Fraction Calculator

Add, subtract, multiply, or divide fractions, mixed numbers, and integers with detailed visual pies and step-by-step math.

Math Audited
Practice Guessing GameChallenge yourself to estimate the simplified answer before it reveals!
Fraction 1
Whole: 0
Numerator: 1
Denominator: 2
Portion (1/2)
Fraction 2
Whole: 0
Numerator: 1
Denominator: 3
Portion (1/3)
Result (Simplified Form)
5
6

The calculation simplifies to the proper fraction 5/6.

Result FormatValue
Improper Fraction5 / 6
Mixed Fraction5/6
Decimal Value0.8333
Result Circle Representation
Portion (5/6)
How is this calculated? (Substituted Steps)
1. Convert mixed numbers to improper fractions:
   Fraction 1: 1/2 = (1 / 2)
   Fraction 2: 1/3 = (1 / 3)
2. Perform operation (+):
(1 * 3 + 1 * 2) / (2 * 3) = (3 + 2) / 6 = 5 / 6
3. Simplify via Greatest Common Divisor (GCD):
   GCD of numerator and denominator is: 1
   Simplified improper fraction: 5 / 6
Mathematical Verification LogChecked and validated against NCTM (National Council of Teachers of Mathematics) guidelines and CCSSM (Common Core State Standards for Mathematics) standards.Last Audited: 2026-07-11
Mathematical Formulas
Addition: \frac{a_1}{b_1} + \frac{a_2}{b_2} = \frac{a_1 b_2 + a_2 b_1}{b_1 b_2}
Subtraction: \frac{a_1}{b_1} - \frac{a_2}{b_2} = \frac{a_1 b_2 - a_2 b_1}{b_1 b_2}
Multiplication: \frac{a_1}{b_1} \times \frac{a_2}{b_2} = \frac{a_1 a_2}{b_1 b_2}
Division: \frac{a_1}{b_1} \div \frac{a_2}{b_2} = \frac{a_1 b_2}{b_1 a_2}
Core Assumptions
  • Fractions are computed using standard exact integer arithmetic to avoid floating point drift before conversion.
  • Mixed fractions are converted to improper forms for operations and simplified back.
Limitations & Exclusions
  • Dividing by zero is mathematically undefined.
  • Does not support irrational numbers, symbolic variables, or infinite repeating decimals.

About the Fraction Calculator

A fraction represents a part of a whole, mathematically expressed as a quotient where a numerator is divided by a non-zero denominator. Fractions serve as the primary representation of rational numbers, which are essential for precision in fields like carpentry, culinary arts, engineering, and music theory. Understanding fraction arithmetic involves distinct rules for each basic operator: addition and subtraction require the computation of a Least Common Denominator (LCD) to align scale divisions; multiplication involves direct horizontal multiplication of numerators and denominators; and division utilizes the "multiply-by-reciprocal" algorithm. Fractions can be classified as proper (numerator smaller than denominator), improper (numerator equal to or larger than denominator), or mixed numbers (combining a whole number with a proper fraction). Modern scientific calculators simplify results by computing the Greatest Common Divisor (GCD) to reduce fractions to their irreducible lowest terms and providing decimal approximations.

Mathematical Formula & Logic

Fraction operations are governed by the following mathematical equations, where a/b and c/d represent two rational terms (b, d ≠ 0): 1. Addition (Common Denominator): a/b + c/d = (a·d + c·b) / (b·d) 2. Subtraction: a/b - c/d = (a·d - c·b) / (b·d) 3. Multiplication: a/b × c/d = (a·c) / (b·d) 4. Division: a/b ÷ c/d = a/b × d/c = (a·d) / (b·c) (where c ≠ 0) 5. Simplification: Reduced_numerator = Numerator / GCD(Numerator, Denominator) Reduced_denominator = Denominator / GCD(Numerator, Denominator) 6. Mixed Number Conversion (for improper fraction N/D where N ≥ D): Whole_part = floor(N / D) Remainder_numerator = N mod D Mixed_form = Whole_part + (Remainder_numerator / D)

Step-by-Step Example

Calculate 5/6 - 2/9 and simplify the result to its lowest terms: 1. Find the Least Common Denominator (LCD) of 6 and 9: - Multiples of 6: 6, 12, 18, 24, 30... - Multiples of 9: 9, 18, 27... - LCD = 18 2. Convert fractions to equivalent forms with denominator 18: - 5/6 = (5 × 3) / (6 × 3) = 15/18 - 2/9 = (2 × 2) / (9 × 2) = 4/18 3. Perform subtraction: 15/18 - 4/18 = (15 - 4) / 18 = 11/18 4. Verify if simplification is possible: GCD(11, 18) = 1. Since the GCD is 1, the fraction 11/18 is already in its simplest form. 5. Compute decimal approximation: 11 / 18 ≈ 0.6111.

Reference Data & Values

operationinputraw resultsimplifieddecimal
Addition1/2 + 1/35/65/60.8333
Subtraction3/4 - 1/820/325/80.6250
Multiplication2/3 × 3/56/152/50.4000
Division4/5 ÷ 2/312/106/5 (1 1/5)1.2000

Frequently Asked Questions

In mathematics, division by zero is undefined. A fraction represents dividing a quantity into a specific number of equal parts; it is logically impossible to divide an object into zero parts. Attempting to calculate a fraction with a denominator of zero yields an undefined or infinity error in computational algorithms.
An improper fraction represents a value greater than or equal to one where the numerator is larger than or equal to the denominator (e.g., 7/4). A mixed number expresses the exact same value using a whole number combined with a proper fraction (e.g., 1 3/4). Both forms are mathematically identical, but mixed numbers are often preferred for readability.
The GCD is the largest positive integer that divides both numbers without leaving a remainder. In calculators, this is computed efficiently using the Euclidean algorithm, which repeatedly replaces the larger number with the remainder of the division of the two numbers until the remainder is zero (e.g., GCD of 12 and 18: 18 mod 12 = 6, then 12 mod 6 = 0, so the GCD is 6).