Lesson video

Thank you for choosing to learn using this video.

My name is Ms. Davis and I'm going to be helping you as you work your way through this lesson.

We're looking at a lot of our algebra skills.

We're also looking at our fraction skills.

You are definitely going to want to write things down as we go through.

You learn best in mathematics when you give things a go.

So make sure that you've got some paper and a pen to hand so that you can try things out.

Of course, I'm going to help you as we go through.

So if there's bits that you're not sure about what the next step would be, have a think yourself, but then watch the video and I will show you how I would do these questions.

As with a lot of things in algebra, there are different ways of doing things.

Just because your way is not the same as my way doesn't mean that you are doing it wrong.

So bear that in mind as you're exploring some of these concepts.

Let's get started then.

Welcome to this lesson on operations with algebraic fractions.

Today we're going to look at multiplying, dividing, adding and subtracting with algebraic fractions.

If you need a reminder of what a reciprocal is or what factorising means, pause the video and read through those keywords now.

We're going to start by multiplying and dividing algebraic fractions.

So we can multiply algebra fractions in the same way as numerical fractions.

So in this example we've got two over three A multiplied by four A over five B.

Just like with a numerical fraction, we can multiply the numerator and the denominators and we get eight A over 15 AB.

How else can we write our final answer? Can you see it? We can simplify because the numerator and denominator have a common factor of A.

So we could write this as eight over 15 B.

Now the basics and multiplying algebraic fractions is straightforward.

However, with trickier fractions, looking for common factors can help us simplify our final answer.

So let's look at three A squared over seven B multiplied by four B over nine A.

So we can absolutely write the numerator as 12 A squared B and the denominator as 63 AB, and then look to simplify our final answer.

Or we can look for common factors before actually writing that product as a single term.

I'll show you now what I mean.

Now we could write that numerator as three times four times A, times A, times B.

The denominator we could write is three times three 'cause nine is three times three, times seven, times A, times B.

Now why do you think I've written it in this way? How does this help us? Right, the individual fractions couldn't be simplified.

They're in simplest form.

However, the numerator and the denominator of the product are going to share common factors.

They're going to have a factor of three of A and of B.

So what that means is before I actually evaluate that final product as a single term, I know that the numerator and the denominator is going to have a common factor of three AB, and therefore I can write this as four A over 21.

Now because we found all the common factors, our answer is already in simplest form.

Time for you to have a go, which is the correct product of five X squared over 12 Y squared times nine Y over 10 X.

Think about what your common factors are going to be in your product to make it easier for you to simplify.

Off you go.

Well done if you said three X over eight Y.

Let's see why.

So numerator can be written as three times three times five, times X, times X, times Y.

The denominator three times four times two times five, times X, times Y, times Y.

Of course you could write those as prime factors if you wished and write the four as two times two.

The reason I've written it this way is that I can see there's going to be a common factor of three, of five of X and of Y.

So remaining I've got three X over eight Y.

So Andeep is looking at this calculation.

"I don't know how to divide algebraic fractions!" "But we do know how to multiply algebraic fractions." What's Laura getting at? What do you think Andeep needs to do? Just like with numerical fractions, dividing by fraction is the same as multiplying by its reciprocal.

So Laura is saying if we know how to multiply, then we can use the same skills to help us divide.

So two A over 15 divided by three B over five A, is the same as two A over 15 multiplied by the reciprocal, which is five A over three B.

Andeep knows how to do that now.

I've just spotted that there's going to be a common factor of five in the numerator and the denominator, which is going to help me with my product and my simplified answer will be two A squared over nine B.

Pause the video if you need to spend some time looking through that.

Izzy has a go at this question.

I'd like you to read through her working.

What mistake has she made? Off you go.

This one is a little bit daft.

She's tried to simplify before dividing.

She's identified that two X and 10 X squared have common factors, but this is a division calculation.

That isn't going to work until you've written it as a multiplication.

She should have written the calculation as a multiplication first and then she could try to identify common factors.

"Thanks," says Izzy, "I have redone the first step now." Have a read of what she's done.

Can you now finish off her working? We want the simplified answer.

Let's have a look then.

I've decided to identify common factors first.

