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Math · Live

Fractions, solved with the working shown.

A free fraction calculator that adds, subtracts, multiplies, and divides any two fractions or mixed numbers, with a full step-by-step solution, automatic simplification, and three equivalent answer forms (fraction, mixed number, decimal).

How it worksReal-time

Inputs

Fractions

Fraction A

Fraction B

Result

add

56
Improper 5/6Decimal 0.8333333333
As a fraction
5/6
As a mixed number
5/6
As a decimal
0.8333333333

Working

Step-by-step

3 steps
  1. 011/2 + 1/3
  2. 02Common denominator 2 × 3 = 6: 3/6 + 2/6
  3. 03Add numerators: (3 + 2)/6 = 5/6

Field guide

How to add, subtract, multiply, and divide fractions.

A fraction is a ratio: a/b means “a parts out of b equal pieces.” Doing arithmetic with fractions is mostly about getting the parts to line up, then combining them and simplifying. Each operation has its own recipe.

Adding fractions

To add two fractions you need a common denominator. The shortcut that always works is to cross-multiply by the other denominator:

a/b + c/d = (a · d + c · b) / (b · d)

Find 1/2 + 1/3 by multiplying out: (1·3 + 1·2) / (2·3) = 5/6. The result is already in lowest terms because gcd(5, 6) = 1.

Subtracting fractions

Same recipe with a minus instead of a plus:

a/b − c/d = (a · d − c · b) / (b · d)

3/4 − 1/6 = (3·6 − 1·4) / (4·6) = (18 − 4) / 24 = 14/24, which simplifies to 7/12 after dividing top and bottom by 2.

Multiplying fractions

The easiest of the four: just multiply the numerators and multiply the denominators.

a/b × c/d = (a · c) / (b · d)

2/3 × 4/5 = 8/15. No common denominator needed — it falls out for free.

Dividing fractions

Dividing is multiplying by the reciprocal of the second fraction:

a/b ÷ c/d = a/b × d/c = (a · d) / (b · c)

3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8, which is 1 7/8 as a mixed number.

Mixed numbers vs improper fractions

A mixed number like 1 1/2 is shorthand for a whole number plus a fraction. To do arithmetic, convert it to an improper fraction first by multiplying the whole part by the denominator and adding the numerator:

mixed → improper:   w n/d = (w · d + n) / d
improper → mixed:   n/d = ⌊n/d⌋ + (n mod d)/d

1 1/2 = (1·2 + 1)/2 = 3/2. Going the other way: 7/4 → ⌊7/4⌋ + (7 mod 4)/4 = 1 3/4.

Simplifying (lowest terms)

A fraction is in lowest terms when its numerator and denominator share no common factor other than 1. To simplify, divide both by their gcd (greatest common divisor):

simplified = num ÷ gcd(num, den), den ÷ gcd(num, den)

14/24 with gcd(14, 24) = 2 reduces to 7/12. The Euclidean algorithm — repeatedly replacing the pair with (b, a mod b) until b hits zero — is what the calculator uses under the hood.

Negative fractions

For arithmetic we keep the sign on the numerator and the denominator strictly positive, so −3/4 stays as −3/4 (not 3/−4). When a result is improper and negative, it's displayed as −1 1/2 rather than (−1) 1/2.

Worked example: a full subtraction

Compute 2 1/3 − 3/4:

  1. Convert: 2 1/3 = (2·3 + 1)/3 = 7/3.
  2. Common denominator 3 × 4 = 12: 7/3 = 28/12 and 3/4 = 9/12.
  3. Subtract: (28 − 9)/12 = 19/12.
  4. Simplify: gcd(19, 12) = 1, already in lowest terms.
  5. Mixed form: 19/12 = 1 7/12.
  6. Decimal: 19/12 ≈ 1.5833….

Disclaimer

This calculator works with rational numbers up to four-digit inputs — plenty for school assignments and everyday use. For arbitrary-precision arithmetic, very large repeating decimals, or symbolic algebra, use a CAS like Wolfram or SymPy.