Calculate and Reduce Two Fractions Exactly

Keep rational arithmetic exact until the decimal preview

A fraction calculator should not convert both inputs to floating-point decimals and hope to reconstruct the result. This tool reads integer numerators and nonzero integer denominators, performs the selected operation with browser BigInt arithmetic, normalizes the sign, and reduces by the greatest common divisor. The fraction result stays exact within the guarded input and output limits.

Use the four operation choices for addition, subtraction, multiplication, or division. The page shows the applicable formula and an unreduced intermediate numerator and denominator before the reduced answer. A decimal preview is optional and explicitly approximate when the expansion does not terminate within the displayed places.

Normalize signs first

A negative sign can appear in the numerator, denominator, or both. The calculator moves any final sign to the numerator and keeps the denominator positive. Thus 1/−2 becomes −1/2, while −1/−2 becomes 1/2. This convention makes equality and reduction easier to read.

Zero is represented as 0/1 after reduction regardless of the entered nonzero denominator. A denominator of zero is never a fraction and returns a field error before an operation begins.

BigInt input uses digit strings rather than a browser number field that could lose integer precision. The interface caps input length and guards intermediate output size so a deliberately enormous multiplication cannot freeze the page.

Addition and subtraction

For a/b + c/d, a common product denominator gives:

(a × d + c × b) / (b × d)

Subtraction changes the plus to minus. The result is reduced by the Euclidean greatest-common-divisor algorithm.

Take 3/4 + 5/6. The unreduced numerator is 3×6 + 5×4 = 38 and the denominator is 24. The GCD of 38 and 24 is 2, so the reduced answer is 19/12. As a mixed number, that is 1 7/12. The decimal preview is approximately 1.583333333333.

The calculator may reduce cross-factors before multiplication to keep intermediates smaller, but the visible derivation remains mathematically equivalent. The result is the same exact rational value.

Multiplication and division

Multiplication uses:

(a × c) / (b × d)

For 7/9 × 3/14, the direct product is 21/126 and reduction yields 1/6. A step view may show that 7 cancels with 14 and 3 cancels with 9 before multiplication. Cancellation is division by shared factors, not removing digits.

Division multiplies by the reciprocal of the second fraction:

(a/b) ÷ (c/d) = (a × d) / (b × c)

For 3/4 ÷ 2/5, the result is 15/8, or 1 7/8. If the second numerator is zero, its fraction is zero and has no reciprocal. The tool returns an undefined-division error instead of creating a zero denominator.

Worked sign example

Enter −2/3 − 5/−9. Sign normalization turns the second fraction into −5/9. The operation becomes −2/3 − (−5/9), which is −6/9 + 5/9 = −1/9. The substituted line keeps parentheses around the negative second value so the two minus signs are not visually collapsed.

Now swap subtraction for addition. The result becomes −2/3 + (−5/9) = −6/9 − 5/9 = −11/9, or −1 2/9. Mixed-number formatting applies the sign to the whole rational value.

Reduction and mixed numbers

The Euclidean algorithm repeatedly replaces a pair with the smaller value and the remainder until the remainder is zero. The final nonzero magnitude is the GCD. Dividing numerator and denominator by it produces lowest terms.

An improper fraction is not an error. It is often the clearest exact form for further arithmetic. The mixed-number line is a reading convenience. For a negative improper fraction, the calculator uses truncation toward zero for the whole component and a positive remainder magnitude, displaying −1 2/9 rather than a confusing signed remainder.

Limits and uses

This page performs two-rational-number arithmetic. It does not parse mixed numbers directly, solve equations, compare a long list, factor symbolic expressions, or claim arbitrary length. Convert a mixed number to an improper fraction before entry: 2 1/3 becomes 7/3.

Fractions in recipes, measurements, grades, probabilities, finance, and engineering may involve units, tolerances, rules, or uncertainty that exact rational arithmetic does not evaluate. A reduced fraction can be mathematically correct while the modeled input is unsuitable.

Frequently asked questions

Enter two fractions, choose the operation, calculate, and compare the unreduced step with the lowest-term result.