Mixed Number Calculator

Calculate fast and accurately with mixed numbers (mixed fractions). Add, subtract, multiply, or divide two mixed numbers, with automatic simplification and clear step-by-step working. Ideal for students, teachers, and professionals who need an authoritative, accessible tool.

Calculator

Operation (choose one)
Mixed Number A

Enter the sign and the parts (whole, numerator, denominator).

Mixed Number B

Enter the sign and the parts (whole, numerator, denominator).

Tip: The tool automatically simplifies results and shows them as improper, mixed, and decimal forms.

Results

Mixed form
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Improper fraction
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Decimal
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Step-by-step

No calculation yet.

Data Source and Methodology

Authoritative Data Source: OpenStax, Prealgebra 2e, Chapter 5: Fractions and Mixed Numbers (2020). Direct link

Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.

Additional references: CalculatorSoup Mixed Numbers for notation alignment.

The Formula Explained

For a mixed number a b/c, the improper fraction is:
$$\frac{n}{d} = \frac{\operatorname{sgn} \cdot (|a|\cdot c + b)}{c}$$

Addition/Subtraction (with LCM L of denominators d1, d2):
$$\frac{n_1}{d_1} \pm \frac{n_2}{d_2} = \frac{n_1 \cdot \frac{L}{d_1} \pm n_2 \cdot \frac{L}{d_2}}{L}$$

Multiplication:
$$\frac{n_1}{d_1} \times \frac{n_2}{d_2} = \frac{n_1 \cdot n_2}{d_1 \cdot d_2}$$

Division:
$$\frac{n_1}{d_1} \div \frac{n_2}{d_2} = \frac{n_1}{d_1} \times \frac{d_2}{n_2},\quad n_2 \neq 0$$

Simplification (GCD g):
$$\frac{n}{d} = \frac{n/g}{d/g} \quad \text{where } g = \gcd(|n|, d)$$

Convert back to mixed:
$$\text{whole} = \left\lfloor \frac{|n|}{d} \right\rfloor,\quad \text{remainder} = |n|\bmod d$$

Glossary of Variables

  • a: whole part of a mixed number (non-negative integer)
  • b: numerator of the fractional part (0 ≤ b < c)
  • c: denominator of the fractional part (integer ≥ 1)
  • sgn: overall sign of the mixed number (+1 or −1)
  • n/d: improper fraction equivalent to the mixed number
  • LCM (L): least common multiple of denominators
  • GCD (g): greatest common divisor used to simplify
  • Result outputs: mixed form, improper fraction, and decimal

Worked Example

How It Works: A Step-by-Step Example

Compute 2 1/3 + 1 3/4.

  1. Convert to improper:
    2 1/3 = (2×3 + 1)/3 = 7/3, 1 3/4 = (1×4 + 3)/4 = 7/4
  2. Common denominator: LCM(3,4) = 12
    7/3 = 28/12, 7/4 = 21/12
  3. Add:
    28/12 + 21/12 = 49/12
  4. Simplify: gcd(49,12) = 1 → already simplest.
  5. Convert back:
    49/12 = 4 1/12 ≈ 4.0833̅

Frequently Asked Questions (FAQ)

What inputs are required?

For each mixed number, provide the sign, whole number, numerator, and denominator (denominator ≥ 1). Numerator must be less than the denominator for the mixed part.

Can I use zero values?

Yes. Whole can be zero, and numerator can be zero. Denominator must be at least 1. If both whole and numerator are zero, the mixed number is 0.

How are negative mixed numbers represented?

Choose the sign (−). The sign applies to the entire mixed number. Internally we keep the sign on the improper numerator.

How do you avoid rounding errors?

All operations are exact on rational numbers. Decimals are shown with up to 12 significant digits for readability; the exact result is the fraction.

What happens if I divide by zero?

The calculator prevents division by a value equal to 0 and explains the error below the relevant field. No calculation is performed until the issue is fixed.

Does the tool show steps for every operation?

Yes. It shows conversion to improper, the core operation (and common denominators where needed), simplification using GCD, and the mixed/decimal forms.

Authorship & Review

Tool developed by Ugo Candido. Content verified by CalcDomain Editorial Team.
Last reviewed for accuracy on: .