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Math calculator

Fractions Calculator

Add, subtract, multiply, or divide fractions and mixed numbers with step-by-step LCD work. Free online tool — no sign-up. Runs locally in your browser.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance
Input interpretation
Enter values to calculate.
Result
—
Assurance
Core
Declared partition coverage
PASS · 2/2 declared partitions (fractions, invalid-domain) · Matrix
Known limitations
  • O2 gold: operate (+/−/×/÷) and simplify
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Binary + − × ÷ on fractions/mixed numbers; simplification; fraction↔decimal conversion with reduced answers and steps.
Scope
Integer numerators/denominators (and whole parts) for fraction modes.
Verification
Engine tested · Source checked · v1.0.5 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.5 · CVP protocol 1.0.0-proposed
CVP identity
2/2 property · digest b258a9be030c
Legacy regression
32/32 tests · Production surface contract 3/3
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
1/1 golden · 8/8 CVP boundary · 6/6 invalid · 2/2 property · 1/1 metamorphic · 1/1 round-trip · 1/1 cross-interface · 1/1 CVP contract · Manifest
Sources
Sources
Methods
  • LCD / LCM addition and subtraction
  • Cross-product identity (equivalent to LCD, used as a check)
  • GCF reduction
  • Reciprocal (keep–change–flip) division
Evidence
20 legacy golden · 8 legacy boundary · legacy regression suite · 1/1 oracle-backed golden · 6/6 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.5 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

LCD addition and subtraction match the step-by-step work. Cross-product form is equivalent after reduction.

AddL = LCM(b, d); a/b + c/d = [a(L/b) + c(L/d)] / L
SubtractL = LCM(b, d); a/b − c/d = [a(L/b) − c(L/d)] / L
Multiplya/b × c/d = (ac) / (bd)
Dividea/b ÷ c/d = a/b × d/c
Simplifya/b = (a÷g)/(b÷g) where g = GCF(a,b)
iAdd and subtract use the LCD (L = LCM of the denominators) — the same chain as the step-by-step work. Equivalent cross-product form: (ad ± bc) / bd, which reduces to the same lowest terms. Divide is keep–change–flip (multiply by the reciprocal). Denominator must not be 0. Negative signs are normalized onto the numerator. Results are reduced and also shown as mixed numbers when |numerator| ≥ denominator.

How to use

1

Choose a mode

Operate on two fractions, simplify one fraction, or convert fraction ↔ decimal.

2

Enter numerators and denominators

Optional whole parts build mixed numbers (e.g. 1 and 1/2). Denominators must be nonzero.

3

Read reduced results and steps

See improper fraction, mixed number, decimal, and the LCD / keep-change-flip working.

Example calculations

Common configurations with formula and result.

ϟ

Add with LCD

3/4 + 1/6

LCD = LCM(4, 6) = 12; 9/12 + 2/12
11/12
ϟ

Subtract

3/4 − 1/6

(9−2)/12
7/12
ϟ

Multiply

3/4 × 1/6

3/24
1/8
ϟ

Divide (keep–change–flip)

3/4 ÷ 1/6

3/4 × 6/1
9/2 = 4 1/2
ϟ

Negative fractions

−3/4 + −5/8

LCD 8
−11/8 = −1 3/8

Quick examples

Common values at a glance.

ProblemResult (lowest terms)Mixed
3/4 + 1/611/12—
3/4 − 1/67/12—
3/4 × 1/61/8—
3/4 ÷ 1/69/24 1/2
−3/4 + −5/8−11/8−1 3/8
Simplify 12/182/3—
0.375 → fraction3/8—
3/8 → decimal0.375—
i “Of” in word problems means multiply: 1/3 of 3/8 = 1/3 × 3/8.

