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. ${\color{green}{{\frac{{a}}{{b}}\div\frac{{c}}{{d}}=\frac{{\frac{{a}}{{b}}}}{{\frac{{c}}{{d}}}}=\frac{{{a}{d}}}{{{b}{c}}}}}}$ .

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 $\frac{{5}}{{12}}$ by $\frac{{11}}{{7}}$ .

$\frac{{\color{blue}{{{5}}}}}{{\color{red}{{{12}}}}}\div\frac{{\color{red}{{{11}}}}}{{\color{blue}{{{7}}}}}=\frac{{{5}\cdot{7}}}{{{12}\cdot{11}}}=\frac{{35}}{{132}}$ .

Fraction is irreducible.

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

Next example.

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

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

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

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

Next example.

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

$\frac{{\frac{{39}}{{7}}}}{{\frac{{5}}{{2}}}}=\frac{{{39}\cdot{2}}}{{{5}\cdot{7}}}=\frac{{78}}{{35}}$ .

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

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

Now, it is time to practice.

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

Answer: $\frac{{24}}{{5}}={4}\frac{{4}}{{5}}$ .

Next exercise.

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

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

Last exercise.

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

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