Multiplying the numerator and denominator of a fraction by the same (non-zero) number results in a fraction that is equivalent to the original fraction. This is true because for any non-zero number $n$, the fraction $\frac{n}{n}=1$. Therefore, multiplying by $\frac{n}{n}$ is equivalent to multiplying by one, and any number multiplied by one has the same value as the original number. By way of an example, start with the fraction $\frac{1}{2}$. When the numerator and denominator are both multiplied by 2, the result is $\frac{2}{4}$, which has the same value (0.5) as $\frac{1}{2}$. To picture this visually, imagine cutting a cake into four pieces; two of the pieces together ($\frac{2}{4}$) make up half the cake ($\frac{1}{2}$). Dividing the numerator and denominator of a fraction by the same non-zero number will also yield an equivalent fraction. This is called reducing or simplifying the fraction. A simple fraction in which the numerator and denominator are coprime (that is, the only positive integer that goes into both the numerator and denominator evenly is 1) is said to be irreducible, in lowest terms, or in simplest terms. For example, $\frac{3}{9}$ is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, $\frac{3}{8}$ is in lowest terms—the only positive integer that goes into both 3 and 8 evenly is 1. Using these rules, we can show that: $$\frac{5}{10}=\frac{1}{2}=\frac{10}{20}=\frac{50}{100}$$

A common fraction can be reduced to lowest terms by dividing both the numerator and denominator by their greatest common divisor. For example, as the greatest common divisor of 63 and 462 is 21, the fraction $\frac{63}{462}$ can be reduced to lowest terms by dividing the numerator and denominator by 21: $$\frac{63}{462}=\frac{63÷21}{462÷21}=\frac{3}{22}$$ The Euclidean algorithm gives a method for finding the greatest common divisor of any two positive integers.

- Equivalent fractions, Fractions (Mathematics), Wikipedia as on 22nd July 2017