Albert Teen YOU ARE LEARNING:  Algebraic Fractions: Division  # Algebraic Fractions: Division ### Algebraic Fractions: Division

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

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Dividing algebraic fractions has one key first step

we need to do that before anything else

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

Which fraction do we need to flip here? $\dfrac{3x}{2} \div \dfrac{5}{7x}$ 4

This means that our division becomes a multiplication:

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

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We then multiply to get our answer. What is $\dfrac{3x}{2} \times\dfrac{7x}{5}$? 6

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

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Let's try another example.

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

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

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

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

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Now we multiply - first let's cancel all the common factors. 5

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

We could have chosen to multiply first.

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

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We then need to simplify $\dfrac{48a}{3a^2}$. What common factor is there in the numerator and denominator? 8

Simplify $\dfrac{48a}{3a^2}$ using the common factor $3a$. 9

Both methods gave the same answer! 😎

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

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

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

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

Now we have $\dfrac{3a}{7}\times \dfrac{14}{5a^2}$, what common factors are there? 4

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

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Which option gives the correct first step? 2

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

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

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Factorise means to find a common factor and add in brackets. What is the common factor in the terms $12x-8$? 4

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

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

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Our fraction is now $\dfrac{4(3x-2)}{3x}\times \dfrac{3}{4x}$. One common factor is $3$, what is the other? 6

We can cancel our common factors $3$ and $4$. What answer do we get? $\dfrac{{\cancel{4}}(3x-2)}{{\cancel{3}}x}\times \dfrac{{\cancel{3}}}{{\cancel{4}}x}$ 7

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 $\dfrac{2x-6}{3}\div \dfrac{2}{x}$. 1

In summary, to divide algebraic fractions

flip the second fraction and multiply.

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

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

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Multiply the numerators and the denominators

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

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

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A trickier one might be:

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

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First we must invert the second fraction.

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

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Then we factorise the numerator in the second fraction.

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

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Then we can see our common factors.

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

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Finally we multiply to give our answer:

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