Fraction Calculator

Add, subtract, multiply, and divide fractions, improper fractions, and mixed numbers, with results automatically simplified to lowest terms and the method shown for each operation.

Basic Fraction Calculator

Add, subtract, multiply, or divide two fractions.

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Mixed Numbers
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Simplify Fractions
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Decimal to Fraction
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Fraction to Decimal
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Calculation Examples

Calculation Case Result
1/2 + 1/4 LCD = 4: 2/4 + 1/4 = 3/4
3/5 × 2/3 6/15, simplified via GCD(6,15) = 3: 2/5
3/4 ÷ 3/8 3/4 × 8/3 = 24/12 = 2
Convert 0.75 to fraction 75/100, GCD(75,100) = 25: 3/4

How to Use the Fraction Calculator

Enter your fractions using the numerator/denominator fields, or type in the format 1/2. Select the operation (addition, subtraction, multiplication, or division) and click "Calculate." The result is displayed in three forms: as a simplified fraction, as a mixed number where applicable, and as a decimal equivalent.

Each operation uses a distinct method: addition and subtraction find the Least Common Denominator (LCD) first, then adjust numerators before combining; multiplication multiplies numerators and denominators directly, \(\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{ac}{bd}\); division multiplies the first fraction by the reciprocal of the second, \(\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} = \dfrac{ad}{bc}\). The final result is then simplified by dividing both numerator and denominator by their Greatest Common Divisor (GCD).

Fraction diagram showing numerator above the fraction bar and denominator below

Understanding Fractions in Mathematics

A fraction \(\dfrac{a}{b}\) represents the ratio of integer \(a\) (the numerator) to non-zero integer \(b\) (the denominator). In the fraction \(\dfrac{3}{8}\), the numerator 3 indicates three parts and the denominator 8 specifies the total number of equal parts: the fraction represents three eighths of a whole. The denominator can never equal zero, as division by zero is mathematically undefined.

Two key operations underpin fraction arithmetic: the Greatest Common Divisor (GCD), the largest integer that divides both numerator and denominator without remainder, used for simplification; and the Least Common Denominator (LCD), the smallest common multiple of the denominators, required for addition and subtraction. For example, \(\dfrac{12}{18}\) simplifies by GCD(12,18) = 6: \(\dfrac{12 \div 6}{18 \div 6} = \dfrac{2}{3}\). For \(\dfrac{1}{4} + \dfrac{1}{6}\), LCD(4,6) = 12: \(\dfrac{3}{12} + \dfrac{2}{12} = \dfrac{5}{12}\).

Simplifying fractions: dividing numerator and denominator by their GCD to reach lowest terms

Useful Tips 💡

  • For division, apply the Keep-Change-Flip method: keep the first fraction unchanged, change the division sign to multiplication, flip (take the reciprocal of) the second fraction. Example: \(\dfrac{2}{3} \div \dfrac{4}{5}\) becomes \(\dfrac{2}{3} \times \dfrac{5}{4} = \dfrac{10}{12} = \dfrac{5}{6}\).
  • When working with mixed numbers, convert them to improper fractions before performing any operation. For \(2\dfrac{1}{4}\): multiply the whole number by the denominator, add the numerator, \(\dfrac{(2 \times 4) + 1}{4} = \dfrac{9}{4}\), then perform the operation and convert back.

📋Steps to Calculate

  1. Enter the numerator and denominator for each fraction, or type in format 1/2.

  2. Select the arithmetic operation: addition (+), subtraction (−), multiplication (×), or division (÷).

  3. Click "Calculate" to see the result as a simplified fraction, mixed number, and decimal equivalent.

Mistakes to Avoid ⚠️

  1. Adding or subtracting fractions without finding a common denominator first. For example, 1/3 + 1/4 does not equal 2/7. The correct approach: LCD(3,4) = 12, so 4/12 + 3/12 = 7/12.
  2. For division, failing to flip the second fraction (the divisor). a/b ÷ c/d must become a/b × d/c. Dividing by a fraction is the same as multiplying by its reciprocal.
  3. Incorrectly converting mixed numbers to improper fractions. For 3 2/5: multiply the whole number by the denominator and add the numerator: (3 × 5) + 2 = 17, giving 17/5, not (3 × 2)/5 = 6/5.
  4. Not simplifying the final answer to lowest terms. 1/2 + 1/2 = 2/2, which must be simplified to 1; it should not be left as 2/2.

Practical Applications📊

  1. Scale recipes up or down precisely: if a recipe calls for ¾ cup of flour and you need to triple it, multiplying \(\dfrac{3}{4} \times 3 = \dfrac{9}{4} = 2\dfrac{1}{4}\) cups gives the exact measurement without rounding that accumulates across multiple ingredients.

  2. Verify manual fraction work in algebra and pre-calculus: rational expressions involve the same LCD and simplification operations as numerical fractions, and checking intermediate steps prevents sign and denominator errors from compounding.

  3. Calculate accurate proportions in construction and woodworking: material dimensions in imperial units (inches) frequently appear as fractions. Adding \(\dfrac{5}{8}\) inch and \(\dfrac{3}{16}\) inch requires finding LCD(8,16) = 16: \(\dfrac{10}{16} + \dfrac{3}{16} = \dfrac{13}{16}\) inch.