Fractions calculator specification

Version 1.0.5 · Engine tested

Calculation status

Review policy · Evidence

Definition
A fraction a/b is a numerator over a nonzero denominator. This calculator adds, subtracts, multiplies, and divides proper/improper fractions and mixed numbers; simplifies to lowest terms; and converts between fractions and decimals.
What it calculates
Binary + − × ÷ on fractions/mixed numbers; simplification; fraction↔decimal conversion with reduced answers and steps.
Inputs
  • Optional whole parts, numerators, denominators
  • Operator (+ − × ÷) in Operate mode
  • Decimal value in From decimal mode
Outputs
  • Reduced improper fraction
  • Mixed number when applicable
  • Decimal approximation
  • Step-by-step working (LCD = LCM, reciprocal, GCF)
Formula
Add/subtract: L = LCM(b,d); [a(L/b) ± c(L/d)] / L (equivalent (ad ± bc)/bd). Multiply ac/bd. Divide a/b × d/c.
Assumptions
  • Integer numerators/denominators (and whole parts) for fraction modes.
  • Results reduced by GCF; signs normalized onto the numerator.
Units
  • Dimensionless
Boundary conditions
  • Denominator 0 → undefined
  • Division by 0/n → undefined
Example
3/4 + 1/6 = 11/12
Validation cases

5 published on this page · 32/32 tests · Production surface contract 3/3 · View evidence

  • 3/4 + 1/6 → 11/12
  • 3/4 ÷ 1/6 → 9/2
  • −3/4 + −5/8 → −11/8
  • 12/18 simplify → 2/3
  • 0.375 → 3/8
Methods
  • LCD / LCM addition and subtraction
  • Cross-product identity (equivalent to LCD, used as a check)
  • GCF reduction
  • Reciprocal (keep–change–flip) division
Sources
  • OpenStax — Prealgebra 2e — 4.5 Add and Subtract Fractions with Different Denominators
    Supports: LCD / LCM method used by this calculator for addition and subtraction
  • OpenStax — Prealgebra 2e — 4.2 Multiply and Divide Fractions
    Supports: GCF simplification, fraction product, and reciprocal (keep–change–flip) division
  • OpenStax — Prealgebra 2e — 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    Supports: Convert mixed numbers to improper fractions, then multiply or divide
Calculation version
1.0.5

Background

Interpretation and common distinctions.

What is a fraction?

A fraction a/b has:

  • numerator a — how many equal parts you have
  • denominator b — how many equal parts make one whole (b ≠ 0)

Examples: 3/8 of a pie is three of eight equal slices. Remaining after eating three of eight is 5/8.

Type Meaning Example
Proper ∣a∣ < b 2/5
Improper ∣a∣ ≥ b 7/3
Mixed whole + proper fraction 2 1/3

Other calculators in this family: Division Calculator, Multiplication Calculator, Scientific Notation Calculator, Subtraction Calculator, Sum Calculator . Explore all Basic Arithmetic.

→ Division Calculator Divide numbers with quotient, remainder, and decimal results for quick arithmetic checks. Runs locally in your browser. Free online tool — no sign-up. › → Multiplication Calculator Multiply two numbers instantly, or see long multiplication with partial products, carrying, decimals, and sign rules. Runs locally in your browser. › → Scientific Notation Calculator Convert numbers to scientific, E, and engineering notation. Free online tool — no sign-up. Mobile-friendly and privacy-first. Runs locally in your browser. › → Subtraction Calculator Subtract minuend − subtrahend to get the difference. Free online tool — no sign-up. Clear formulas and worked examples. Interactive calculations run locally in your browser. › → Sum Calculator Add a list of numbers (sum), see the mean, or work long addition with place-value steps. Paste values separated by commas, spaces, or new lines — decimals and negatives. ›
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Frequently asked questions

Key distinctions behind the calculation.

What is a fraction?

A number that represents equal parts of a whole: numerator (parts you have) over denominator (parts in one whole). The denominator cannot be zero.

How do I add or subtract fractions?

Rewrite with a common denominator (often the LCM of the denominators), then add or subtract numerators and keep that denominator. Reduce the answer.

How do I multiply fractions?

Multiply numerators together and denominators together, then simplify. No common denominator is required.

How do I divide fractions?

Keep the first fraction, change ÷ to ×, and flip the second fraction (use its reciprocal), then multiply and simplify.

What is a mixed number?

A whole number plus a proper fraction, like 1 3/8. Improper fractions (numerator ≥ denominator) convert to mixed form by dividing.

How do I simplify a fraction?

Divide numerator and denominator by their greatest common factor (GCF / HCF).

How do I convert a decimal to a fraction?

Write the decimal over the matching power of 10 (0.375 = 375/1000), then simplify — or use the calculator’s From decimal mode for a reduced fraction.

What does “of” mean with fractions?

In phrases like “1/3 of 3/8,” of means multiply.