Dividing Fractions

To divide fraction by a fraction multiply numerator of the first fraction by the denominator of the second fraction and denominator of the first fraction by the numerator of the second fraction, i.e. ab÷cd=abcd=adbc{\color{green}{{\frac{{a}}{{b}}\div\frac{{c}}{{d}}=\frac{{\frac{{a}}{{b}}}}{{\frac{{c}}{{d}}}}=\frac{{{a}{d}}}{{{b}{c}}}}}}.

After this you, possibly, need to reduce fraction.

Note! Rules for determining sign of the result are same as when dividing integers.

Example 1. Divide 512\frac{{5}}{{12}} by 117\frac{{11}}{{7}}.

512÷117=571211=35132\frac{{\color{blue}{{{5}}}}}{{\color{red}{{{12}}}}}\div\frac{{\color{red}{{{11}}}}}{{\color{blue}{{{7}}}}}=\frac{{{5}\cdot{7}}}{{{12}\cdot{11}}}=\frac{{35}}{{132}}.

Fraction is irreducible.

Answer: 35132\frac{{35}}{{132}}.

Next example.

Example 2. Divide 1611\frac{{16}}{{11}} by 83\frac{{8}}{{3}}.

161183=163118=4888\frac{{\frac{{\color{blue}{{{16}}}}}{{\color{red}{{{11}}}}}}}{{\frac{{\color{red}{{{8}}}}}{{\color{blue}{{{3}}}}}}}=\frac{{{16}\cdot{3}}}{{{11}\cdot{8}}}=\frac{{48}}{{88}}.

Reduce fraction: 4888=611\frac{{48}}{{88}}=\frac{{6}}{{11}}.

Answer: 611\frac{{6}}{{11}}.

Next example.

Example 3. Divide 397\frac{{39}}{{7}} by 52\frac{{5}}{{2}}.

39752=39257=7835\frac{{\frac{{39}}{{7}}}}{{\frac{{5}}{{2}}}}=\frac{{{39}\cdot{2}}}{{{5}\cdot{7}}}=\frac{{78}}{{35}}.

Convert to mixed number if needed: 7835=2835\frac{{78}}{{35}}={2}\frac{{8}}{{35}}.

Answer: 7835=2835\frac{{78}}{{35}}={2}\frac{{8}}{{35}}.

Now, it is time to practice.

Exercise 1. Divide 25\frac{{2}}{{5}} by 112\frac{{1}}{{12}}.

Answer: 245=445\frac{{24}}{{5}}={4}\frac{{4}}{{5}}.

Next exercise.

Exercise 2. Divide 15 by 57-\frac{{5}}{{7}}.

Answer: -21. Hint: 15=151{15}=\frac{{15}}{{1}}.

Last exercise.

Exercise 3. Divide 215-\frac{{21}}{{5}} by 354-\frac{{35}}{{4}}.

Answer: 1225\frac{{12}}{{25}}.