Dividing Fractions Using the Inverse Operation

Dividing Fractions Using the Inverse Operation exercise

• Find the reciprocal.

  Hints

  To find the reciprocal of a fraction swap the numerator and denominator, i.e. the reciprocal of $\frac{6}{7}$ is $\frac{7}{6}$.

  The reciprocal of a unit fraction is a whole number. For example, the reciprocal of $\frac{1}{6}$ is $6$.

  The reciprocal of a whole number is a unit fraction. For example, the reciprocal of $2$ is $\frac{1}{2}$.

  Solution

  The reciprocal of $\frac{2}{3}$ is $\bf{\frac{3}{2}}$.

  The reciprocal of $\frac{4}{5}$ is $\bf{\frac{5}{4}}$.

  The reciprocal of $3$ is $\bf{\frac{1}{3}}$.

  The reciprocal of $\frac{7}{2}$ is $\bf{\frac{2}{7}}$.

  The reciprocal of $\frac{1}{4}$ is $\bf{4}$.

• Use the inverse operation to solve the division questions.

  Hints

  The inverse of multiplication is division.

  Remember to use the reciprocal (swap the numerator and denominator). For example, the reciprocal of $\frac{3}{8}$ is $\frac{8}{3}$.

  For example:

  $\frac{2}{7} \div \frac{3}{4} = \frac{2}{7} \times \frac{4}{3} = \frac{8}{21}$

  Solution

  $\frac{2}{5} \div \frac{1}{4} = \frac{2}{5} \bf{\times \frac{4}{1}}$ $= \frac{8}{5}$

  ${}$

  $\frac{5}{6} \div \frac{2}{7} = \frac{5}{6} \bf{\times \frac{7}{2}}$ $= \frac{35}{12}$

  ${}$

  $\frac{3}{4} \div \frac{5}{3} = \frac{3}{4} \bf{\times \frac{3}{5}}$ $= \frac{9}{20}$

  ${}$

  $\frac{4}{7} \div 5 = \frac{4}{7} \bf{\times \frac{1}{5}}$ $= \frac{4}{35}$

  ${}$

  $8 \div \frac{3}{5} = 8 \bf{\times \frac{5}{3}}$ $=\frac{40}{3}$

• What is $\frac{2}{5} \div \frac{4}{9}$?

  Hints

  Multiply the first fraction by the reciprocal of the second fraction.

  We can also say use the inverse operation - swap the numerator and denominator and multiply.

  For example, $\frac{3}{8} \div \frac{5}{4} = \frac{3}{8} \times \frac{4}{5}$

  There are two correct answers, one is simplified and the other is unsimplified.

  For example, hint two has two possible answers.

  $\frac{3}{8} \div \frac{5}{4} = \frac{3}{8} \times \frac{4}{5} = \bf{\frac{12}{40}}$ or $\bf{\frac{3}{10}}$

  Solution

  $\frac{2}{5} \div \frac{4}{9}$

  $= \frac{2}{5} \times \frac{9}{4}$

  $= \bf{\frac{18}{20}} = \bf{\frac{9}{10}}$

• Divide the fractions.

  Hints

  The reciprocal of a unit fraction is a whole number. For example, the reciprocal of $\frac{1}{6}$ is $6$.

  The reciprocal of a whole number is a unit fraction. For example, the reciprocal of $2$ is $\frac{1}{2}$.

  Remember when multiplying a fraction by a whole number we only multiply the numerator. For example, $\frac{2}{3} \times 4 = \frac{8}{3}$.

  Solution

  $\frac{3}{8} \div \frac{2}{5} = \frac{3}{8} \times \frac{5}{2} =$ $\bf{\frac{15}{16}}$

  $\frac{3}{4} \div 5 = \frac{3}{4} \times \frac{1}{5} =$ $\bf{\frac{3}{20}}$

  $6 \div \frac{7}{2} = 6 \times \frac{2}{7} =$ $\bf{\frac{12}{7}}$

  $\frac{5}{9} \div \frac{1}{4} = \frac{5}{9} \times \frac{4}{1} =$ $\bf{\frac{20}{9}}$

• What is $\frac{2}{3} \div \frac{1}{5}$?

  Hints

  Multiply the first fraction by the reciprocal of the second fraction.

  To find the reciprocal swap the numerator and denominator. For example, the reciprocal of $\frac{3}{8}$ is $\frac{8}{3}$.

  An example:

  $\frac{3}{4} \div \frac{2}{3} = \frac{3}{4} \times \frac{3}{2} = \frac{9}{8}$.

  Solution

  $\frac{2}{3} \div \frac{1}{5} = \frac{2}{3} \times \frac{5}{1} = \frac{10}{3}$

• What is $\frac{5}{6} \div \frac{5}{9}$?

  Hints

  Multiply the first fraction by the reciprocal of the second fraction.

  Simplify your answer before converting it to a mixed number.

  A mixed number has a whole number part and a fractional part, i.e. $3 \frac{2}{5}$.

  Solution

  $\frac{5}{6} \div \frac{5}{9} = \frac{5}{6} \times \frac{9}{5} = \frac{45}{30} = \frac{3}{2} = 1\frac{1}{2}$