Fraction Calculator - Fraction Arithmetic & Mixed Number Solver

Fraction Calculator – Arithmetic & Mixed Number Guide

Working with fractions requires applying specific cross-multiplication and reduction rules depending on whether you are adding, subtracting, multiplying, or dividing. This calculator solves fractional equations, returning a reduced proper fraction, its mixed-number form, and its decimal equivalent.

How Fraction Calculations Work

Addition and subtraction cross-multiply to find a common denominator:

(a / b) + (c / d) = (ad + bc / bd) qquad (a / b) - (c / d) = (ad - bc / bd)

Multiplication and division combine numerators and denominators directly (division multiplies by the reciprocal of the second fraction):

(a / b) × (c / d) = (ac / bd) qquad (a / b) ÷ (c / d) = (ad / bc)

Reduction and mixed numbers. The raw result is reduced by dividing both the numerator and denominator by their Greatest Common Divisor (GCD). If the reduced numerator's absolute value is at least as large as the denominator, the calculator also expresses the result as a mixed number: the whole-number part is the truncated quotient, and the remaining fractional part uses the remainder over the same denominator.

Worked Examples

Example 1: Addition — 1/2 + 2/3

Numerator = 1 × 3 + 2 × 2 = 7. Denominator = 2 × 3 = 6. Raw result = 7/6, already in lowest terms since gcd(7, 6) = 1. As a mixed number: 7 ÷ 6 = 1 remainder 1, so 1 tfrac16. Decimal: 7 / 6 ≈ 1.166667.

Example 2: Division — 3/4 ÷ 2/5

Numerator = 3 × 5 = 15. Denominator = 4 × 2 = 8. Raw result = 15/8, already reduced since gcd(15, 8) = 1. As a mixed number: 15 ÷ 8 = 1 remainder 7, so 1 tfrac78. Decimal: 15 / 8 = 1.875.

Example 3: Subtraction — 5/6 − 1/4

Numerator = 5 × 4 - 1 × 6 = 20 - 6 = 14. Denominator = 6 × 4 = 24. Raw result = 14/24, and since gcd(14, 24) = 2, the reduced fraction is 7/12. Because the numerator (7) is smaller than the denominator (12), this is already a proper fraction — no whole-number part is shown. Decimal: 7 / 12 ≈ 0.583333.

Denominator Restrictions

Frequently Asked Questions

How do you add fractions with different denominators?

Cross-multiply the numerators by the opposite denominators, sum them for the new numerator, and multiply the denominators together: (a/b) + (c/d) = (ad + bc) / bd.

How does the calculator reduce fractions to lowest terms?

The calculator computes the Greatest Common Divisor (GCD) of the resulting numerator and denominator using the Euclidean algorithm, then divides both numbers by that factor.

What happens if a denominator is entered as zero?

Division by zero is mathematically undefined. The calculator validates both denominator fields and will not compute a result if either denominator is zero, and it also blocks division operations where the second numerator is zero, since that would divide by zero.

Why does the reduced fraction sometimes have no whole-number part?

A fraction only converts to a mixed number when the absolute value of its reduced numerator is greater than or equal to its denominator (an improper fraction). As the subtraction example (7/12) shows, a proper fraction — where the numerator is smaller than the denominator — displays with a whole-number part of 0 and no separate mixed format.