section_path: Arithmetic with fractions/Comparing fractions
section_name: Comparing fractions
paragraph_text: Comparing fractions
  
  Comparing fractions with the same positive denominator yields the same result as comparing the numerators:
  
  If the equal denominators are negative, then the opposite result of comparing the numerators holds for the fractions:
  
  If two positive fractions have the same numerator, then the fraction with the smaller denominator is the larger number. When a whole is divided into equal pieces, if fewer equal pieces are needed to make up the whole, then each piece must be larger. When two positive fractions have the same numerator, they represent the same number of parts, but in the fraction with the smaller denominator, the parts are larger.
  
  One way to compare fractions with different numerators and denominators is to find a common denominator. To compare a b {\displaystyle {\tfrac {a}{b}}} and c d {\displaystyle {\tfrac {c}{d}}}, these are converted to a ⋅ d b ⋅ d {\displaystyle {\tfrac {a\cdot d}{b\cdot d}}} and b ⋅ c b ⋅ d {\displaystyle {\tfrac {b\cdot c}{b\cdot d}}} (where the dot signifies multiplication and is an alternative symbol to ×). Then bd is a common denominator and the numerators ad and bc can be compared. It is not necessary to determine the value of the common denominator to compare fractions – one can just compare ad and bc, without evaluating bd, e.g., comparing 2 3 {\displaystyle {\tfrac {2}{3}}} ? 1 2 {\displaystyle {\tfrac {1}{2}}} gives 4 6 > 3 6 {\displaystyle {\tfrac {4}{6}}>{\tfrac {3}{6}}}.
  
  For the more laborious question 5 18 {\displaystyle {\tfrac {5}{18}}} ? 4 17, {\displaystyle {\tfrac {4}{17}},} multiply top and bottom of each fraction by the denominator of the other fraction, to get a common denominator, yielding 5 × 17 18 × 17 {\displaystyle {\tfrac {5\times 17}{18\times 17}}} ? 18 × 4 18 × 17 {\displaystyle {\tfrac {18\times 4}{18\times 17}}}. It is not necessary to calculate 18 × 17 {\displaystyle 18\times 17} – only the numerators need to be compared. Since 5×17 (= 85) is greater than 4×18 (= 72), the result of comparing is 5 18 > 4 17 {\displaystyle {\tfrac {5}{18}}>{\tfrac {4}{17}}}.
  
  Because every negative number, including negative fractions, is less than zero, and every positive number, including positive fractions, is greater than zero, it follows that any negative fraction is less than any positive fraction. This allows, together with the above rules, to compare all possible fractions.