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Fraction Calculator

Fraction arithmetic follows well-defined rules: to add or subtract fractions, find a common denominator; to multiply, multiply across numerators and denominators; to divide, multiply by the reciprocal. All results are automatically reduced to lowest terms using the greatest common divisor (GCD). This calculator also converts fractions to decimals and decimals back to fractions.

Quick Answer

1/3 + 1/4 = 7/12 (approximately 0.583333). To add fractions with different denominators, cross-multiply and add: (1x4 + 1x3) / (3x4) = 7/12.

Fraction Arithmetic

Simplify a Fraction

Decimal to Fraction

Common Examples

Input Result
1/2 + 1/3 5/6 (0.8333)
3/4 - 1/6 7/12 (0.5833)
2/5 x 3/7 6/35 (0.1714)
3/4 / 2/5 15/8 (1.875)
Simplify 24/36 2/3

How It Works

Addition and Subtraction

To add or subtract fractions with different denominators, first find a common denominator. The general formula uses cross-multiplication:

a/b + c/d = (a x d + c x b) / (b x d)

a/b - c/d = (a x d - c x b) / (b x d)

The result is then simplified by dividing both the numerator and denominator by their GCD.

Multiplication

Multiply the numerators together and the denominators together:

a/b x c/d = (a x c) / (b x d)

Division

Multiply the first fraction by the reciprocal (flipped version) of the second:

a/b / c/d = (a x d) / (b x c)

This works because dividing by a fraction is the same as multiplying by its reciprocal.

Simplifying

A fraction is in simplest form when the numerator and denominator share no common factor other than 1. To simplify, divide both by their greatest common divisor (GCD).

Simplified = (n / GCD(n, d)) / (d / GCD(n, d))

Decimal to Fraction

To convert a decimal to a fraction, count the number of decimal places, place the digits over the corresponding power of 10, and simplify. For example, 0.75 = 75/100. GCD(75, 100) = 25, so 75/100 simplifies to 3/4.

Worked Example

To compute 1/3 + 1/4: Common denominator = 3 x 4 = 12. Numerator = (1 x 4) + (1 x 3) = 7. Result = 7/12. GCD(7, 12) = 1, so the fraction is already in simplest form. As a decimal, 7/12 = 0.583333.

To simplify 24/36: GCD(24, 36) = 12. Numerator = 24/12 = 2. Denominator = 36/12 = 3. Simplified fraction = 2/3.

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Frequently Asked Questions

How do I add fractions with different denominators?
Find a common denominator by multiplying the two denominators, then adjust each numerator accordingly. For 1/3 + 1/4, the common denominator is 12. Rewrite as 4/12 + 3/12 = 7/12.
Why do I flip and multiply when dividing fractions?
Dividing by a fraction is equivalent to multiplying by its reciprocal. The reciprocal of c/d is d/c. So (a/b) / (c/d) becomes (a/b) x (d/c). This is a consequence of the definition of division as the inverse of multiplication.
What does it mean to simplify a fraction?
Simplifying (or reducing) a fraction means dividing the numerator and denominator by their greatest common divisor so that the fraction is expressed in the smallest possible whole numbers. For example, 8/12 simplifies to 2/3.
Can fractions have negative numbers?
Yes. A fraction is negative if exactly one of its numerator or denominator is negative. For example, -3/4 and 3/-4 both equal -0.75. By convention, the negative sign is placed on the numerator.
How do I convert a repeating decimal to a fraction?
For a simple repeating decimal like 0.333..., let x = 0.333..., then 10x = 3.333..., so 9x = 3 and x = 3/9 = 1/3. This calculator handles terminating decimals directly. For repeating decimals, enter a close approximation (e.g., 0.333333).

Learn More

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How to Add, Subtract, and Multiply Fractions

Step-by-step instructions for adding, subtracting, multiplying, and dividing fractions. Includes finding common denominators and simplifying results.