1.5: Dividing Fractions

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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.

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