Algebra

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Algebraic Fractions: Division

Algebraic fractions can also be divided together to form new fractions containing algebra.

Dividing algebraic fractions has one key first step

we need to do that before anything else

We want to divide these fractions $\dfrac{3x}{2} \div \dfrac{5}{7x}$ . What is the first step we need to do?

Which fraction do we need to flip here? $\dfrac{3x}{2} \div \dfrac{5}{7x}$

This means that our division becomes a multiplication:

$\dfrac{3x}{2} \div \dfrac{5}{7x}=\dfrac{3x}{2} \times\dfrac{7x}{5}$

We then multiply to get our answer. What is $\dfrac{3x}{2} \times\dfrac{7x}{5}$ ?

We've found our final answer!

$\dfrac{3x}{2} \div \dfrac{5}{7x}=\dfrac{21x^2}{10}$

Let's try another example.

$\dfrac{8a}{3}\div \dfrac{a^2}{6}$

What is the first step we need to take? $\dfrac{8a}{3}\div \dfrac{a^2}{6}$

We flip, or invert, the second fraction and the division becomes multiplication.

$\dfrac{8a}{3}\div \dfrac{a^2}{6}=\dfrac{8a}{3}\times \dfrac{6}{a^2}$

Now we multiply - first let's cancel all the common factors.

Now we have cancelled our common factors, what is the final answer? $\dfrac{8{\cancel{a}}}{{\cancel{3}}1}\times \dfrac{{\cancel{6}}2}{a^{\cancel{2}}}$

We could have chosen to multiply first.

$\dfrac{8a}{3}\times \dfrac{6}{a^2}=\dfrac{48a}{3a^2}$

We then need to simplify $\dfrac{48a}{3a^2}$ . What common factor is there in the numerator and denominator?

Simplify $\dfrac{48a}{3a^2}$ using the common factor $3a$ .

Both methods gave the same answer! 😎

$\dfrac{8a}{3}\div \dfrac{a^2}{6}=\dfrac{16}{a}$

Give this one a try. We'll go through it step by step.

$\dfrac{3a}{7}\div \dfrac{5a^2}{14}$

First we invert the second fraction. $\dfrac{3a}{7}\div \dfrac{5a^2}{14}$

Now we have $\dfrac{3a}{7}\times \dfrac{14}{5a^2}$ , what common factors are there?

Now we have the common factors, $7$ and $a$ , what is $\dfrac{3a}{7}\times \dfrac{14}{5a^2}$ in its simplest terms?

Try this one - remember to make sure your answer is fully simplified. $\dfrac{3y^2}{4}\div \dfrac{4y}{3}$

This one is a little harder $\dfrac{12x-8}{3x}\div \dfrac{4x}{3}$ .

Which option gives the correct first step?

We now have $\dfrac{12x-8}{3x}\times \dfrac{3}{4x}$ .

Our next step is to factorise the numerator of the first fraction.

Factorise means to find a common factor and add in brackets. What is the common factor in the terms $12x-8$ ?

The common factor in $12x-8$ is $4$ .

We can now write this as $4(3x-2)$ .

Our fraction is now $\dfrac{4(3x-2)}{3x}\times \dfrac{3}{4x}$ . One common factor is $3$ , what is the other?

We can cancel our common factors $3$ and $4$ . What answer do we get? $\dfrac{{\cancel{4}}(3x-2)}{{\cancel{3}}x}\times \dfrac{{\cancel{3}}}{{\cancel{4}}x}$

Well done! There were lots of steps in this one.

$\dfrac{12x-8}{3x}\div \dfrac{4x}{3}=\dfrac{3x-2}{x^2}$ .

Try another one! Simplify $\dfrac{2x-6}{3}\div \dfrac{2}{x}$ .

In summary, to divide algebraic fractions

flip the second fraction and multiply.

It must be the second fraction that you flip, or invert

$\dfrac{3x}{10}\div {\color{orange}{\dfrac{2}{5x}}}=\dfrac{3x}{10}\times {\color{orange}{\dfrac{5x}{2}}}$

Multiply the numerators and the denominators

$\dfrac{3x}{10} \times \dfrac{5x}{2}=\dfrac{15x^2}{20}$

Remember to check if your answer can simplify.

$\dfrac{15x^2}{20}=\dfrac{3x^2}{4}$

A trickier one might be:

$\dfrac{4}{3x}\div \dfrac{2}{3x-12}$

First we must invert the second fraction.

$\dfrac{4}{3x}\div \dfrac{2}{3x-12}=\dfrac{4}{3x}\times \dfrac{3x-12}{2}$

Then we factorise the numerator in the second fraction.

$\dfrac{4}{3x}\times \dfrac{3x-12}{2}=\dfrac{4}{3x}\times \dfrac{3(x-4)}{2}$

Then we can see our common factors.

$\dfrac{{\cancel{4}}2}{{\cancel{3}}x}\times \dfrac{{\cancel{3}}(x-4)}{{\cancel{2}}}$

Finally we multiply to give our answer:

$\dfrac{{\cancel{4}}2}{{\cancel{3}}x}\times \dfrac{{\cancel{3}}(x-4)}{{\cancel{2}}}=\dfrac{2x-8}{x}$