Last updated: 2026-05-25

Understanding fraction arithmetic

Fraction arithmetic is the set of rules for adding, subtracting, multiplying, and dividing numbers written as a ratio of two integers, $\frac{a}{b}$, where $b \neq 0$. Each operation produces another fraction, which is then reduced to lowest terms by dividing the numerator and denominator by their greatest common divisor (GCD). This calculator applies those rules to two values and returns the exact reduced result.

Addition and subtraction

Fractions can only be added or subtracted when they share a denominator. The standard approach rewrites both over a common denominator — typically the least common denominator (LCD) — then combines the numerators:

The numerator and denominator of the result are then divided by their GCD to reduce it to lowest terms.

As an example, take $\dfrac{1}{4} + \dfrac{1}{3}$:

Multiplication

Multiplication needs no common denominator. The numerators multiply together and the denominators multiply together, after which the product is reduced:

For $\dfrac{2}{3} \times \dfrac{3}{4} = \dfrac{6}{12} = \dfrac{1}{2}$.

Division

Division uses the reciprocal: dividing by a fraction is the same as multiplying by its inverse. Inverting the second fraction and multiplying gives:

For $\dfrac{3}{4} \div \dfrac{1}{2} = \dfrac{3}{4} \times \dfrac{2}{1} = \dfrac{6}{4} = \dfrac{3}{2}$.

Worked examples

Mixed numbers

A mixed number such as 1 3/4 (one and three-quarters, equal to $\frac{7}{4}$) is entered as whole numerator/denominator. The calculator converts the mixed number to an improper fraction before computing, then converts the result back to mixed form when its absolute value exceeds 1.

Fractions versus decimals

Fractions preserve exact values that decimals cannot express. For example, $\frac{1}{3} = 0.\overline{3}$ is a repeating decimal that loses accuracy if it is truncated. In contexts such as cooking, woodworking, and music, the exact ratio matters, and a result of $\frac{7}{8}$ carries more information than its decimal approximation 0.875.

Using the calculator

Enter the first value as a decimal (0.5), a simple fraction (1/2), or a mixed number (1 3/4), select the operation, and enter the second value in the same format. The result field shows the exact reduced fraction.