section_path: Arithmetic with fractions/Addition/Adding unlike quantities
section_name: Adding unlike quantities
paragraph_text: Adding unlike quantities
  
  To add fractions containing unlike quantities (e.g. quarters and thirds), it is necessary to convert all amounts to like quantities. It is easy to work out the chosen type of fraction to convert to; simply multiply together the two denominators (bottom number) of each fraction. In case of an integer number apply the invisible denominator 1.
  
  For adding quarters to thirds, both types of fraction are converted to twelfths, thus:
  
  Consider adding the following two quantities:
  
  First, convert 3 5 {\displaystyle {\tfrac {3}{5}}} into fifteenths by multiplying both the numerator and denominator by three: 3 5 × 3 3 = 9 15 {\displaystyle {\tfrac {3}{5}}\times {\tfrac {3}{3}}={\tfrac {9}{15}}}. Since 3 / 3 equals 1, multiplication by 3 / 3 does not change the value of the fraction.
  
  Second, convert 2 / 3 into fifteenths by multiplying both the numerator and denominator by five: 2 3 × 5 5 = 10 15 {\displaystyle {\tfrac {2}{3}}\times {\tfrac {5}{5}}={\tfrac {10}{15}}}.
  
  Now it can be seen that:
  
  is equivalent to:
  
  This method can be expressed algebraically:
  
  This algebraic method always works, thereby guaranteeing that the sum of simple fractions is always again a simple fraction. However, if the single denominators contain a common factor, a smaller denominator than the product of these can be used. For example, when adding 3 4 {\displaystyle {\tfrac {3}{4}}} and 5 6 {\displaystyle {\tfrac {5}{6}}} the single denominators have a common factor 2, and therefore, instead of the denominator 24 (4 × 6), the halved denominator 12 may be used, not only reducing the denominator in the result, but also the factors in the numerator.
  
  The smallest possible denominator is given by the least common multiple of the single denominators, which results from dividing the rote multiple by all common factors of the single denominators. This is called the least common denominator.