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½ Fraction Calculator

Add, subtract, multiply, and divide fractions; simplify, convert to decimals, and compare rational numbers.

Fraction Arithmetic With Simplification

BrainyCalculators editorial insight — unique to this tool

Adding ⅓ + ¼ requires LCD 12 → 7/12. Recipe scaling and construction fractions (⅝" plywood) need reliable LCD math. Converts to decimal for comparison — ⅔ ≈ 0.667 vs 0.66 rounded.

When to use this calculator

Use for fraction + − × ÷ and simplification. For decimal conversion only, use Fraction to Decimal (if available) or enter here.

Percent change or “20% of 150” style problems?

This page works with fractions. For general percent math, use the Percentage Calculator →

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What is a Fraction Calculator?

A fraction calculator operates on rational numbers: simplify, convert to mixed numbers, and perform arithmetic with common denominators shown step by step.

Use this page when inputs are true fractions or ratios. For percent-of problems and percent change, use the Percentage Calculator.

Markup and discount pages focus on retail price language rather than pure rational arithmetic.

Fraction Operation Formulas

Add: a/b + c/d = (a×d + b×c) / (b×d)
Subtract: a/b − c/d = (a×d − b×c) / (b×d)
Multiply: a/b × c/d = (a×c) / (b×d)
Divide: a/b ÷ c/d = (a×d) / (b×c)

How to Use

  1. 1
    Select Operation
    Click Add, Subtract, Multiply, or Divide at the top.
  2. 2
    Enter Fractions
    Type the numerator and denominator for each fraction. Negative numerators are allowed.
  3. 3
    Calculate
    Press Calculate or simply type — results update automatically.
  4. 4
    Read the Result
    See the simplified fraction, mixed number (if applicable), decimal, and full step-by-step workings.

Example — Adding Fractions

1/3 + 1/4 = ?

Step 1: Cross-multiply numerators: (1×4) + (3×1) = 4 + 3 = 7
Step 2: Multiply denominators: 3 × 4 = 12
Step 3: Result = 7/12
Step 4: GCD(7, 12) = 1 → already simplified
Decimal = 0.5833

How the Fraction Calculator Works

Formula, assumptions, and calculation steps for this math tool.

Formula Used

a/b + c/d = (a*d + c*b) / (b*d), then simplify by GCF

Methodology

Finds a common denominator for add/subtract, cross-multiplies for compare, or multiplies numerators and denominators directly for multiply/divide, then reduces by the greatest common factor.

Calculation Steps

  1. Read the values and operation selected.
  2. Normalize signs, decimals, fractions, or units if needed.
  3. Apply the mathematical rule or formula.
  4. Format the answer and any intermediate values for checking.

Assumptions and Limits

  • Inputs must be within the supported domain of the operation.
  • Decimal answers may be rounded for readability.
  • Symbolic simplification is limited to the calculator scope.

Frequently Asked Questions

Use the formula a/b + c/d = (a×d + b×c) / (b×d). This finds a common denominator and adds the numerators. Then simplify using the GCD.

Divide both the numerator and denominator by their Greatest Common Divisor (GCD). For example, 6/8 → GCD(6,8)=2 → 3/4.

A mixed number combines a whole number with a proper fraction, e.g. 7/4 = 1¾. An improper fraction (numerator ≥ denominator) can always be converted to a mixed number.

Simply divide the numerator by the denominator. For example, 3/4 = 3 ÷ 4 = 0.75.

Real-World Applications

🍳
Recipe Scaling
Add 3/4 cup + 1/3 cup of flour when doubling a recipe — fraction addition gives an exact measurement without needing a decimal conversion.
📐
Carpentry Measurements
Add 5/8" + 3/16" when calculating combined material thickness for framing and cladding — standard lumber measurements use fractions.
📊
Probability Calculations
Combine event probabilities: if P(A) = 1/6 and P(B) = 1/4, add or multiply these fractions to find compound probability.
💊
Medication Dosing
Divide a fractional tablet dose across multiple days or combine partial doses for paediatric calculations.
🏗️
Engineering Ratios
Gear ratios, mix ratios (concrete, fuel), and flow ratios are naturally expressed and calculated as fractions.
🎓
Algebra & Pre-Calculus
Simplify rational expressions, combine fractional coefficients, and solve equations with fractional terms.

Common Mistakes

1
Adding numerators AND denominators
1/2 + 1/3 ≠ 2/5. Fractions require a common denominator before adding numerators — the correct answer is 3/6 + 2/6 = 5/6.
2
Not converting mixed numbers to improper fractions
2½ + 1¾ must be converted to 5/2 + 7/4 before adding — operating directly on the whole and fractional parts separately leads to errors.
3
Cancelling incorrectly across addition
(a + b)/c ≠ a + b/c — the entire numerator is divided by the denominator, not just the last term. Always use parentheses when simplifying.
4
Not simplifying the final result
A result of 6/8 should always be simplified to 3/4 by dividing both numerator and denominator by their GCD (2 in this case).
5
Inverting the wrong fraction when dividing
a/b ÷ c/d = a/b × d/c — you invert and multiply the SECOND fraction (divisor), not the first. Inverting the wrong one gives a reciprocal result.

Fraction Operations Quick Reference

Operation Formula Example
Addition a/b + c/d = (ad+bc)/bd 1/3 + 1/4 = 7/12
Subtraction a/b − c/d = (ad−bc)/bd 3/4 − 1/3 = 5/12
Multiplication a/b × c/d = ac/bd 2/3 × 3/5 = 6/15 = 2/5
Division a/b ÷ c/d = ad/bc 2/3 ÷ 4/5 = 10/12 = 5/6
Simplification Divide by GCD(a,b) 12/18 ÷ 6/6 = 2/3
Mixed to Improper (n×d + num) / d 2¾ = (2×4+3)/4 = 11/4

References

  1. National Council of Teachers of Mathematics. Developing Essential Understanding of Fractions. NCTM, 2013.
  2. Van de Walle, John A. et al. Elementary and Middle School Mathematics. Pearson, 2019.
  3. Wu, Hung-Hsi. Understanding Numbers in Elementary School Mathematics. AMS, 2011.
  4. Larson, Ron and Boswell, Laurie. Big Ideas Math: Algebra 1. Big Ideas Learning, 2022.
  5. Common Core State Standards Initiative. Mathematics Standards: Number and Operations — Fractions. NGA/CCSSO, 2010.

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