Fraction Calculator

Add, subtract, multiply, and divide fractions and mixed numbers, with every result simplified to lowest terms and the method shown.

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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 by GCD(6, 15) = 3: 2/5
3/4 ÷ 3/8 3/4 × 8/3 = 24/12 = 2
Convert 0.75 to a fraction 75/100, simplified by GCD(75, 100) = 25: 3/4
Mihail Doncenco
Mihail Doncenco Site Creator & Calculator Expert

Founder and developer of CalcMate. Builds and maintains every calculator on this site, focusing on formula accuracy and clear, practical results.

How to Use the Fraction Calculator

Enter each fraction in the numerator and denominator fields, or type it as 1/2. Select the operation (addition, subtraction, multiplication, or division) and click "Calculate." The result appears in three forms: a simplified fraction, a mixed number where applicable, and a decimal.

Each operation uses its own method. Addition and subtraction first find the Least Common Denominator (LCD), then adjust the numerators before combining them. Multiplication multiplies numerators and denominators directly: \(\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{ac}{bd}\). Division multiplies by the reciprocal of the second fraction: \(\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} = \dfrac{ad}{bc}\). The last step divides numerator and denominator by their Greatest Common Divisor (GCD) to reach lowest terms.

Example: \(\dfrac{2}{3} + \dfrac{3}{4} = \dfrac{8}{12} + \dfrac{9}{12} = \dfrac{17}{12} = 1\dfrac{5}{12} \approx 1.4167\).

Fraction diagram showing numerator above the fraction bar and denominator below

Understanding Fractions in Mathematics

A fraction \(\dfrac{a}{b}\) is the ratio of an integer \(a\) (the numerator) to a non-zero integer \(b\) (the denominator). In \(\dfrac{3}{8}\), the denominator says the whole is cut into 8 equal parts and the numerator counts 3 of them. The denominator can never be zero, because division by zero is undefined. Fractions matter well beyond arithmetic: the U.S. National Mathematics Advisory Panel (2008) named fluency with fractions a critical foundation for algebra, and Siegler et al. (2012, Psychological Science) found that fraction knowledge at age 10 predicts math achievement in high school.

Two operations drive fraction arithmetic. The Greatest Common Divisor (GCD) is the largest integer that divides both numerator and denominator without remainder, and it handles simplification. The Least Common Denominator (LCD) is the smallest common multiple of the denominators, and it is required for addition and subtraction. Example 1: \(\dfrac{12}{18}\) simplifies by \(\gcd(12, 18) = 6\), giving \(\dfrac{12 \div 6}{18 \div 6} = \dfrac{2}{3}\). Example 2: for \(\dfrac{1}{4} + \dfrac{1}{6}\), the LCD of 4 and 6 is 12, so \(\dfrac{3}{12} + \dfrac{2}{12} = \dfrac{5}{12}\).

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

Useful Tips 💡

  • For division, use Keep-Change-Flip: keep the first fraction, change ÷ to ×, and flip the second fraction. Example: 2/3 ÷ 4/5 becomes 2/3 × 5/4 = 10/12 = 5/6.
  • Convert mixed numbers to improper fractions before any operation. For 2 1/4, multiply the whole number by the denominator and add the numerator: (2 × 4) + 1 = 9, giving 9/4. Convert back at the end.
  • Sanity-check results by estimating with decimals. 7/8 + 5/6 is about 0.875 + 0.833 = 1.708, which matches the exact answer 41/24 = 1 17/24.

📋Steps to Calculate

  1. Enter the numerator and denominator for each fraction, or type it as 1/2.

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

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

Mistakes to Avoid ⚠️

  1. Adding or subtracting fractions without a common denominator. 1/3 + 1/4 does not equal 2/7. The LCD of 3 and 4 is 12, so 4/12 + 3/12 = 7/12.
  2. Forgetting to flip the divisor in division. a/b ÷ c/d must become a/b × d/c, because dividing by a fraction is the same as multiplying by its reciprocal.
  3. Converting mixed numbers incorrectly. For 3 2/5, the numerator is (3 × 5) + 2 = 17, giving 17/5. Multiplying the whole number by the old numerator, (3 × 2)/5 = 6/5, is wrong.
  4. Leaving the answer unsimplified. 1/2 + 1/2 = 2/2, which must be reduced to 1.

Practical Applications📊

  1. Scale recipes exactly: tripling 3/4 cup of flour gives 3/4 × 3 = 9/4 = 2 1/4 cups. Working in fractions avoids the rounding errors that pile up when you convert every ingredient to decimals.

  2. Check algebra and pre-calculus homework: rational expressions use the same LCD and simplification steps as numeric fractions, so verifying a numeric version catches sign and denominator errors early.

  3. Add measurements in construction and woodworking: imperial dimensions are fractions of an inch. Adding 5/8 inch and 3/16 inch needs LCD(8, 16) = 16, so 10/16 + 3/16 = 13/16 inch.

