Example 1: Dividing Fractions

Evaluate \dfrac{4}{11}\div \dfrac{5}{9}.

So, we want to keep the first fraction as it is, change the \div to a \times, and flip the second fraction. Doing this, our calculation becomes

Example 2: Dividing Whole Numbers by Fractions

Evaluate 6\div\dfrac{5}{9}.

To divide a whole number by a fraction (or vice versa), remember that any number divided by 1 is just itself. So, we can write

6=\dfrac{6}{1}

Therefore, we can write

6\div \dfrac{5}{9}=\dfrac{6}{1}\div\dfrac{5}{9}

Now this just looks like a normal fraction division. So, apply the ‘keep, change, flip’ rule and doing the calculation, we get

\begin{aligned}\dfrac{6}{1}\div\dfrac{5}{9} &= \dfrac{6}{1}\times\dfrac{9}{5} \\ &=\dfrac{6\times 9}{1\times 5} \\ &=\dfrac{54}{5}\end{aligned}

Example 3: Dividing Mixed Fractions

Evaluate 4\frac{1}{3}\div\dfrac{5}{6}. Write your answer in its simplest form.

To divide a mixed number by a fraction (or vice versa), first convert the mixed number to an improper fraction.

4\frac{1}{3}=\dfrac{(4\times3)+1}{3}=\dfrac{13}{3}

So, we can now write the calculation to look like a normal division:

4\frac{1}{3}\div\dfrac{5}{6}=\dfrac{13}{3}\div\dfrac{5}{6}

Now we can apply the ‘keep, change, flip’ rule and do the calculation as usual.

\begin{aligned}\dfrac{13}{3}\div\dfrac{5}{6} &= \dfrac{13}{3}\times\dfrac{6}{5} \\ &=\dfrac{13\times 6}{3\times 5} \\ &=\dfrac{78}{15}\end{aligned}

Now, we must simply the fraction. Cancelling a factor of 3 from top and bottom, we get

\dfrac{78}{15}=\dfrac{26}{5}

This cannot be simplified further, so we’re done.

Note: if negative numbers are involved, normal rules of dividing apply:

1. If you divide one positive fraction with one negative fraction (or vice versa), the answer should be negative.

2. If you divide one negative fraction by another negative fraction, the answer should be positive.