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Fraction Calculator

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Result

5/6

Decimal
0.8333333333
Equation
1/2 + 1/3

Estimates for educational purposes — not financial, medical, or legal advice. See terms.

Fraction arithmetic follows consistent rules regardless of the numbers involved. This calculator applies those rules and shows the simplified result along with its decimal and mixed-number equivalents.

How fraction arithmetic works

Addition and subtraction: finding a common denominator

Two fractions can only be added or subtracted directly when they share a denominator. To add fractions with different denominators, convert both to an equivalent form that uses the same denominator.

The most reliable method is to multiply the two denominators together to form the common denominator:

ab+cd=ad+bcbd\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

For example, adding 1/3 and 1/6:

13+16=1⋅6+1⋅33⋅6=918=12\frac{1}{3} + \frac{1}{6} = \frac{1 \cdot 6 + 1 \cdot 3}{3 \cdot 6} = \frac{9}{18} = \frac{1}{2}

Subtraction follows the same pattern:

ab−cd=ad−bcbd\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}

Multiplication: numerator times numerator, denominator times denominator

Multiplication does not require a common denominator:

ab×cd=acbd\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}

For example: 2/5 × 5/8 = 10/40 = 1/4. You can simplify either before or after multiplying.

Division: multiply by the reciprocal

Dividing by a fraction is equivalent to multiplying by its reciprocal (the fraction flipped):

ab÷cd=ab×dc=adbc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}

For example: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6.

Simplification via GCD

After each operation, the result is reduced to lowest terms using the greatest common divisor (GCD). The GCD is the largest integer that divides both the numerator and denominator without a remainder. For raw GCD or LCM of an integer list, the GCD / LCM calculator is the dedicated tool.

The Euclidean algorithm finds the GCD efficiently:

gcd⁡(a,b)=gcd⁡(b, a mod b)until b=0\gcd(a, b) = \gcd(b,\ a \bmod b) \quad \text{until } b = 0

For example, gcd(12, 8): 12 mod 8 = 4, then gcd(8, 4): 8 mod 4 = 0, so gcd = 4. Thus 12/8 simplifies to 3/2.

Example: recipe scaling

A recipe calls for 3/4 cup of sugar, but you want to make 2/3 of the recipe.

How much sugar do you need?

34×23=612=12 cup\frac{3}{4} \times \frac{2}{3} = \frac{6}{12} = \frac{1}{2} \text{ cup}

You used 1/4 cup — how much is left from the original 3/4?

34−14=24=12 cup\frac{3}{4} - \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \text{ cup}

Mixed numbers

An improper fraction (where the numerator is larger than the denominator) can be expressed as a mixed number: a whole part and a proper fraction. For 7/2, dividing 7 by 2 gives 3 with remainder 1, so the mixed form is 3 1/2.

The calculator shows the result in all three forms: simplified fraction, decimal, and mixed number when the result is greater than 1. If you want the result as a percentage, the percentage calculator takes it from here; for ratio form, the ratio calculator handles simplification and proportion solving.

Frequently asked questions

How do you add fractions with different denominators?

Find a common denominator by multiplying the two denominators together (or using their least common multiple). Convert each fraction to the common denominator, then add the numerators. Finally, simplify by dividing numerator and denominator by their greatest common divisor. For example: 1/3 + 1/6. Common denominator is 6. Convert: 2/6 + 1/6 = 3/6 = 1/2.

How do you multiply fractions?

Multiply the numerators together and the denominators together, then simplify. For example: 2/3 × 3/4 = (2×3)/(3×4) = 6/12 = 1/2. Unlike addition, you do not need a common denominator when multiplying.

How do you divide fractions?

Multiply the first fraction by the reciprocal of the second (flip the second fraction, then multiply). For example: 3/4 ÷ 3/8 = 3/4 × 8/3 = 24/12 = 2. This works because dividing by a fraction is the same as multiplying by its inverse.

What is a mixed number?

A mixed number combines a whole number and a proper fraction, like 3 1/2 (three and one-half). It is equivalent to the improper fraction 7/2. Mixed numbers are often easier to read, while improper fractions are easier to use in arithmetic.

How do you simplify a fraction?

Divide both the numerator and denominator by their greatest common divisor (GCD). For 12/18, the GCD is 6, so 12/18 = 2/3. A fraction is fully simplified when the only common factor of its numerator and denominator is 1.