So I've written the numerator and denominator like this.

I can see there's a common factor of five, so I can write my final answer as four X cubed over three Y squared.

Laura says, "I don't see the need to identify common factors before multiplying, it is easy enough to just simplify at the end." Izzy says, "Sometimes that's true, but have you tried a calculation like this yet?" "Oh, I don't think multiplying those numerators is going to be as easy." How could Laura spot the common factors to help her? What does she need to do? Factorise of course.

So if we factorised the expression, so Izzy's factorised the expressions in the first fraction.

Laura's going to do the second one.

Can you do the second one before her? Give it a go.

So there's the expressions in the second fraction factorised.

What common factors will the numerator and the denominator have when we multiply these together? What can you spot? So hopefully you've spotted that the numerator and the denominator will have a common factor of X plus five and a common factor of X minus two when we multiply them together.

This simplifies then to X plus three X minus one all over four X.

Or if we wanted to expand the numerator and write it as X squared plus two X minus three, that's fine as well.

Laura's right, factorising is proving to be a really useful skill.

Let's see if we can apply that to a division question.

I'm going to do one on the left, then you are going to have go this one on the right, there's quite a few steps, so we're going to do a little bit at a time.

So I'm going to factorised the numerator of the first fraction first.

That's going to be three lots of two X minus one, and then I'm going to factorise the quadratic.

That's going to be X plus seven and X minus two.

I'd like you to do the same for the first fraction in your example.

Good factorising skills.

Hopefully you got two lots of X plus three over X plus four X minus three.

Don't forget to write the whole question out.

We've still got divided by and then that second fraction.

Right, now I'm going to factorise the second fraction.

The numerator doesn't factorise in this case.

The denominator I can write is five X lots of X minus two.

You have a go for the second fraction in your calculation.

Well done.

Hopefully you got four lots of X plus three over X lots of X plus four.

Right, now we're going to write this as a multiplication.

We could have started by writing it as a multiplication.

It doesn't really matter.

So dividing by a fraction is the same as multiplying by its reciprocal.

Have a go at writing yours as a multiplication.

Lovely.

So at this stage we're going to look for common factors before we try and write our fraction as a single expression.

So I can see that there's going to be a common factor of X minus two and a common factor of two X minus one when I multiply these together.

So that means I can write my answer as 15 X over X plus seven.

Look at what I've done, then see if you can do the same with your question.

So you are going to have common factors of X plus four and X plus three.

If you spotted that you'll have a common factor of two as well, well done, that was quite a tricky spot.

Doesn't matter if you don't spot it straight away, you should be able to spot it in your next step.

So you get a final answer of X over two X minus six or X over two lots of X minus three.

Time for practise then.

I'd like you to express the results of each calculation in its simplest form.

Give it a go.

Well done.

Same for this second one is going to make your life a lot easier if you look to factorise to find any common factors.

Give this one a go.

Right last challenge, pay attention to whether you are multiplying and dividing for each one.

We want our answers in our simplest form.

Good effort.

Pause the video and check you've got the same answers.

Notice A and B, we have the same fractions.

First we're multiplying and then dividing.

Dividing actually ended up simplifying a lot more than when we multiplied them.

Once you're happy with your answers we'll look at the next set.

So this second set we're looking at how factorising could help us.

So pause the video and read through my answers.

Remember you can leave your final answer in factorised form or expanded form at this stage.

And finally, we had lots of factorised of quadratics to do here.

So again, check, you've got the correct final answer.

Some of them simplified a lot, some of them didn't simplify quite so much.

And there's C.

So now we're going to have a look at adding and subtracting algebraic fractions.

So, Sofia is trying this calculation.

"The answer will have a common factor of X.

So the final answer will be five over two." What do you think? Maybe like Laura, you're going to politely remind Sofia that we're adding these fractions.

I think she has multiplied them.

Yep, Sofia didn't read the question properly.

It's a really easy thing to do when you are rushing is to forget what you are actually doing in your question.

So just think carefully.

Are you adding, are you multiplying, are you dividing, are you subtracting? And then you can apply the correct method.

So how do we add fractions? Well of course we do them just like numerical fractions.

