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Compare fractions with unlike denominators

Comparison of fractions with unlike denominators requires interpreting them as portions of the same whole and determining their relative values by renaming them as equivalent fractions with a common denominator, or by using a common numerator or benchmark such as 12\tfrac12 or 1. Number lines and area models clarify why a larger denominator does not by itself indicate a larger fraction; the focus is positive fractions, including mixed numbers when relevant, not algebraic rational expressions or generalized inequality proofs.

Detailed Explanation: Compare fractions with unlike denominators

Fractions must describe parts of the same-sized whole before you compare them. When the denominators are different, rename the fractions with a common denominator.

Compare:

34and56\frac{3}{4}\quad\text{and}\quad\frac{5}{6}

  1. Find a common denominator.
    A number that both 44 and 66 divide into is 1212.

  2. Rename each fraction using denominator 1212:

34=3×34×3=912\frac{3}{4}=\frac{3\times3}{4\times3}=\frac{9}{12} 56=5×26×2=1012\frac{5}{6}=\frac{5\times2}{6\times2}=\frac{10}{12}

  1. Compare the numerators. Since 10>910>9,

1012>912\frac{10}{12}>\frac{9}{12}

  1. Write the comparison using the original fractions:

56>34\boxed{\frac{5}{6}>\frac{3}{4}}

The denominator 66 is larger than 44, but that does not mean 56\frac{5}{6} is smaller. The fractions must first be renamed so they show the same-sized pieces. Here, both fractions are measured in twelfths, and 1010 twelfths is greater than 99 twelfths.

Learn by doing: Compare fractions with unlike denominators

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    Fraction Comparison - Mixed - Two Changed Denominators

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