  4. Combine probabilities: the chance of rolling two sixes in a row is 1/6 × 1/6 = 1/36, about 2.78%.

Questions and Answers

Does the calculator handle negative fractions?

Yes, negative fractions follow the standard rules of signed arithmetic: one negative sign makes the result negative, and two negative signs make it positive. Example: \(\left(-\dfrac{2}{3}\right) \times \left(-\dfrac{3}{4}\right) = \dfrac{6}{12} = \dfrac{1}{2}\). The minus sign can sit on the numerator, on the denominator, or in front of the fraction, and all three forms are equal: \(\dfrac{-a}{b} = \dfrac{a}{-b} = -\dfrac{a}{b}\).

How do you add fractions with different denominators?

To add fractions with different denominators, convert both to the Least Common Denominator (LCD), then add the numerators and keep the LCD. The LCD is the smallest number divisible by every denominator. Example: for \(\dfrac{1}{4} + \dfrac{1}{6}\), the LCD is 12, so \(\dfrac{3}{12} + \dfrac{2}{12} = \dfrac{5}{12}\). Subtraction works the same way, with the numerators subtracted. A shortcut that skips the LCD search is \(\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad + bc}{bd}\), but it often needs extra simplification: \(\dfrac{1}{4} + \dfrac{1}{6} = \dfrac{10}{24} = \dfrac{5}{12}\).

How do you convert a decimal to a fraction?

To convert a terminating decimal to a fraction, write the digits over \(10^n\), where \(n\) is the number of decimal places, then simplify by the GCD. Example: 0.75 has two decimal places, so it equals \(\dfrac{75}{100}\), and since \(\gcd(75, 100) = 25\), the result is \(\dfrac{3}{4}\). Likewise \(0.125 = \dfrac{125}{1000} = \dfrac{1}{8}\). For a repeating decimal such as \(0.\overline{3}\), let \(x = 0.\overline{3}\). Then \(10x = 3.\overline{3}\), subtracting gives \(9x = 3\), and \(x = \dfrac{3}{9} = \dfrac{1}{3}\).

How do you multiply fractions?

To multiply fractions, multiply the numerators together and the denominators together: \(\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{ac}{bd}\). You do not need a common denominator. Example: \(\dfrac{3}{5} \times \dfrac{2}{3} = \dfrac{6}{15}\), and since \(\gcd(6, 15) = 3\), the simplified result is \(\dfrac{2}{5}\). To save a step, cross-cancel first: the 3 in the numerator and the 3 in the denominator cancel, leaving \(\dfrac{1}{5} \times \dfrac{2}{1} = \dfrac{2}{5}\).

How do you simplify a fraction?

To simplify a fraction, divide the numerator and denominator by their Greatest Common Divisor (GCD), the largest integer that divides both without remainder. Find the GCD with the Euclidean algorithm (Euclid, Elements, Book VII): divide the larger number by the smaller, then repeat with the smaller number and the remainder until the remainder is 0. The last non-zero remainder is the GCD. Example: \(18 = 1 \times 12 + 6\) and \(12 = 2 \times 6 + 0\), so \(\gcd(12, 18) = 6\) and \(\dfrac{12}{18} = \dfrac{12 \div 6}{18 \div 6} = \dfrac{2}{3}\). Simplifying never changes the value of the fraction.

What does a fraction calculator do?

A fraction calculator adds, subtracts, multiplies, and divides proper fractions, improper fractions, and mixed numbers, then returns the result in lowest terms. It uses the Least Common Denominator (LCD) for addition and subtraction, multiplies numerators and denominators directly for multiplication, and multiplies by the reciprocal for division. Every result is simplified with the Greatest Common Divisor (GCD) and shown as a fraction, a mixed number, and a decimal. Example: \(\dfrac{3}{4} \div \dfrac{3}{8} = \dfrac{3}{4} \times \dfrac{8}{3} = \dfrac{24}{12} = 2\).

What formulas does the fraction calculator use?

The calculator uses the four standard algorithms of rational number arithmetic, and simplifies every result by the GCD. Addition and subtraction: \(\dfrac{a}{b} \pm \dfrac{c}{d} = \dfrac{ad \pm bc}{bd}\). Multiplication: \(\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{ac}{bd}\). Division: \(\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} = \dfrac{ad}{bc}\). The GCD comes from the Euclidean algorithm. These are the foundational operations taught in standard algebra curricula.

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

A mixed number is a whole number plus a proper fraction, such as \(2\dfrac{3}{4}\). The calculator first converts mixed numbers to improper fractions, where the numerator is larger than the denominator: for \(2\dfrac{3}{4}\), \((2 \times 4) + 3 = 11\), giving \(\dfrac{11}{4}\). It then runs the operation on the improper fractions and simplifies. If the result is an improper fraction, it converts back to a mixed number: \(\dfrac{11}{4} = 2\dfrac{3}{4}\).
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.