Fraction Calculator
Add, subtract, multiply or divide two fractions and get an exact answer in lowest terms — shown as an improper fraction, a mixed number and a decimal.
Calculator
- Add, subtract, multiply, divide
- Exact lowest-terms result
- Mixed-number output
- Negative denominators normalized
- Inverse-operation check
Result & calculation receipt
- Method
- Exact rational arithmetic with sign normalization and gcd reduction
- Formula
a/b + c/d = (a·d + c·b) / (b·d)- Substitution
- 1/2 + 1/3 = 5/6
- Steps
- Fraction 1 = 1/2, Fraction 2 = 1/3 (reduced and sign-normalized).
- Common denominator: 2 × 3 = 6.
- Numerator: 1·3 + 1·2 = 5.
- Simplify 5/6 by dividing by gcd = 1 → 5/6.
- Verification
- 5/6 − 1/3 = 1/2 (should equal 1/2).
- Exact result
- 5/6
- Decimal result
- 0.8333333333
- Engine
- math-solvers v1.4.0
How this calculator works
Each fraction is first sign-normalized (any negative denominator is moved to the numerator) and reduced. For addition and subtraction the calculator puts both over a common denominator; for multiplication it multiplies straight across; for division it multiplies by the reciprocal. The result is reduced by its greatest common factor, and the answer is verified by running the inverse operation back to the first fraction.
Worked example
To add 1/2 and 1/3 the calculator forms the common denominator 6, rewrites them as 3/6 and 2/6, and adds the numerators to get 5/6. To divide 2/3 by 4/9 it flips the divisor and multiplies: 2/3 × 9/4 = 18/12 = 3/2 = 1½. Dividing by 0/… is refused, because the reciprocal would be undefined.
Key insight
Fractions are exact where decimals are not: 1/3 is precise, while 0.333… is only an approximation. Keeping arithmetic in fractions until the very end — then converting once — avoids the rounding errors that creep in when you switch to decimals too early.
Frequently asked questions
How do I add fractions with different denominators?
Rewrite both fractions over a common denominator (the product of the two denominators always works, the least common multiple keeps numbers small), add the numerators, then reduce. The calculator shows the common denominator it used.
How do I divide one fraction by another?
Multiply the first fraction by the reciprocal of the second — flip the divisor and multiply. So 2/3 ÷ 4/9 = 2/3 × 9/4 = 3/2. Dividing by a fraction equal to zero is undefined and is reported as an error.
What happens with a negative denominator?
A negative sign in the denominator is moved to the numerator so the denominator is always positive, e.g. 1/−2 is stored as −1/2. The value is unchanged; the form is standardized.
What is an improper fraction versus a mixed number?
An improper fraction has a numerator at least as large as its denominator, like 7/2; the same value as a mixed number is 3½. The calculator shows both forms of the answer.
How is the answer verified?
By the inverse operation: for a sum, subtracting the second fraction from the result should return the first; for a quotient, multiplying the result by the divisor should return the dividend. That check is shown in the receipt.
Methodology & verification
Fractions are stored as sign-normalized, gcd-reduced integer pairs (denominator always positive). Addition and subtraction use a common denominator, multiplication multiplies numerators and denominators, and division multiplies by the reciprocal; a zero denominator or division by a zero fraction returns a structured error. The result is reduced by its greatest common factor and expressed as an improper fraction, a mixed number and a decimal. Verification runs the inverse operation and confirms it recovers the first operand.
Engine: math-solvers v1.4.0 · Last reviewed: · Every result is computed client-side and never sent to a server. See the verification methodology and the scientific calculator validation.