section_path: Forms of fractions/Historical notions/Complex and compound fractions
section_name: Complex and compound fractions
paragraph_text: Complex and compound fractions
  
  In a complex fraction, either the numerator, or the denominator, or both, is a fraction or a mixed number, corresponding to division of fractions. For example, 1 / 2 1 / 3 {\displaystyle {\tfrac {1/2}{1/3}}} and (12 3 4) / 26 {\displaystyle {\bigl (}12{\tfrac {3}{4}}{\bigr)}{\big /}26} are complex fractions. To interpret nested fractions written "stacked" with a horizontal fraction bars, treat shorter bars as nested inside longer bars. Complex fractions can be simplified using multiplication by the reciprocal, as described below at § Division. For example:
  
  A complex fraction should never be written without an obvious marker showing which fraction is nested inside the other, as such expressions are ambiguous. For example, the expression 5 / 10 / 20 {\displaystyle 5/10/20} could be plausibly interpreted as either 5 10 / 20 = 1 40 {\displaystyle {\tfrac {5}{10}}{\big /}20={\tfrac {1}{40}}} or as 5 / 10 20 = 10. {\displaystyle 5{\big /}{\tfrac {10}{20}}=10.} The meaning can be made explicit by writing the fractions using distinct separators or by adding explicit parentheses, in this instance (5 / 10) / 20 {\displaystyle (5/10){\big /}20} or 5 / (10 / 20). {\displaystyle 5{\big /}(10/20).}
  
  A compound fraction is a fraction of a fraction, or any number of fractions connected with the word of, corresponding to multiplication of fractions. To reduce a compound fraction to a simple fraction, just carry out the multiplication (see § Multiplication). For example, 3 4 {\displaystyle {\tfrac {3}{4}}} of 5 7 {\displaystyle {\tfrac {5}{7}}} is a compound fraction, corresponding to 3 4 × 5 7 = 15 28 {\displaystyle {\tfrac {3}{4}}\times {\tfrac {5}{7}}={\tfrac {15}{28}}}. The terms compound fraction and complex fraction are closely related and sometimes one is used as a synonym for the other. (For example, the compound fraction 3 4 × 5 7 {\displaystyle {\tfrac {3}{4}}\times {\tfrac {5}{7}}} is equivalent to the complex fraction 3 / 4 7 / 5 {\displaystyle {\tfrac {3/4}{7/5}}}.)
  
  Nevertheless, "complex fraction" and "compound fraction" may both be considered outdated and now used in no well-defined manner, partly even taken synonymously for each other or for mixed numerals. They have lost their meaning as technical terms and the attributes "complex" and "compound" tend to be used in their every day meaning of "consisting of parts".