So to combine algebraic fractions by adding or subtracting, we need a common denominator.

So in this case, our common denominator will be a multiple of two and of X.

In this case, the lowest common multiple will be two X.

So let's think about what X over two looks like.

If I want it to have a denominator of two X, I've multiplied the denominator by X.

So I need to multiply the numerator by X as well.

So that's equivalent to X squared over two X.

If I want the second fraction to have a denominator of two X, I need to multiply the numerator and the denominator by two, which gives me 10 over two X.

And now I can combine them by adding X squared plus 10 over two X.

Doesn't look particularly simple at the moment, but being able to write the sum of a fraction as a single fraction can be quite a useful tool.

Sofia's noticed that this time we are subtracting, she's making sure she knows what she's doing before she starts.

Like numerical fraction using the lowest common multiple will be most efficient.

So the lowest common multiple of five X and three X squared is going to be 15 X squared.

Okay, so we need to multiply the numerator of the first fraction by three X and we need to make sure we're multiplying the whole of that numerator.

So we've got three X times X and three X times one.

Three X squared plus three X.

The second fraction, we need to multiply the numerator and the denominator by five.

So that gives us 10 over 15 X squared.

Now that we've got a common denominator, we can subtract them.

Sofia reminded us we are subtracting.

We've got three X squared plus three X minus 10 all over 15 X squared.

Right, I'm going to show you one on the left and then you are going to have go at one on the right.

So the most efficient way is to find the lowest common multiple of those denominators.

The lowest common multiple of six A and four is 12 A.

So we need to multiply the numerator and denominator of the first fraction by two and the numerator and the denominator of the second fraction by three A.

We're going to make sure we multiply that whole expression.

So we get 10 over 12 A, minus and then we've got three A squared plus six A all over 12 A.

Again, we need to make sure we're subtracting that whole second fraction.

So it's 10 A subtract three A squared, subtract six A all over 12 A.

Right, time for you to have a go.

Think carefully about those key points.

Make sure you're multiplying the entire numerator and denominator by the same value to get an equivalent fraction.

And then when you are subtracting, make sure you are subtracting all the terms in that second fraction.

Give that a go and then we'll go through it together.

So the lowest common multiple is six A.

That means we can multiply the numerator and denominator of the first fraction by three and then the second fraction, the numerator and denominator multiplied by two gives us nine over six A subtracts two A minus 10 over six A.

That means to make sure we're subtracting both of those terms in the bracket.

So we're subtracting two A and we're subtracting negative 10.

So nine subtract negative 10 is the same as nine add 10 or 19.

That gets us 19 subtract two A, all over six A.

We've already touched on this a little bit, making sure using our brackets so that we are getting the correct products.

So what would a common denominator be for these fractions? Well, it needs to be a multiple of X plus one and X plus two.

We can write that as X plus one in brackets X plus two in brackets, the product of two binomials.

Now if we're going to convert these fractions over the common denominator, we need to multiply the numerator and denominator of the first fraction by X plus two and we want that whole expression.

And the second fraction, the numerator and denominator need to be multiplied by X plus one.

And now we just need to be careful when expanding our brackets and collecting like terms. If we wanted to, we could write our denominator in expanded form.

So Andeep and Sofia are discussing this question.

"To get a common denominator we can simply multiply the two denominators together." "But that would mean multiplying a quadratic trinomial by a binomial.

I don't think that is simple." What would you do? I'm inclined to agree with Sofia.

I don't really want to expand a quadratic trinomial multiplied by a binomial if I can help it.

It might depend what format we need our answer in.

So let's look at some methods.

We could do what Andeep said to do multiply X squared plus five X plus six and three X plus six to get our denominator and then obviously make sure we've got equivalent fractions by multiplying our numerators as required.

For now, I'm not going to expand that denominator.

Let's leave it as it is.

I'm going to expand the numerators, subtract and collect like terms. So this is what Andeep was proposing we do.

Now I can't see at the moment whether that is in simplest form.

In order to check if it's in simplest form, I'm going to need to factorise the numerator as well, factorise the remaining part of the denominator, and look for common factors.

Now that's a bit of a faff.

Factorising when you've got a coefficient of 11 is not going to be the easiest thing in the world.

