ 2/5 = 15/8.

Mixed and improper forms

w n/d = (wd+n)/d

Example: 2 1/3 = 7/3; 11/4 = 2 3/4.

Compare and find a common denominator

Compare exact cross-products. The least common denominator is lcm(b,d).

Example: 2/3 = 8/12 and 3/4 = 9/12, so 2/3 < 3/4.

How to use this calculator

Choose a task in Calculation type, such as Simplify a fraction, Add fractions, or Mixed number to improper fraction.
Enter each requested Numerator and Denominator as a whole number. A denominator must not be zero.
For a mixed number, choose Sign and enter Whole number, Numerator, and Denominator.
Select Calculate, then read the exact reduced Result and its decimal display when that mode provides one.
Open Show calculation steps to inspect normalization and reduction; use Reset before starting an unrelated calculation.

Symbols and units

Symbol	Meaning	Unit or domain
`a/b`	First fraction: numerator a divided by denominator b	a is an integer; b is a nonzero integer
`c/d`	Second fraction: numerator c divided by denominator d	c is an integer; d is a nonzero integer
`w n/d`	Mixed number with whole magnitude w and proper fractional part n/d	w and n are nonnegative magnitudes; d is positive; Sign sets the overall sign
`gcd`	Greatest common divisor used to reduce numerator and denominator	Positive integer divisor of both magnitudes
`lcm`	Least common multiple used as the least common denominator	Positive integer multiple of both denominator magnitudes
`±`	Addition or subtraction chosen by Calculation type	Exact integer arithmetic

Checked examples

Add one third and one sixth

Inputs: Calculation type: Add fractions; First numerator: 1; First denominator: 3; Second numerator: 1; Second denominator: 6.

Calculation: Use a common denominator: 1/3 = 2/6. Add numerators: 2/6 + 1/6 = 3/6. Reduce by gcd(3, 6) = 3: 3/6 = 1/2.

Result: 1/2; decimal 0.5.

The exact sum is one half.

Check: tests/fraction-general.test.ts — adds, subtracts, multiplies and divides exactly

Convert a negative mixed number

Inputs: Calculation type: Mixed number to improper fraction; Sign: Negative; Whole number: 2; Numerator: 1; Denominator: 3.

Calculation: Multiply the whole magnitude by the denominator: 2 × 3 = 6. Add the fractional numerator: 6 + 1 = 7. Apply the selected negative sign: −7/3.

Result: −7/3; decimal approximately −2.3333333333.

Negative 2 1/3 is exactly negative seven thirds.

Check: tests/fraction-general.test.ts — converts mixed and improper fractions

Assumptions, rounding, and limits

Inputs are base-10 integers with at most 128 digits; decimal fraction inputs are not accepted.
Every result is normalized to a positive denominator and reduced by the greatest common divisor.
Core fraction operations are exact. The decimal display is secondary and rounded to at most 10 displayed digits using half-up formatting.
Division by a zero fraction, a zero denominator, and invalid mixed-number fields fail without a result.
Comparison uses exact cross-products, avoiding floating-point comparison error.
Large exact integers are supported, but an extremely long value may be impractical to read or reuse.

Frequently asked questions

Why must a denominator be nonzero?

Division by zero is undefined, so a fraction with denominator zero cannot represent a number and the calculator returns an error.

Why does the calculator change a negative denominator?

It moves the sign to the numerator so equivalent results use one consistent form with a positive denominator.

Is the decimal result exact?

Terminating decimals can be exact. Repeating or longer decimals are labeled approximate and rounded for display; the reduced fraction remains the exact result.

Sources and review

OpenStax: Key Concepts: Fractions — Accessed 2026-10-25
National Institute of Standards and Technology: Binary GCD — Accessed 2026-10-25

Formula version: fraction.general@1.0.0
Formula review: Formula review: Equatara formula review
Editorial review: Editorial review: Equatara editorial review
Last verified: 2026-10-25
Applicability: Global, locale-independent fraction arithmetic using exact integers. No jurisdiction-specific rule is used.
Next review: On material formula/source change, or by 2028-10-25.

Found a problem?

Report a formula, result, example, source, or accessibility issue through the calculator correction process. Do not include sensitive input values.

Calculate with fractions