Division of Fractions and Integers

A fraction is a division operation. $\frac{1}{2}=0.5$ because $1÷2=0.5$. But you can also divide fractions by fractions! Below is the rule for what to do when you have one fraction divided by another fraction.

Formula

Keep,Change,Flip

 $\genfrac{}{}{1.0pt}{}{\phantom{\rule{0.17em}{0ex}}\frac{a}{b}\phantom{\rule{0.17em}{0ex}}}{\phantom{\rule{0.17em}{0ex}}\frac{c}{d}\phantom{\rule{0.17em}{0ex}}}=\frac{a}{b}×\frac{d}{c}=\frac{ad}{bc}$

What happens is that you flip the bottom fraction, and replace the division sign with a multiplication sign. The math terminology is that you are “multiplying by the inverse fraction”. I call it “Keep, Change, Flip Formula”!

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Example 1

Compute

 $\genfrac{}{}{1.0pt}{}{\phantom{\rule{0.17em}{0ex}}\frac{2}{6}\phantom{\rule{0.17em}{0ex}}}{\phantom{\rule{0.17em}{0ex}}\frac{2}{3}\phantom{\rule{0.17em}{0ex}}}$

Using the flip formula, you get

 $\genfrac{}{}{1.0pt}{}{\phantom{\rule{0.17em}{0ex}}\frac{2}{6}\phantom{\rule{0.17em}{0ex}}}{\phantom{\rule{0.17em}{0ex}}\frac{2}{3}\phantom{\rule{0.17em}{0ex}}}=\frac{2}{6}×\frac{3}{2}=\frac{\text{2}×3}{6×\text{2}}=\frac{3}{6}=\frac{1}{2}$

Note! When you divide an integer or fraction by a proper fraction (which is less than $1$), you’ll find that the answer you get is actually larger than the integer or fraction you started with. This is different than normal division with whole numbers where the answer is smaller.

If you divide a fraction by an integer, or an integer by a fraction, it may be easier to use the trick you apply on integers multiplied by fractions—that is, to write the integer as a fraction.

Example 2

Compute

 $5÷\frac{2}{7}=\genfrac{}{}{1.0pt}{}{\phantom{\rule{0.17em}{0ex}}5\phantom{\rule{0.17em}{0ex}}}{\phantom{\rule{0.17em}{0ex}}\frac{2}{7}\phantom{\rule{0.17em}{0ex}}}$

Since $5=\frac{5}{1}$, you get

 $\genfrac{}{}{1.0pt}{}{\phantom{\rule{0.17em}{0ex}}5\phantom{\rule{0.17em}{0ex}}}{\phantom{\rule{0.17em}{0ex}}\frac{2}{7}\phantom{\rule{0.17em}{0ex}}}=\genfrac{}{}{1.0pt}{}{\phantom{\rule{0.17em}{0ex}}\frac{5}{1}\phantom{\rule{0.17em}{0ex}}}{\phantom{\rule{0.17em}{0ex}}\frac{2}{7}\phantom{\rule{0.17em}{0ex}}}=\frac{5}{1}×\frac{7}{2}=\frac{35}{2}$

Example 3

 $\frac{4}{3}÷3=\genfrac{}{}{1.0pt}{}{\phantom{\rule{0.17em}{0ex}}\frac{4}{3}\phantom{\rule{0.17em}{0ex}}}{\phantom{\rule{0.17em}{0ex}}3\phantom{\rule{0.17em}{0ex}}}$

Since $3=\frac{3}{1}$, you get

 $\genfrac{}{}{1.0pt}{}{\phantom{\rule{0.17em}{0ex}}\frac{4}{3}\phantom{\rule{0.17em}{0ex}}}{\phantom{\rule{0.17em}{0ex}}3\phantom{\rule{0.17em}{0ex}}}=\genfrac{}{}{1.0pt}{}{\phantom{\rule{0.17em}{0ex}}\frac{4}{3}\phantom{\rule{0.17em}{0ex}}}{\phantom{\rule{0.17em}{0ex}}\frac{3}{1}\phantom{\rule{0.17em}{0ex}}}=\frac{4}{3}×\frac{1}{3}=\frac{4}{9}$