Equivalents Fractions

Equivalent fractions

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 ${\tfrac {n}{n}}$ equals $1$. Therefore, multiplying by ${\tfrac {n}{n}}$ is the same as 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 ${\tfrac {1}{2}}$. When the numerator and denominator are both multiplied by 2, the result is ${\tfrac {2}{4}}$, which has the same value (0.5) as ${\tfrac {1}{2}}$. To picture this visually, imagine cutting a cake into four pieces; two of the pieces together (${\tfrac {2}{4}}$) make up half the cake (${\tfrac {1}{2}}$).

Simplifying (reducing) fractions

Dividing the numerator and denominator of a fraction by the same non-zero number yields an equivalent fraction: if the numerator and the denominator of a fraction are both divisible by a number (called a factor) greater than 1, then the fraction can be reduced to an equivalent fraction with a smaller numerator and a smaller denominator. For example, if both the numerator and the denominator of the fraction ${\tfrac {a}{b}}$ are divisible by $c,$ then they can be written as $a=cd$ and $b=ce,$ and the fraction becomes ${\tfrac {cd}{ce}}$, which can be reduced by dividing both the numerator and denominator by $c$ to give the reduced fraction ${\tfrac {d}{e}}.$

If one takes for c the greatest common divisor of the numerator and the denominator, one gets the equivalent fraction whose numerator and denominator have the lowest absolute values. One says that the fraction has been reduced to its lowest terms.

If the numerator and the denominator do not share any factor greater than 1, the fraction is already reduced to its lowest terms, and it is said to be irreducible, reduced, or in simplest terms. For example, ${\tfrac {3}{9}}$ is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, ${\tfrac {3}{8}}$ 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 ${\tfrac {5}{10}}={\tfrac {1}{2}}={\tfrac {10}{20}}={\tfrac {50}{100}}$, for example.

As another example, since the greatest common divisor of 63 and 462 is 21, the fraction ${\tfrac {63}{462}}$ can be reduced to lowest terms by dividing the numerator and denominator by 21:

${\tfrac {63}{462}}={\tfrac {63\,\div \,21}{462\,\div \,21}}={\tfrac {3}{22}}$

The Euclidean algorithm gives a method for finding the greatest common divisor of any two integers.

Licensing

Content obtained and/or adapted from:

• Fraction, Wikipedia under a CC BY-SA license