Dividing Fractions

How to Divide Fractions?

Dividing fractions involves these four steps –

Take the reciprocal of the divisor (second fraction). Meaning, interchange the top and bottom numbers.
Multiply the top numbers (numerators).
Multiply the bottom numbers (denominators).
Simplify the resulting fraction (if possible).

Alright, now let’s use these steps to solve a couple of examples.

Then, we’ll look at a few special cases. And also at a way to make divisions simpler.

Example

Simplify :

\frac{4}{5} \div \frac{6}{10}

Solution

Step 1. Start by taking the reciprocal of the divisor (the fraction after the division symbol) – turn it upside down.

As you do that, change the division symbol into a multiplication symbol.

\frac{4}{5} \div \frac{ {\color{Red} 6} }{ {\color{Teal} 10} } = \frac{4}{5} \times \frac{ {\color{Teal} 10} }{ {\color{Red} 6} }

Step 2 & 3. Multiply the top numbers. And the bottom numbers.

\begin{align*} \frac{4}{5} \times \frac{10}{6} \hspace{0.2em} &= \hspace{0.2em} \frac{4 \times 10}{5 \times 6} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{40}{10} \hspace{0.25em}=\hspace{0.25em} \frac{4}{3} \end{align*}

Step 4. Simplify the fraction, if possible (it is in this case).

Example

Simplify :

\frac{3}{7} \div \frac{4}{9}

Solution

Just as we did in the previous example,

\begin{align*} \frac{3}{7} \div \frac{ {\color{Red} 4} }{ {\color{Teal} 9} } \hspace{0.2em} &= \hspace{0.2em} \frac{3}{7} \times \frac{ {\color{Teal} 9} }{ {\color{Red} 4} } \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{3 \times 9}{7 \times 4} \hspace{0.25em}=\hspace{0.25em} \frac{27}{28} \end{align*}

Done. This time, the answer was already in its lowest terms. So, no question of further simplification.

Dividing Fractions by a Whole Number

For divisions involving a fraction and a whole number, convert the whole number into a fraction (put 1 as its denominator). Then proceed as usual.

Example

Simplify -

(a) \hspace{0.3cm} \frac{4}{7} \div 3

(b) \hspace{0.3cm} 8 \div \frac{9}{10}

Solution (a)

We start by writing the whole number (3) as a fraction – 3 over 1. From there on, it’s just like the previous examples.

\begin{align*} \frac{4}{7} \div {\color{Red} 3} \hspace{0.2em} &= \hspace{0.2em} \frac{4}{7} \div {\color{Red} \frac{3}{1}} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{4}{7} \times \frac{1}{3} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{4 \times 1}{7 \times 3} \hspace{0.25em}=\hspace{0.25em} \frac{4}{21} \end{align*}

Solution (b)

\begin{align*} {\color{Red} 8} \div \frac{9}{10} \hspace{0.2em} &= \hspace{0.2em} {\color{Red} \frac{8}{1}} \div \frac{9}{10} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{8}{1} \times \frac{10}{9} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{8 \times 10}{1 \times 9} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{4}{21} \hspace{0.25em}=\hspace{0.25em} 8\frac{8}{9} \end{align*}

See how I converted the improper (top-heavy) fraction into a mixed number, in the last step?

It wasn’t necessary. But unless the question uses improper fraction(s), it is preferred that we don’t leave improper fractions in our answers either. So, convert them into mixed numbers.

Dividing Fractions by a Mixed Number

When doing divisions involving a mixed number, rewrite the mixed number as an improper fraction and proceed.

Example

Simplify -

(a) \hspace{0.3cm} \frac{1}{2} \div 1 \frac{2}{3}

(b) \hspace{0.3cm} \frac{5}{8} \div 5 \frac{1}{2}

Solution (a)

We convert the mixed number into an improper fraction – in the very first step. And then, it’s the same old story.

\begin{align*} \frac{1}{2} \div {\color{Red} 1 \frac{2}{3}} \hspace{0.2em} &= \hspace{0.2em} \frac{1}{2} \div \frac{5}{3} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{1}{2} \times \frac{3}{5} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{1 \times 3}{2 \times 5} \hspace{0.25em} = \hspace{0.25em} \frac{3}{10} \end{align*}

Solution (b)

\begin{align*} \frac{5}{8} \div {\color{Red} 5 \frac{1}{2}} \hspace{0.2em} &= \hspace{0.2em} \frac{5}{8} \div \frac{11}{2} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{5}{8} \times \frac{2}{11} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{5 \times 2}{8 \times 11} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{10}{88} \hspace{0.25em} = \hspace{0.25em} \frac{5}{44} \end{align*}

Canceling Out Common Factors Early

In all our previous examples, we waited till the end to cancel the factors common between the top and bottom numbers. But often, there’s a much better way.

When dividing fractions, after you have taken the reciprocal, you can split any of the numbers into smaller factors at any stage and cancel out the common factors. You can cancel them out in one step or multiple steps, no problem.

Example

Simplify :

\frac{15}{48} \div \frac{36}{25}

Solution

As usual, we begin by taking the reciprocal and turning it into a multiplication problem.

\frac{15}{48} \div \frac{25}{36} = \frac{15}{48} \times \frac{36}{25}

But from here on we are free to split the numbers into smaller factors and cancel the factors that are common between the top and bottom parts.

\begin{align*} \frac{15}{48} \times \frac{36}{25} \hspace{0.2em} &= \hspace{0.2em} \frac{3 \times \cancel{5}}{\cancel{6} \times 8} \times \frac{\cancel{6} \times 6}{\cancel{5} \times 5} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{3 \times 6}{8 \times 5} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{18}{40} \hspace{0.25em}=\hspace{0.25em} \frac{9}{20} \end{align*}

See how canceling common factors early makes our task easier down the line? Try doing it like we did earlier examples and you’ll know what I mean.

Also, it’s not compulsory that we remove all common factors at once. In the present example, there was a common factor, 2, left after the initial cancelation. We canceled it in the last step to simplify our answer.

Give it some time and practice. And you’ll get better at spotting and canceling common factors early.

Example

Simplify :

\frac{42}{16} \div \frac{7}{4}

Solution

Nothing different here. Let’s apply the same method and get the answer quickly.

\begin{align*} \frac{42}{16} \div \frac{7}{4} \hspace{0.2em} &= \hspace{0.2em} \frac{42}{16} \times \frac{4}{7} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{\cancel{2} \times 3 \times \cancel{7}}{\cancel{2} \times 2 \times \cancel{4}} \times \frac{\cancel{4}}{\cancel{7}} \\[1.5em] \hspace{0.2em} &= \hspace{0.2em} \frac{3}{2} \end{align*}

Important – Don’t split or cancel before you have taken the reciprocal.

Why Take the Reciprocal of the Dividing Fraction?

Division is the inverse (opposite) of multiplication. Multiplying something by a number and then dividing it by the same number must give back the original number. The two operations – those of multiplication and division are supposed to cancel each other out.^*

Now the inverse (opposite) of a fraction is its reciprocal. If you multiply them together, they will cancel each other out.

\frac{ {\color{Red} 2} }{ {\color{Teal} 3} } \times \frac{ {\color{Teal} 3} }{ {\color{Red} 2} } = 1

Combining the two ideas from above, you can see how dividing by a fraction would be the same as multiplying by its reciprocal.

* One exception to this would be dividing by zero since division by zero is not defined.

And with that, we come to the end of this tutorial. Until next time.