Fraction Calculator
Compute accurate results for adding, subtracting, multiplying, or dividing fractions and mixed numbers. Built for students, educators, and professionals, this tool simplifies results, converts to mixed-number and decimal forms, and can show the full working for maximum clarity.
Calculator
Press Calculate or simply start typing; results update automatically when inputs are valid.
Results
Working
Data Source and Methodology
Authoritative Source: OpenStax, Prealgebra 2e, Section 5 “Fractions” (2015; updated 2022). Direct link: https://openstax.org/details/books/prealgebra-2e. All calculations are rigorously based on the formulas and procedures described in this source.
All computations are performed exactly on rational numbers using integer arithmetic (no floating-point rounding during fraction operations). Results are reduced using the greatest common divisor (GCD).
The Formula Explained
Addition/Subtraction:
\( \displaystyle \frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{\,bd\,} \)
Multiplication:
\( \displaystyle \frac{a}{b} \times \frac{c}{d} = \frac{ac}{\,bd\,} \)
Division (with \(c \neq 0\)):
\( \displaystyle \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{\,bc\,} \)
Simplification using GCD \(g=\gcd(n, d)\):
\( \displaystyle \frac{n}{d} = \frac{n/g}{\,d/g\,} \)
Mixed number from improper fraction \(n/d\):
\( \displaystyle n = qd + r,\quad 0 \le r < d \Rightarrow \frac{n}{d} = q \;\frac{r}{d} \)
Glossary of Variables
| Term | Definition |
|---|---|
| Fraction A, Fraction B | Input values. Accept integers, simple fractions, mixed numbers (e.g., 2 1/3), or decimals. |
| Operation | The arithmetic operation: Add, Subtract, Multiply, or Divide. |
| Raw (Unsimplified) | Intermediate result before reduction, as a single fraction. |
| Simplified Fraction | The result fully reduced by dividing numerator and denominator by their GCD. |
| Mixed Number | Whole number plus a proper fraction equivalent to the simplified fraction. |
| Decimal | Decimal representation of the exact rational value, rounded for display. |
How It Works: A Step-by-Step Example
Suppose we want to compute: 1 1/2 + 3/4.
- Normalize inputs to improper fractions: 1 1/2 = 3/2, and 3/4 is already proper.
- Apply the addition formula: \( \frac{3}{2} + \frac{3}{4} = \frac{3\cdot4 + 3\cdot2}{2\cdot4} = \frac{12 + 6}{8} = \frac{18}{8} \).
- Simplify using GCD(18, 8) = 2 → \( \frac{18}{8} = \frac{9}{4} \).
- Convert to mixed number: \( \frac{9}{4} = 2\ \frac{1}{4} \). Decimal: 2.25.
Frequently Asked Questions (FAQ)
What formats can I enter?
Use integers (e.g., -3), fractions (e.g., 7/8), mixed numbers (e.g., 1 3/4), or decimals (e.g., 0.125). Use a space between whole and fractional parts.
How do you avoid rounding errors?
All fraction arithmetic is performed using integer numerators and denominators. Decimal inputs are converted to exact rational numbers before computation, and the result is reduced by GCD.
Why is division by zero blocked?
Division by zero is undefined in mathematics. If the second operand equals 0 and you choose Divide, the tool shows a clear error and prevents calculation.
Can I disable simplification?
Yes. Toggle “Auto-simplify” to view the raw unsimplified fraction (useful for checking intermediate work).
Will it show the steps?
Enable “Show steps” to see normalization, the operation with substituted values, and the simplification path. Formulas are rendered in LaTeX for clarity.
Does it support negative mixed numbers?
Yes. Enter them as “-3 1/2”. The negative sign applies to the entire mixed number.
Is the tool accessible?
Yes. It is keyboard-friendly, has visible focus states, descriptive labels, ARIA attributes, and meets WCAG 2.1 AA contrast and interaction standards.