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Algebraic Fractions: Multiplication

# Algebraic Fractions: Multiplication

### Algebraic Fractions: Multiplication

Fractions containing algebra can be multiplied together, just like fractions containing integers.

1

An algebraic fraction is just like any other fraction except it contains at least one letter - normally $x$ or $y$

We call that letter a variable

2

Which of these fractions is an algebraic fraction?

3

Algebraic fractions can contain letters in the numerator or the denominator or both.

$\dfrac{2}{x}$, $\dfrac{3y}{10}$ and $\dfrac{a^2}{b^4}$ are all algebraic fractions.

1

When we multiply a numerical fraction, we multiply the numerators and then we multiply the denominators.

$\dfrac{3}{5}\times \dfrac{3}{4}=\dfrac{9}{20}$

2

We do the same with algebraic fractions. What is $\dfrac{3}{x}\times \dfrac{2}{5x}$?

3

Try another one. What is $\dfrac{a}{4}\times \dfrac{5a}{3}$?

1

As with numerical fractions, we often find there are common factors in algebraic fractions.

We need to cancel these common factors to simplify our answer.

2

Let's look at this one. What is $\dfrac{b}{4}\times \dfrac{2b}{3}$ without cancelling the common factor?

3

What is the common factor in the numerator and denominator of our answer $\dfrac{2b^2}{12}$.

4

We have a common factor of $2$ in $\dfrac{2b^2}{12}$. What is the simplified version of this fraction?

5

Try this one - what is the unsimplified answer to $\dfrac{x^2}{5}\times \dfrac{3}{x}$?

6

What is the common factor in the numerator and denominator in this answer? $\dfrac{3x^2}{5x}$

7

This time our common factor is the variable $x$. What is the simplified version of $\dfrac{3x^2}{5x}$?

Multiply $\dfrac{x^2}{y} \times \dfrac{4}{x}$ then cancel any common factors.

1

In our previous examples, we have simplified after multiplying out.

You can also simplify, or cancel, before multiplying.

2

Let's look at $\dfrac{3x}{4}\times \dfrac{8x}{5}$.

We can cancel terms diagonally across the fractions.

3

Looking at the denominator in the first fraction and the numerator in the second, what is the common factor? $\dfrac{3x}{{\color{orange}{4}}}\times \dfrac{{\color{orange}{8x}}}{5}$

4

Now we have spotted our common factor of $4$ we can cancel as follows:

$\dfrac{3x}{{\cancel{4}}1}\times \dfrac{2{\cancel{8}}x}{5}$

5

Sometimes this can get a little messy!

$\dfrac{3x}{{\cancel{4}}1}\times \dfrac{2{\cancel{8}}x}{5}$ , note we have replaced the cancelled figures with the simplified one.

6

What is the final answer for $\dfrac{3x}{{\cancel{4}}1}\times \dfrac{2{\cancel{8}}x}{5}$?

7

Our final answer is $\dfrac{6x^2}{5}$.

We cannot simplify this further! 👍

8

You can choose whether to cancel before or after multiplying.

But always check that your final answer can't be simplified further.

What is $\dfrac{3a}{4b} \times \dfrac{5b}{4}$? You can choose which method you prefer, either cancel first and then multiply, or multiply then simplify.

1

Let's try a slightly trickier one.

$\dfrac{7}{2x+10}\times \dfrac{4}{3x}$

2

With $\dfrac{7}{2x+10}\times \dfrac{4}{3x}$ the denominator of the first fraction is $2x+10$.

If there are any common factors, they must divide into both parts of the denominator.

3

To make it easier, let's see if there is a common factor for the terms in the first denominator, $2x+10$. What can both these terms be divided by?

4

Now we have found the common factor in the denominator, let's re-write the question.

$\dfrac{7}{2x+10}\times \dfrac{4}{3x}=\dfrac{7}{2(x+5)}\times \dfrac{4}{3x}$

5

What common factor is there across the fractions? $\dfrac{7}{2(x+5)}\times \dfrac{4}{3x}$

6

By cancelling the common factor $2$ we get $\dfrac{7}{{\cancel{2}}(x+5)}\times \dfrac{{\cancel{4}}2}{3x}$. What is the result of this?

7

Our final answer is $\dfrac{7}{3x^2+15x}$.

Good work - this was a tricky question! 👍🏽

1

To summarise, an algebraic fraction is a fraction containing a letter.

$\dfrac{4}{x}$ and $\dfrac{7}{a-1}$ are examples of algebraic fractions.

2

To multiply an algebraic fraction, multiply the numerator then multiply the denominators.

$\dfrac{4x}{5}\times \dfrac{2x}{3}=\dfrac{8x^2}{15}$

3

Look out for common factors which may be letters of numbers.

$\dfrac{4x}{5}\times \dfrac{10}{x^2}$ has common factors $5$ and $x$.

4

You can cancel common factors either before multiplying:

$\dfrac{4{\cancel{x}}}{{\cancel{5}}}\times \dfrac{{\cancel{10}}2}{x^{\cancel{2}}}=\dfrac{8}{x}$

5

Or multiply first then cancel:

$\dfrac{4x}{5}\times \dfrac{10}{x^2}=\dfrac{40x}{5x^2}=\dfrac{8}{x}$