Page 6.1.5: Dividing Fractions

WTRM 201 Introduction to Water, Wastewater Technology · LibreTexts

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    From LibreTexts, under CC BY 1.0, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    When dividing fractions you invert (flip upside down) the fraction on the right side of the equation (the dividend). Then it becomes a multiplication problem. Invert and multiply!

    Example \(\PageIndex{1}\)

    \(\dfrac{2}{5} \div \dfrac{1}{2} \rightarrow \dfrac{2}{5} \times \dfrac{2}{1}=\dfrac{4}{5}\)

    Example \(\PageIndex{2}\)

    \(\dfrac {\dfrac{4}{9}}{\dfrac{8}{9}}\)

    This example reads \(\dfrac{4}{9}\) divided by \(\dfrac{8}{9}\).

    However, after you “invert and multiply,” it becomes:

    \[\dfrac{4}{9} \times \dfrac{9}{8} \rightarrow \dfrac{1}{1} \times \dfrac{1}{2}=\dfrac{1}{2} \nonumber \]

    After inverting the fraction, the same rules apply as previously mentioned when multiplying fractions. You need to change mixed numbers into improper fractions; you can cross cancel, and always remember to reduce if necessary.

    Example \(\PageIndex{3}\)

    \(\dfrac{3}{8} \div \dfrac{12}{6} \rightarrow \dfrac{3}{8} \times \dfrac{6}{12} \rightarrow \dfrac{\not{3}}{\not {8}} \times \dfrac{\not {6}}{\not {12}}=\dfrac{1}{4} \times \dfrac{3}{4}=\dfrac{3}{16}\)

    In the Example \(\PageIndex{3}\) above the 3 divided into itself once and into the 12 four times. Similarly, 2 divided into 6, three times and into 8, four times. Can you see a different way to cross cancel?

    If the dividend is a whole number write it as a fraction before inverting. Always remember to cross cancel and reduce if necessary.

    Example \(\PageIndex{4}\)

    \(\dfrac{5}{8} \div 10 \rightarrow \dfrac{5}{8} \div \dfrac{10}{1} \rightarrow \dfrac{5}{8} \times \dfrac{1}{10} \rightarrow \dfrac{\not{5}}{8} \times \dfrac{1}{\not {10}} \rightarrow \dfrac{1}{8} \times \dfrac{1}{2}=\dfrac{1}{16}\)

    Exercise 1.5

    Divide the following and reduce if necessary.

    1. \(\dfrac{1}{2} \div \dfrac{2}{4}\)
    2. \(\dfrac{5}{22} \div 7 \dfrac{2}{4}\)
    3. \(\dfrac{6}{8} \div \dfrac{9}{12}\)
    4. \(3 \dfrac{1}{3} \div 10\)
    5. \(5 \dfrac{1}{4} \div 8 \dfrac{1}{2}\)
    6. \(\dfrac{8}{16} \div \dfrac{16}{8}\)
    7. \(4 \div 3 \dfrac{2}{3}\)
    8. \(\dfrac{2}{5} \div 5 \dfrac{6}{9}\)
    9. \(\dfrac{9}{13} \div 9\)
    10. \(\dfrac{11}{22} \div \dfrac{1}{2}\)
    11. \(2 \dfrac{9}{20} \div 5 \dfrac{2}{5}\)
    12. \(3 \dfrac{1}{2} \div 9 \dfrac{1}{2}\)
    13. \(5 \div 25\)
    14. \(\dfrac{10}{3} \div \dfrac{1}{6}\)
    15. \(\dfrac{13}{33} \div \dfrac{39}{3}\)
    16. \(100 \dfrac{1}{2} \div 10 \dfrac{5}{6}\)

    This page titled Page 6.1.5: Dividing Fractions was last modified on Tue, 07 Oct 2025 23:06:04 GMT and is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by Catherine Miettinen Ames.