And Sofia is correct.

"It would've been easier to factorise at the beginning." We're going to factorised here.

We might as well factorise before we had the trickier quadratics to deal with.

So let's go back to our original.

So now we've factorised our denominators.

We can see they share a common factor.

What that means is our lowest common multiple is not going to be the product of those two denominators.

The lowest common multiple is three lots of X plus two X plus three.

That means our first fraction, we can multiply the numerator and the denominator by three.

Our second fraction, we only need to multiply the numerator and the denominator by X plus three.

That gives us 15 X.

Subtract four X plus 12 all over three lots of X plus two X plus three.

It's a lot simpler than what we had before.

We need to make sure we're subtracting both terms. So 15 X subtract four X is 11 X and then we're subtracting 12.

And the good thing about the fact that the denominator's been factorised is we can see that it's in simplest form.

The numerator doesn't factorise and it doesn't share any common factors with the denominator.

If we wanted to, we can now expand that denominator so that it is a quadratic.

Quick check then, which of these is the lowest common multiple of six lots of X minus four and three lots of X minus four X plus one.

What do you think? Make sure you didn't confuse your common multiple with common factors.

The lowest common multiple is six lots of X plus one X minus four.

That would be multiplying the first one by X plus one and multiplying the second one by two.

Time for you to have a practise then.

Andeep has had a go at these two algebraic fractions questions.

I'd like you to identify and correct any of his mistakes.

Off you go.

Now you know what mistakes to look out for I'd like you to have a go yourself.

I'd like you to match the calculations with their simplified answers.

Give those a go.

Finally, you have freedom to do these in any way you like.

I would like you to write your answer as a single fraction and in expanded form.

So expand any brackets in your final answer.

Give it a go.

And for this final set, you're going to want to think about factorising where possible to keep your answers as simple as possible because your final answer needs to be a single fraction in expanded form.

Give it a go.

Let's have a look then.

So for the first one, Andeep has not multiplied the whole expression by X, needs to put brackets around his X minus two to remind him to multiply the X by X and then negative two by X.

For the second one, he actually did a really, really good method.

He just didn't realise he's supposed to be subtracting rather than adding.

Make sure when you're copying a question down that you're copying it down carefully and that you are answering the question that you've actually been asked.

So match up time.

I'd like you to pause the video and check you've matched these up correctly.

Let's have a look at our methods here then.

So for the first one, the lowest common multiple is 12.

So we get negative four X minus three all over 12.

If you wanted to, you could write that as negative four X plus three over 12.

Either of those are absolutely fine.

For the second one, if you multiply the numerator and denominator of the first fraction by three and the second fraction by X plus one, then when you add and simplify, you get eight X plus 20 over three X plus three.

The third one, our common denominator is going to be XY.

So we get Y squared minus X squared over XY.

Well done if you got this far, there was lots of questions that you've been working at now and these ones are getting a little bit tricky.

So factorising helps us spot that our lowest common multiple is going to be three lots of two X plus five.

And our final answer is five X minus four over six X plus 15.

For the second one factorising again helps us spot our lowest common multiple is going to be seven lots of X plus 10 X minus three.

Converting over the common denominator and rearranging gives us two X minus 78 over seven X squared plus 49 X minus 210.

And finally, lots of steps to this last one.

So we're going to need to write one as a fraction over a common denominator.

The denominator that will be most sensible will be X plus two X minus four.

We can write that as X plus two X minus four over X plus two X minus four.

Now all our fractions have a common denominator, write them in expanded form is going to help us subtract.

And then we need to make sure we're subtracting all the terms. I've done it in two stages to help me.

I've subtracted the first two fractions to give me negative two X minus seven over X plus two X minus four, and then I've subtracted the final fraction.

Our final answer is negative five X minus 13 over X squared minus two X minus eight.

You can write that in other forms as well.

That form is absolutely fine.

Well done.

We were pulling together lots of skills today.

Some of those questions at the end, were getting quite hard to manipulate.

We're not doing anything that we haven't done before.

We've got to be really careful with our working that we're not missing steps, we're not making silly mistakes.

Thank you for joining us today.

I really look forward to seeing you again.