Questions and Answers

What does a fraction calculator do?

A fraction calculator performs the four arithmetic operations, addition, subtraction, multiplication, and division, on proper fractions, improper fractions, and mixed numbers. It applies the correct algorithm for each operation: Least Common Denominator (LCD) for addition and subtraction; direct numerator-by-numerator and denominator-by-denominator multiplication for multiplication; and multiplication by the reciprocal for division. Results are automatically simplified to lowest terms using the Greatest Common Divisor (GCD) and expressed as a simplified fraction, a mixed number, and a decimal equivalent.

How do you add fractions with different denominators?

To add fractions with unlike denominators, find the Least Common Denominator (LCD), the smallest integer divisible by all denominators. Convert each fraction to an equivalent fraction with the LCD as its denominator by multiplying numerator and denominator by the appropriate factor, then add the numerators and retain the LCD as the denominator. Example: \(\dfrac{1}{4} + \dfrac{1}{6}\): LCD(4,6) = 12; \(\dfrac{1}{4} = \dfrac{3}{12}\), \(\dfrac{1}{6} = \dfrac{2}{12}\); sum equals \(\dfrac{5}{12}\). Subtraction follows the same process, substituting subtraction for addition of the numerators.

How do you simplify a fraction?

Simplifying a fraction means reducing it to lowest terms by dividing both numerator and denominator by their Greatest Common Divisor (GCD), the largest integer that divides both without remainder. The GCD can be found using the Euclidean algorithm: repeatedly divide the larger number by the smaller and take the remainder until the remainder is zero; the last non-zero remainder is the GCD. Example: GCD(12, 18): 18 = 1 × 12 + 6; 12 = 2 × 6 + 0; GCD = 6. Therefore \(\dfrac{12}{18} = \dfrac{12 \div 6}{18 \div 6} = \dfrac{2}{3}\). Simplification does not change the value of the fraction.

How do you multiply fractions?

To multiply two fractions, multiply the numerators to get the new numerator and multiply the denominators to get the new denominator: \(\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{a \times c}{b \times d}\). No common denominator is required. Simplify the result to lowest terms. Example: \(\dfrac{3}{5} \times \dfrac{2}{3} = \dfrac{6}{15}\); GCD(6,15)=3; simplified: \(\dfrac{2}{5}\). For efficiency, you can cross-cancel before multiplying: in \(\dfrac{3}{5} \times \dfrac{2}{3}\), the 3 in the numerator and denominator cancel, giving \(\dfrac{1}{5} \times \dfrac{2}{1} = \dfrac{2}{5}\) directly.

What formulas does the fraction calculator use?

The calculator applies four standard algorithms of rational number arithmetic. Addition/Subtraction: \(\dfrac{a}{b} \pm \dfrac{c}{d} = \dfrac{ad \pm bc}{bd}\) (then simplified by GCD). Multiplication: \(\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{ac}{bd}\) (then simplified by GCD). Division: \(\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} = \dfrac{ad}{bc}\) (then simplified by GCD). Simplification uses the Euclidean algorithm to find GCD(numerator, denominator). These are the foundational operations of rational number arithmetic as defined in standard algebra curricula.

How do you convert a decimal to a fraction?

For a terminating decimal, count the number of decimal places, \(n\), and write the decimal digits as the numerator over \(10^n\) as the denominator. Then simplify by GCD. Example: 0.75 has two decimal places, \(\dfrac{75}{100}\); GCD(75,100) = 25; simplified: \(\dfrac{3}{4}\). For a repeating decimal such as \(0.\overline{3}\), let \(x = 0.\overline{3}\); then \(10x = 3.\overline{3}\); subtracting, \(9x = 3\); \(x = \dfrac{3}{9} = \dfrac{1}{3}\).

What is a mixed number and how does the calculator handle it?

A mixed number consists of a whole number and a proper fraction, such as \(2\dfrac{3}{4}\). To perform arithmetic on mixed numbers, the calculator first converts them to improper fractions (where the numerator exceeds the denominator): for \(2\dfrac{3}{4}\), \((2 \times 4) + 3 = 11\), giving \(\dfrac{11}{4}\). The operation is then performed on the improper fractions using the standard algorithms. After simplification, if the result is an improper fraction, it is converted back to a mixed number: \(\dfrac{11}{4} = 2\dfrac{3}{4}\).

Does the calculator handle negative fractions?

Yes. Negative fractions follow the standard rules of signed arithmetic: a negative numerator with a positive denominator (or vice versa) produces a negative fraction; two negatives produce a positive. For example, \(-\dfrac{2}{3} \times -\dfrac{3}{4} = \dfrac{6}{12} = \dfrac{1}{2}\). Negative signs can be placed on the numerator, the denominator, or in front of the fraction; all three representations are mathematically equivalent and produce identical results: \(\dfrac{-a}{b} = \dfrac{a}{-b} = -\dfrac{a}{b}\).

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Disclaimer: This calculator is designed to provide helpful estimates for informational purposes. While we strive for accuracy, financial (or medical) results can vary based on local laws and individual circumstances. We recommend consulting with a professional advisor for critical decisions.