section_path: Arithmetic with fractions/Equivalent fractions/Simplifying (reducing) fractions
section_name: Simplifying (reducing) fractions
paragraph_text: 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 a b {\displaystyle {\tfrac {a}{b}}} are divisible by c {\displaystyle c}, then they can be written as a = c d {\displaystyle a=cd}, b = c e {\displaystyle b=ce}, and the fraction becomes cd / ce, which can be reduced by dividing both the numerator and denominator by c to give the reduced fraction 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, 3 9 {\displaystyle {\tfrac {3}{9}}} is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, 3 8 {\displaystyle {\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 5 / 10 = 1 / 2 = 10 / 20 = 50 / 100, for example.
  
  As another example, since the greatest common divisor of 63 and 462 is 21, the fraction 63 / 462 can be reduced to lowest terms by dividing the numerator and denominator by 21:
  
  The Euclidean algorithm gives a method for finding the greatest common divisor of any two integers.