Fraction Calculator
Add, subtract, multiply or divide two fractions.
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Working With Fractions
A fraction represents part of a whole and is written as a numerator over a denominator, such as 3/4. The numerator counts how many parts you have, and the denominator tells how many equal parts make up the whole. Calculating with fractions follows four core operations: addition, subtraction, multiplication, and division.
Addition and subtraction require the fractions to share a common denominator before you can combine them. Multiplication and division do not, which makes them mechanically simpler. In every case, the final answer should be reduced to lowest terms so it is expressed as cleanly as possible.
This calculator performs any of the four operations on two fractions, automatically finds a common denominator when needed, and simplifies the result. Below are the formulas and worked examples for each operation.
Adding and Subtracting Fractions
To add or subtract, the denominators must match. The general formula using cross-multiplication works for any two fractions:
a/b + c/d = (a×d + c×b) / (b×d)
a/b − c/d = (a×d − c×b) / (b×d)
This rewrites both fractions over the common denominator b×d, combines the numerators, and leaves the result ready to simplify. If the denominators are already the same, you simply add or subtract the numerators and keep the shared denominator.
For tidier numbers you can use the least common denominator (LCD) instead of the product b×d, but the cross-multiplication method always works and gives an equivalent answer once reduced. After combining, always check whether the result can be simplified.
Multiplying and Dividing Fractions
Multiplication is the most direct operation: multiply the numerators together and the denominators together. No common denominator is needed.
a/b × c/d = (a×c) / (b×d)
Division is handled by multiplying the first fraction by the reciprocal of the second — that is, flip the second fraction and then multiply. This is the well-known "keep, change, flip" rule:
a/b ÷ c/d = a/b × d/c = (a×d) / (b×c)
The reciprocal works because dividing by a number is the same as multiplying by its inverse. As with the other operations, reduce the final fraction to lowest terms once you have the result. A common shortcut is to cancel any common factor between a numerator and a denominator before multiplying, which keeps the numbers small and often removes the need to simplify at the end.
Simplifying and Worked Examples
To simplify a fraction, divide the numerator and denominator by their greatest common divisor (GCD). For example, 8/12 has a GCD of 4, so it reduces to 2/3. A fraction is in lowest terms when the numerator and denominator share no common factor other than 1. Reducing does not change the value of the fraction; it only writes that value in its simplest, most readable form, which is why every worked example below ends with a simplification step.
Addition: 1/2 + 1/3 = (1×3 + 1×2)/(2×3) = (3 + 2)/6 = 5/6. Already in lowest terms.
Subtraction: 3/4 − 1/6 = (3×6 − 1×4)/(4×6) = (18 − 4)/24 = 14/24, which reduces by 2 to 7/12.
Multiplication: 2/3 × 3/5 = (2×3)/(3×5) = 6/15, which reduces by 3 to 2/5.
Division: 3/4 ÷ 2/5 = 3/4 × 5/2 = (3×5)/(4×2) = 15/8, or 1 7/8 as a mixed number.
Frequently Asked Questions
No. A common denominator is only required for addition and subtraction. To multiply, you simply multiply the numerators together and the denominators together. For example, 2/3 × 4/5 = 8/15, with no common denominator needed.
Multiply the first fraction by the reciprocal of the second, meaning you flip the second fraction upside down and then multiply. This is the "keep, change, flip" rule. For example, 1/2 ÷ 3/4 = 1/2 × 4/3 = 4/6, which simplifies to 2/3.
Divide both the numerator and the denominator by their greatest common divisor (GCD). For example, 12/18 has a GCD of 6, so dividing both by 6 gives 2/3. The fraction is fully reduced when the top and bottom share no common factor besides 1.
The reciprocal of a fraction is that fraction turned upside down, swapping the numerator and denominator. The reciprocal of 3/4 is 4/3. Multiplying a fraction by its reciprocal always gives 1, which is why reciprocals are used to divide fractions.
Not directly. You first need to rewrite them over a common denominator. The cross-multiplication formula a/b + c/d = (ad + cb)/(bd) handles this automatically, after which you add the numerators and simplify the result.
Divide the numerator by the denominator. The whole-number quotient is the integer part, and the remainder over the original denominator is the fractional part. For example, 15/8 gives 1 with a remainder of 7, so it equals the mixed number 1 7/8.