Fraction Calculator

A fraction represents an exact ratio of an integer numerator to a nonzero integer denominator. This calculator performs arithmetic, simplifies results, works with mixed numbers and converts finite decimals and percentages.

The exact rational result is kept separate from its decimal display. For example, 17/12 remains the answer even when a six-place decimal is shown as approximately 1.416667. Changing display precision does not change the fraction or a comparison result.

1. Select a calculation mode. Fraction Arithmetic is the starting mode.
2. Enter integer numerators above the divider and nonzero integer denominators below it. For two fractions, choose addition, subtraction, multiplication or division.
3. In mixed mode, select the sign of the entire number, then enter a nonnegative whole part and a proper fractional part. Choose arithmetic or conversion of the first number.
4. Review the exact result, mixed form, decimal and percentage. Select 2, 4, 6 or 10 maximum decimal places as needed.
5. Expand Show Calculation Steps to inspect the actual working. Copy the steps, print the calculation or export its values as CSV.

Results update as you type. Reset restores 2/3 + 3/4 and six-place decimal display. Each integer field supports up to 200 digits; finite decimal inputs support up to 200 digits in total. Inputs stay in this browser tab.

With a common denominator, keep the denominator and add numerators. When denominators differ, first write equivalent fractions with a shared denominator.

a/b + c/d = (a × d + c × b)/(b × d). A least common denominator can keep the intermediate values smaller. For 2/3 + 3/4, use 12: 2/3 = 8/12 and 3/4 = 9/12. Adding gives 17/12 = 1 5/12.

Negative numerators keep their signs through the calculation. Two equal denominators do not get added together; they describe the size of the same fractional units.

Subtraction follows the same common-denominator method: a/b − c/d = (a × d − c × b)/(b × d).

For 5/6 − 1/4, the least common denominator is 12. Rewrite as 10/12 − 3/12, then subtract to get 7/12. If the second fraction is greater, the result is negative; its sign is preserved rather than removed.

Multiply numerators together and denominators together: a/b × c/d = (a × c)/(b × d). Then reduce the result.

For 3/5 × 10/9, multiplication gives 30/45. Their GCD is 15, so the answer is 2/3. Cross-cancellation offers another route: cancel a factor of 3 between 3 and 9, and a factor of 5 between 10 and 5, leaving (1/1) × (2/3).

The Fraction of a Number mode applies this same operation to a finite decimal or whole-number quantity. For example, 3/4 of 12 is 3/4 × 12/1 = 9.

Divide by multiplying by the reciprocal of a nonzero second fraction: a/b ÷ c/d = (a × d)/(b × c), where c must not be zero.

For 3/4 ÷ 2/5, use 3/4 × 5/2 = 15/8 = 1 7/8. Division by a negative fraction preserves the appropriate sign. A second numerator of zero makes the divisor zero and produces a validation message.

The greatest common divisor, or GCD, is the largest positive integer dividing both parts. Dividing numerator and denominator by it produces an equivalent fraction in lowest terms.

For 24/36, GCD(24, 36) = 12. Divide both parts by 12 to obtain 2/3. The calculator also normalizes denominator signs: 2/−3 becomes −2/3, while −2/−3 becomes 2/3.

A zero numerator with any nonzero denominator is normalized to 0/1. A zero denominator is rejected before reduction.

The least common denominator is the least common multiple of the positive normalized denominators. For b and d, LCM(b, d) = (b × d)/GCD(b, d).

For denominators 6 and 4, GCD is 2 and LCM is 12. Multiplying both denominators gives 24, which is also a common denominator, but 12 is the smallest. The solution steps show the chosen denominator and the exact equivalent numerators.

Equivalent fractions come from multiplying both parts by the same nonzero integer: 1/2 = 2/4 = 3/6 = 4/8. Open the equivalent-fractions section to see examples generated from your current result.

With a positive denominator, a proper fraction has a numerator whose magnitude is smaller than the denominator. An improper fraction has a numerator whose magnitude is at least the denominator. A mixed number expresses a whole part and a proper fractional remainder.

For 17/12, integer division gives one whole and remainder five, so the mixed form is 1 5/12. If there is no remainder, the mixed/whole display shows only the whole number. The exact rational form retains denominator 1, making integer and zero results consistent.

Convert a positive mixed number w n/d to (w × d + n)/d. For 2 1/3, that is (2 × 3 + 1)/3 = 7/3. Arithmetic then follows the ordinary fraction rules.

A negative mixed sign applies to everything: −1 1/2 means −(1 + 1/2) = −3/2. It does not mean −1 + 1/2. The separate sign control makes that interpretation explicit, including when the whole part is zero.

Mixed inputs require nonnegative whole and fractional parts, with numerator smaller than denominator. Enter an improper fraction directly in Fraction Arithmetic or Simplify Fractions instead.

A decimal equivalent comes from numerator ÷ denominator. For 3/8, the result is exactly 0.375. For 1/3, the digits repeat forever; a finite displayed value is therefore rounded.

After reduction, a decimal terminates precisely when its denominator contains only factors of 2 and 5. The calculator distinguishes exact terminating decimals, terminating decimals rounded to the selected precision, and rounded repeating decimals.

The selected precision is a maximum; unnecessary trailing zeros are removed. Halfway cases round away from zero, so 1/8 at two places displays approximately 0.13 and −1/8 displays approximately −0.13. A very small nonzero fraction may round to an approximate zero at low precision, while its exact fraction remains visible.

Write the decimal digits over the matching power of ten, then reduce. For 0.125, use 125/1000; dividing by 125 gives 1/8. For 0.1, use 1/10 exactly.

The converter reads the decimal string as integers. It does not first approximate the value with binary floating-point arithmetic. A finite input such as 0.333333 stays 333333/1000000; it is not changed to 1/3.

Use a decimal point and optional leading sign. Scientific notation, thousands separators and explicit repeating notation are outside the accepted input syntax.

Multiply a fraction by 100 to express its value as a percentage. For 3/8, (3 × 100)/8 = 37.5%.

To reverse the process, divide the entered percentage by 100. Thus 37.5% = 37.5/100 = 375/1000 = 3/8. The percentage converter accepts an optional trailing percent sign and preserves the exact finite decimal value.

For fractions a/b and c/d with positive denominators, compare a × d and c × b. For 3/4 and 2/3, the products are 9 and 8, so 3/4 is greater than 2/3.

This method also handles negative numerators. Signs are normalized first so a negative denominator cannot reverse the inequality unexpectedly. Rounded decimal values never determine the comparison, even if the displayed decimals happen to match.

Fractions use BigInt numerators and denominators. Every rational value is reduced, its denominator is positive and zero becomes 0/1. Display rounding uses integer division and remainder, with halfway cases away from zero.

Inputs are limited to 200 digits per integer or finite decimal. Mixed fractional parts must be proper. Results and generated steps use the same exact engine; copy, print and CSV include the actual values and explanations. Calculations require no external service.

• OpenStax: fraction simplification, multiplication and division
• OpenStax: common denominators, addition and subtraction
• MDN: exact integer arithmetic with BigInt