Fraction Calculator - Add, Subtract, Multiply & Divide Fractions

What is a Fraction Calculator?

A Fraction Calculator is an exact arithmetic and algebraic computation tool designed to perform addition, subtraction, multiplication, and division on fractions, mixed numbers, and integers with step-by-step simplification. Unlike floating-point decimal calculators that induce rounding inaccuracies when representing non-terminating values such as $$\frac{1}{3} = 0.333333\dots$$, fractional mathematics preserves absolute precision.

Whether reducing complex engineering fractions, balancing culinary ingredient proportions, or completing academic homework proofs, understanding the underlying rules of rational numbers ($$\mathbb{Q}$$) is an essential mathematical foundation.

The Core Algebraic Rules of Fractional Arithmetic

Fraction operations follow rigorous algebraic identity theorems based on common denominators and reciprocals:

1. Addition and Subtraction

Fractions with different denominators must be converted to equivalent fractions sharing the Least Common Denominator ($$\text{LCD}$$):

\frac{a}{b} \pm \frac{c}{d} = \frac{(a \times d) \pm (b \times c)}{b \times d}

2. Multiplication

Multiply numerators together directly and denominators together directly:

\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}

3. Division (The Reciprocal Rule)

Dividing by a fraction is equivalent to multiplying by its reciprocal (inverting the divisor):

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c} \quad (c, d \neq 0)

4. Reduction to Lowest Terms via Greatest Common Divisor

Divide both numerator and denominator by their $$\gcd(N, D)$$, computed via Euclid's Algorithm:

\frac{N}{D} = \frac{N / \gcd(N, D)}{D / \gcd(N, D)}

Step-by-Step Practical Fraction Example

Evaluate the fractional addition of $$2\frac{1}{4}$$ and $$\frac{5}{6}$$:

Convert the Mixed Number to an Improper Fraction: $$2\frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{9}{4}$$
Find the Least Common Denominator ($$\text{LCD}$$): $$\text{Denominators: } 4 \text{ and } 6. \quad \text{LCM}(4, 6) = 12$$
Convert Fractions to Equivalent Denominators of 12: $$\frac{9}{4} = \frac{9 \times 3}{4 \times 3} = \frac{27}{12}, \qquad \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$$
Add the Numerators Over the Common Denominator: $$\frac{27}{12} + \frac{10}{12} = \frac{27 + 10}{12} = \frac{37}{12}$$
Simplify and Convert to Mixed Number and Decimal: $$37 \div 12 = 3 \text{ with a remainder of } 1 \implies 3\frac{1}{12}$$ $$\text{Decimal Equivalent} = \frac{37}{12} = 3.08333\dots$$

Conclusion: The exact simplified sum is $$\frac{37}{12}$$, which evaluates to the mixed number $$3\frac{1}{12}$$ or approximately 3.0833.